2d:null 2e:["$","title",null,{"children":"Workshop Notebook: Statistical Inference (iJS 1a) | Athan | Observable"}] 33:T6a2,md` Before we dive in, we need to cover some preliminary terminology. In statistics, a distinction is made between _populations_ and _samples_. A _population_ is defined as the entire set of entities (or observations) under study, and a _sample_ is defined as a **subset** of a population used to draw inferences about that population. As a matter of convention, when we want to refer to the total number of observations, we use the uppercase letter _N_, and, when we want to refer to the number of observations in a subset, we use the lowercase letter _n_. As a basic example, suppose a planet exists in a galaxy far, far away on which inhabits an alien race. To us, these aliens are a bit peculiar. According to ancient lore, the planet was colonized by 10 gods (5 male and 5 female) who paired off and evenly divided the planet amongst themselves forming 5 kingdoms. Upon doing so, they made a pact: in order to preserve their lineage, every newborn boy would take the name of the father and every newborn girl would take the name of the mother. Thus, the peculiarity: while inhabitants number in the thousands, there exist only 10 unique names. Were we to perform statistical inference, the set containing all the inhabitants would constitute a _population_. In a similar vein, the set of inhabitant names would also constitute a _population_, as would the set of all inhabitant genders. Were we to take the subset of male names (_0 ≤ n ≤ N_), that subset would constitute a _sample_. Similarly, if we created a subset of all names beginning with the letter _a_, that, too, would be a _sample_, which may or may not share elements in the subset of male names and where _0 ≤ n ≤ N_. `34:T522,md` > **Aside**: at this point, you might be asking yourself: why do we need to sample in the first place? After all, _N = 10_, so the task is trivial. But what about when _N >>> 10_? E.g., _N = 1e10_ or _N = 1e15_? For _N = 1e10_, if each loop iteration takes _1_ nanosecond, then counting names and computing the ratio will take at least _10_ seconds. For _N = 1e15_, _1e6_ seconds, or more than _115_ days. > Representative _sampling_ becomes critically important as data size increases, as it enables us to solve particular engineering challenges at scale. When performing exploratory data analysis (a subject of a later workshop notebook), we often don't want to (and shouldn't need to) spin up a large cluster just to load, transform, manipulate, and visualize our data. For fast iteration, more often than not, we'd prefer having the ability to load just enough data on our personal workstations to help us get a "feel" for the data, understand its qualities and characteristics (such as presence of missing values, summary statistics, and more), and generate testable models. Only once we are comfortable with our data should we feel compelled to invest considerable resources in analyzing what may be petabytes of data. Judicious use of sampling techniques allows us to satisfy this aim. `35:T993,md` To further demonstrate how sample size affects parameter estimation, let's perform a brief simulation. To begin, we'll draw [pseudorandom numbers](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/random/base/normal) from a normal distribution with a known mean ${tex`\mu`} and standard deviation ${tex`\sigma`}. **Exercise 6**: generate an array containing, say, 10,000 [pseudorandom numbers](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/random/base/normal) drawn from a normal distribution with ${tex`\mu = 10.0`} and ${tex`\sigma = 3.0`}. Once you have generated the array of random variates, [plot](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/plot/ctor) the data using the boilerplate [plot](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/plot/ctor) invocation below. > **Hint**: see the documentation for [normally distributed pseudorandom numbers](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/random/base/normal) (available as [\`stdlib.base.random.normal(\`${tex`\mu, \sigma`}\`)\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/random/base/normal) in this notebook) and [inmap](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/utils/inmap) (available as [\`stdlib.inmap(collection, fcn)\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/utils/inmap) in this notebook). > **Hint**: for a reproducible pseudorandom number sequence, you may want to use a seeded [pseudorandom number generator](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/random/base/normal). > **Hint**: to generate a linearly spaced array for the plot \`x\` values, see [linspace](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/math/utils/linspace) (available as [\`stdlib.linspace(a, b, n)\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/math/utils/linspace) in this notebook). `36:T848,md` Now that we have a collection of [pseudorandom numbers](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/random/base/randn), let's proceed to compute the sample mean for progressively larger sample sizes. A naive approach to achieving this might be to repeatedly compute the sample mean starting from index 0 until an index _n_, where _0 ≤ n < N_ and _N_ is the length of our collection. If we vary _n_ from _0_ to _N-1_, we'll have _N_ estimates of the sample mean. For example, consider the collection ${tex.block`x = [ 1, 2, 3]`} From this collection, we can compute 3 estimates of the sample mean ${tex.block`\begin{aligned}\bar{x_0} &= \tfrac{1}{1} \cdot (1) = 1 \\ \bar{x_1} &= \tfrac{1}{2} \cdot (1 + 2) = 1.5 \\ \bar{x_2} &= \tfrac{1}{3} \cdot (1 + 2 + 3) = 2\end{aligned}`} While the naive approach works, its performance is less than optimal. > **Question**: what is the algorithmic complexity of the naive approach in Big ${tex`{O}`} notation? Fortunately, efficient [algorithms](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/incr) exist for computing a running estimate of the sample mean (and other [summary statistics](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/incr)), ensuring linear runtime. **Exercise 7**: use the function [\`stdlib.incrmean()\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/incr/mean) to compute the sample mean for progressively larger sample sizes. > **Hint**: a one-liner can be achieved via [inmap](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/utils/inmap) (available as [\`stdlib.inmap(collection, fcn)\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/utils/inmap) in this notebook). Just be sure to slice (i.e., copy) the raw simulated data **before** invocation! `37:T4f1,md` A _model_ is an attempt to understand and represent a particular phenomenon (e.g., page clicks, human heights, solar flares, etc) via a specific representation, be it architectural (e.g., a scaled model of a building), biological (e.g., a description of gene regulation), or mathematical (e.g., mapping home size to its selling price). A model typically consists of what are deemed essential elements, making qualifying assumptions and abstractions. When we refer to a _mathematical model_, we refer to a collection of symbols and expressions which describe relationships between variable quantities. The expressions are typically general enough that they include parameters whose values we seek to infer from observed data. But how do we even build a model in the first place? The truth is that building a model is part art and part science and entails an iterative process in which one is forced to make, clarify, and refine assumptions about the underlying structure of reality. And while we'd like to say that there exists a single template and standard by which we make those choices and refinements and how we explain them, no such standard exists. We must rely on a combination of common sense, domain expertise, and rigorous testing and evaluation. `38:T121e,md` Over time, as you encounter new datasets and problems, you'll build an arsenal of potential statistical models, gaining an intuition as to their strengths, general applicability, and, perhaps most importantly, limitations. Undoubtedly, what you'll find is that some of the fundamental building blocks of these models are _probability distributions_. Before the advent of modern digital computers, scientists observed real-world phenomenon, recorded their observations, and noticed that certain mathematical shapes kept recurring. To approximate these shapes, scientists wrote mathematical equations whose parameters could be estimated from the data. A classic example is the normal distribution, a bell-shaped curve commonly referred to as the Gaussian distribution, which is used to approximate the distribution of human heights. While, in reality, there exists an infinite number of probability distributions, we often encounter a more limited set of canonical (named) distributions, which are simply a set of shapes capable of being described mathematically and which occur often enough that someone decided they deserved names. For example, the normal distribution may be expressed ${tex.block`N(x|\mu,\sigma) \sim \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}`} where the parameter ${tex`\mu`} is the mean and, as a normal distribution is symmetric, specifies the distribution center. The parameter ${tex`\sigma`} is the standard deviation, which quantifies how spread out (wide) a distribution is. When we want to describe a phenomenon with a normal distribution, these parameters are realized as actual numbers, whose values we estimate from observed data in a manner similar to when we computed the sample mean above. Probability distributions come in two flavors: **discrete** and **continuous**. A _discrete probability distribution_ describes the probability of occurrence for each value of a discrete random variable, where a _discrete random variable_ is a random variable that has countable values (e.g., a list of nonnegative integers). A _continuous probability distribution_ describes the probability of occurrence for a range of continuous random variables, where a _continuous random variable_ is a random variable whose set of possible values is infinite and uncountable. To help familiarize ourselves with common probability distributions, let's complete the following exercises. **Exercise 9**: create an interactive widget immediately following this cell which plots the [probability density function](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/normal/pdf) (PDF) for a [normal distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/normal) and includes two sliders, one for ${tex`\mu`} and another for ${tex`\sigma`}, which, when updated, updates the plot. A boilerplate [plot](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/plot/ctor) instance has already been provided below. > **Hint**: to generate a linearly spaced array for the \`x\` values (i.e., the values at which to evaluate the PDF), see [linspace](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/math/utils/linspace) (available as [\`stdlib.linspace(a, b, n)\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/math/utils/linspace) in this notebook). > **Hint**: the function to evaluate the [probability density function](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/normal/pdf) for a [normal distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/normal) is available as [\`stdlib.base.dists.normal.pdf(x, \`${tex`\mu`}, ${tex`\sigma`}\`)\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/normal/pdf) in this notebook. > **Hint**: for interactive HTML elements, see the documentation on [tagged template literals](https://beta.observablehq.com/@mbostock/standard-library) in Observable. > **Hint**: you may find it useful to fix the x-axis limits by setting the \`xMin\` and \`xMax\` [plot](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/plot/ctor) properties. `39:T43d,md` **Exercise 15**: repeat the previous exercise for the [beta distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/beta/pdf) with parameters ${tex`\alpha`} and ${tex`\beta`}. ${tex.block`f(x|\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha) + \Gamma(\beta)}{x^{\alpha-1}(1-x)^{\beta-1}} & \text{ for } x \in (0,1) \\ 0 & \text{ otherwise } \end{cases}`} > **Hint**: the function to evaluate the [probability density function](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/beta/pdf) for a [beta distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/beta) is available as [\`stdlib.base.dists.beta.pdf(\`${tex`x`}, ${tex`\alpha`}, ${tex`\beta`}\`)\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/beta/pdf) in this notebook. `3a:T41b,md` **Exercise 16**: repeat the previous exercise for the [Weibull distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/weibull/pdf) with parameters ${tex`k`} and ${tex`\lambda`}. ${tex.block`f(x|\lambda,k) = \begin{cases} \frac{k}{\lambda}\left (\frac{x}{\lambda} \right)^{k-1}e^{-(x/\lambda)^k} & x \geq 0 \\ 0 & x < 0\end{cases}`} > **Hint**: the function to evaluate the [probability density function](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/weibull/pdf) for a [Weibull distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/weibull) is available as [\`stdlib.base.dists.weibull.pdf(\`${tex`x`}, ${tex`k`}, ${tex`\lambda`}\`)\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/weibull/pdf) in this notebook. `3b:T407,md` **Exercise 18**: repeat the previous exercise for the [Poisson distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/poisson/pmf) with parameter ${tex`\lambda`}. ${tex.block`f(x|\lambda) = P(X=x|\lambda) = \begin{cases} \tfrac{\lambda^x}{x!}e^{-\lambda} & \text{ for } x = 0,1,2,\ldots \\ 0 & \text{ otherwise} \end{cases}`} > **Hint**: the function to evaluate the [probability mass function](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/poisson/pmf) for a [Poisson distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/poisson) is available as [\`stdlib.base.dists.poisson.pmf(\`${tex`x`}, ${tex`\lambda`}\`)\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/poisson/pmf) in this notebook. `3c:T53c,md` A _statistical hypothesis_ is a testable hypothesis concerning an observed process modeled according to a set of random variables. A _statistical hypothesis test_ is a statistical inference method in which two (or more) datasets are compared based on their statistical properties. Commonly, such a test involves the statistical comparison of two datasets or of a _sample_ against a simulated dataset generated according to an idealized statistical model. Often a _statistical hypothesis_ proposes a statistical relationship between two (or more) datasets and serves as an alternative to a **null hypothesis** that proposes no relationship. A comparison is considered _statistically significant_ if the relationship between the datasets has a high likelihood of being due to a real systematic difference and a low likelihood of occurring by chance alone. Whether a relationship is likely is determined according to a threshold probability--the _significance level_. To make hypothesis testing more concrete, imagine the following scenario: you have two landing pages for your website and would like to know which one is more effective in converting visitors. Suppose you have collected the following two data sets, where \`1\` denotes that a visitor signed up and \`0\` denotes that a visitor left the page without taking any action. `3d:T92b,md` Provided the proportions are different, is the difference also statistically significant? Let's find out! A common statistical hypothesis test for testing whether two datasets are different is the [Student's _t_-test](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/ttest2). > A [_t_-test](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/ttest2) is any statistical hypothesis test in which the test statistic follows a [Student's _t_-distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/t/pdf) under the **null hypothesis**. When dataset sample sizes and population variances may **not** be equal, the test statistic for determining whether the population means of two datasets differ may be expressed as ${tex.block`t = \frac{\bar{x_1} - \bar{x_2}}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}`} where ${tex`\bar{x_i}`} is the sample mean, ${tex`s_i`} is the unbiased estimate of the sample variance, and ${tex`n_i`} is the number of observations in dataset ${tex`i`}. **Exercise 20**: perform a two-sample [_t_-test](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/ttest2) for the datasets introduced above. > **Hint**: the function to perform a two-sample [_t_-test](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/ttest2) is available as [\`stdlib.ttest2(\`${tex`x`}, ${tex`y`}\`)\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/ttest2) in this notebook. > **Hint**: the returned test result has a \`rejected\` property which is a boolean indicating whether the **null hypothesis** of the test was rejected. The **null hypothesis**, often denoted ${tex`H_0`}, denotes the case where a difference is **not** statistically significant. The **alternative hypothesis**, often denoted ${tex`H_1`}, denotes the case where a difference **is** statistically significant. > **Hint**: for pretty-printed output, use the \`print()\` method, which will print the exact definition of the **alternative hypothesis**. `3e:T460,md` **Exercise 21**: by default, [\`ttest2()\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/ttest2) and the other [stdlib](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats) statistical tests test for a non-directional effect. This is called a _two-sided_ test. When you have prior beliefs about which group has a larger effect, you should consider running a _one-sided_ test. For illustration purposes, rerun the test for the landing pages using one-sided tests and observe how the alternative hypothesis and test result change. Keep in mind, however, that you should choose a test direction **before** looking at the data. > **Hint**: to run a one-sided [_t_-test](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/ttest2), set the \`alternative\` option when invoking [\`stdlib.ttest2()\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/ttest2). `3f:Ta93,md` **Challenge 3**: create an interactive widget immediately following this cell for simulating landing page data and performing a two-sample [_t_-test](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/ttest2) on the simulated datasets. The widget should contain four sliders: one for adjusting the number of simulated visitors to the first landing page ${tex`n_1`}, a second for adjusting the number of simulated visitors to the second landing page ${tex`n_2`}, another for adjusting the probability ${tex`p_1`} that a user converts when visiting the first landing page, and a fourth for adjusting the probability ${tex`p_2`} that a user converts when visiting the second landing page. > **Hint**: the probabilities ${tex`p_1`} and ${tex`p_2`} should be both limited to the interval [0,1]. > **Hint**: to generate binary pseudorandom numbers (i.e., random values which can be either 0 or 1), you can generate a [uniformly distributed pseudorandom number](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/random/base/randu) on the interval [0,1) and apply a threshold corresponding to probability ${tex`p`}. If a pseudorandom number is greater than ${tex`p`}, then the binary pseudorandom number is 1 and otherwise 0. The function for generating [uniformly distributed pseudorandom numbers](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/random/base/randu) on the interval [0,1) is available as [\`stdlib.base.random.randu()\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/random/base/randu) in this notebook. > **Hint**: to generate reproducible pseudorandom number sequences, you may want to create a seeded pseudorandom number generator. > **Hint**: to sanity check the simulation, [plot](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/plot/ctor) the simulated datasets (sorted in descending order) and visually confirm that the relative proportion of conversions accords with expectation. A boilerplate [plot](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/plot/ctor) instance has already been provided below. > **Hint**: for pretty-printed output, use the \`print()\` method, which will print the exact definition of the **alternative hypothesis**. > **Hint**: for interactive HTML elements, see the documentation on [tagged template literals](https://beta.observablehq.com/@mbostock/standard-library) in Observable. `40:T9fb,md` **Challenge 4**: generate a [scatterplot](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/plot/ctor) which explores how many visitors we would need to sample in order to ascertain whether a difference in conversion proportions is statistically significant. To do so, for a fixed probability ${tex`p_1`} and a varying probability ${tex`p_2`}, where, for simplicity, ${tex`p_2 >= p_1`}, systematically generate simulated datasets of varying length ${tex`n`} until you find a statistically significant difference (i.e., rejection of the **null hypothesis**) according to a two-sample two-sided [_t_-test](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/ttest2). For each ${tex`\Delta p = p_1 - p_2`}, store the length at which a statistically significant difference was found, and [plot](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/plot/ctor) ${tex`|\Delta p| = |p_1 - p_2|`} against ${tex`n`}. A boilerplate [plot](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/plot/ctor) instance has already been provided below. > **Hint**: for simplicity, set ${tex`p_1 = 0.1`} and then vary ${tex`p_2`} on the interval ${tex`[p_1,1]`}. > **Hint**: to generate a linearly spaced array for the ${tex`p_2`} (and/or ${tex`\Delta p`}) values, see [linspace](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/math/utils/linspace) (available as [\`stdlib.linspace(a, b, n)\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/math/utils/linspace) in this notebook). > **Hint**: in order to avoid infinite loops, you will likely want to set a maximum ${tex`n`} at which, if a statisticially significant difference is **not** found, you stop searching. This is particularly relevant when ${tex`\Delta p \approx 0`}. Where appropriate, you can omit such occurrences from plotted results. > **Hint**: in order to make results more robust, you may want to consider repeating each dataset simulation and corresponding two-sample two-sided [_t_-test](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/ttest2) a specified number of times and then calculating the average length at which a statistically significant difference was found. `41:T10bd,md` We can decide whether a **null hypothesis** should be rejected by using the [_p_-value](https://en.wikipedia.org/wiki/P-value) of a statistical test. A smaller [_p_-value](https://en.wikipedia.org/wiki/P-value) indicates that a **null hypothesis** fails to adequately explain the data. As a decision rule, we reject the **null hypothesis** whenever the [_p_-value](https://en.wikipedia.org/wiki/P-value) is lower than the chosen significance level ${tex`\alpha`}, which, by convention, is often set to \`0.05\`. Recall that ${tex`\alpha`} denotes the probability that you reject the **null hypothesis** even though the **null hypothesis** is true. By setting ${tex`\alpha = 0.05`}, we are basically saying that we are content with erroneously rejecting the **null hypothesis** in 5% of tests. Why is ${tex`\alpha`} equal to this error probability? We can mathematically prove that, if the **null hypothesis** is true, then the [_p_-value](https://en.wikipedia.org/wiki/P-value) follows a [uniform distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/uniform/pdf) on the interval ${tex`[0,1]`}. Hence, over the course of many experiments, we would expect 5% of [_p_-values](https://en.wikipedia.org/wiki/P-value) to be smaller than ${tex`0.05`}. Instead of a mathematical proof, let's simulate some data to confirm that this is, in fact, the case. **Exercise 22**: create an interactive widget immediately following this cell for generating an array whose elements are values drawn from a [normal distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/normal/pdf) with mean ${tex`\mu`} and standard deviation ${tex`\sigma`}. The widget should include three sliders: one for adjusting the number of elements in the simulated array, another for adjusting ${tex`\mu`}, and a third for adjusting ${tex`\sigma`}. And further, the widget should update a [plot](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/plot/ctor) displaying the generated numbers. For initial values, use ${tex`\mu = 0`} and ${tex`\sigma = 3`}. A boilerplate [plot](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/plot/ctor) instance has already been provided below. > **Hint**: the function to generate [pseudorandom numbers](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/random/base/normal) drawn from a [normal distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/random/base/normal) is available as [\`stdlib.base.random.normal(\`${tex`\mu`}, ${tex`\sigma`}\`)\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/random/base/normal) in this notebook. > **Hint**: generating an array of random variates can be done in a single line using a combination of the PRNG \`factory()\` method and [inmap](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/utils/inmap) (available as [\`stdlib.inmap(collection, fcn)\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/utils/inmap) in this notebook). > **Hint**: to better understand how adjusting parameters affects results, you may want to specify a pseudorandom number generator seed, thus allowing reproducible generation of pseudorandom numbers. > **Hint**: based on the range of possible values for ${tex`\mu`} and ${tex`\sigma`}, you may want to explicitly set the y-axis limits via the \`yMin\` and \`yMax\` [plot](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/plot/ctor) properties. Doing so will provide visual constancy which may better convey how adjusting ${tex`\mu`} and ${tex`\sigma`} affects simulated output. > **Hint**: for interactive HTML elements, see the documentation on [tagged template literals](https://beta.observablehq.com/@mbostock/standard-library) in Observable. `42:T5d4,md` **Challenge 5**: combining the solutions for the previous two exercises, create an interactive widget immediately following this cell for exploring the relationship between dataset length ${tex`n`} and the proportion of [_p_-values](https://en.wikipedia.org/wiki/P-value) which are less than ${tex`\alpha`}. The widget should include three sliders: one for adjusting ${tex`\mu`}, another for adjusting ${tex`\sigma`}, and a third for adjusting ${tex`\alpha`}. Upon adjusting a slider, the widget should 1. generate \`M\` simulated datasets of length ${tex`n`} (where ${tex`n`} is varies over an implementation-defined interval). 2. perform a one-sample [Student's _t_-test](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/ttest) for each simulated dataset. 3. calculate the proportion of [_p_-values](https://en.wikipedia.org/wiki/P-value) which less than ${tex`\alpha`}. 4. update a [scatterplot](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/plot/ctor) displaying the relationship between dataset length and the proportion of [_p_-values](https://en.wikipedia.org/wiki/P-value) less than ${tex`\alpha`}. Let ${tex`n`} exist on the interval (0,1000] and \`M = 100\`. A boilerplate [plot](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/plot/ctor) instance has already been provided below. `43:T86e,md` To formally test whether computed [_p_-values](https://en.wikipedia.org/wiki/P-value) are drawn from a [uniform distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/uniform/pdf) on the interval [0,1], we can use the [Kolmogorov-Smirnov test](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/kstest), a statistical hypothesis test which tests a sample against a specified continuous reference distribution. **Exercise 24**: using the same approach as in previous exercises, generate \`M\` simulated datasets. For each simulated dataset, perform a one-sample [Student's _t_-test](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/ttest), which tests the hypothesis that the true mean of the distribution from which we drew pseudorandom numbers is equal to zero. Store the [_p_-value](https://en.wikipedia.org/wiki/P-value) of each test in an array of length \`M\`. Perform a [Kolmogorov-Smirnov test](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/kstest) to determine whether the [_p_-values](https://en.wikipedia.org/wiki/P-value) follow a [uniform distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/uniform/pdf). > **Hint**: the function to perform a [Kolmogorov-Smirnov test](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/kstest) is available as [\`stdlib.kstest(\`${tex`x`}, ${tex`y`}\`)\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/kstest) in this notebook. > **Hint**: the [\`stdlib.kstest(\`${tex`x`}, ${tex`y`}\`)\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/kstest) documentation examples demonstrate how to test against a ${tex`U(0,1)`} distribution. `44:Td83,md` The [Student's _t_-test](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/ttest) is based on the assumption that data originates from a [normal distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/normal/pdf). If this assumption is violated, [Student's _t_-test](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/ttest) results may not be trusted under certain circumstances. **Exercise 25**: create an interactive widget immediately following this cell for generating an array whose elements are values drawn from a [gamma distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/gamma/pdf) with shape parameter ${tex`\alpha`} and rate parameter ${tex`\beta`}. The widget should include three sliders: one for adjusting the number of elements in the simulated array, another for adjusting ${tex`\alpha`}, and a third for adjusting ${tex`\beta`}. And further, the widget should update a [plot](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/plot/ctor) displaying the generated numbers. For initial values, use ${tex`\alpha = 1.0`} and ${tex`\beta = 2.0`}. A boilerplate [plot](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/plot/ctor) instance has already been provided below. > **Hint**: the function to generate [pseudorandom numbers](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/random/base/gamma) drawn from a [gamma distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/random/base/gamma) is available as [\`stdlib.base.random.gamma(\`${tex`\alpha`}, ${tex`\beta`}\`)\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/random/base/gamma) in this notebook. > **Hint**: generating an array of random variates can be done in a single line using a combination of the PRNG \`factory()\` method and [inmap](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/utils/inmap) (available as [\`stdlib.inmap(collection, fcn)\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/utils/inmap) in this notebook). > **Hint**: to better understand how adjusting parameters affects results, you may want to specify a pseudorandom number generator seed, thus allowing reproducible generation of pseudorandom numbers. > **Hint**: based on the range of possible values for ${tex`\alpha`} and ${tex`\beta`}, you may want to explicitly set the y-axis limits via the \`yMin\` and \`yMax\` [plot](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/plot/ctor) properties. Doing so will provide visual constancy which may better convey how adjusting ${tex`\alpha`} and ${tex`\beta`} affects simulated output. > **Hint**: for interactive HTML elements, see the documentation on [tagged template literals](https://beta.observablehq.com/@mbostock/standard-library) in Observable. `45:T49b,md` **Exercise 26**: using the same approach as in the previous exercise, generate \`M\` simulated datasets. For each simulated dataset, perform a one-sample [Student's _t_-test](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/ttest), which tests the hypothesis that the true mean of the distribution from which we drew pseudorandom numbers is equal to ${tex`\alpha \cdot \beta`} (where ${tex`\alpha`} is the shape parameter and ${tex`\beta`} is the scale parameter of a [gamma distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/gamma/pdf)). Store the [_p_-value](https://en.wikipedia.org/wiki/P-value) of each test in an array of length \`M\`. Calculate the proportion of [_p_-values](https://en.wikipedia.org/wiki/P-value) which are less than a specified significance level. > **Hint**: to specify the mean to test against when performing a one-sample [Student's _t_-test](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/ttest), set the \`mu\` option. `46:T45a,md` Instead of using a [_p_-value](https://en.wikipedia.org/wiki/P-value), we can alternatively use a confidence interval for the parameter of interest. Confidence intervals are often used to convey uncertainty in model estimates and are useful even when one does not have a concrete hypothesis to test. For a ${tex`1-\alpha`} confidence interval, one has the guarantee that, for a large number of such confidence intervals, ${tex`1-\alpha`} of them will contain the true parameter. **Exercise 27**: verify the above guarantee by generating \`M\` simulated datasets of length ${tex`n`} whose values are drawn from a [normal distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/random/base/normal) with ${tex`\mu = 0`}. For each dataset, perform a one-sample [Student's _t_-test](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/ttest) and extract the confidence interval from the test results. Count the number of times in which the confidence interval contains ${tex`\mu`}. `47:T6f8,md` All too often, you encounter newspaper articles and blogs presenting the results of some newly published study which announce an association between eating some kind of food and a desired health outcome. Chances are high that these bold announcements are untenable and incapable of replication in controlled studies. By mining data sets for correlations between different variables, especially those of high dimensionality, we'll inevitably "discover" correlations. Even if variables are all independent, simply due to the sheer number of comparisons that can be made, one is going to find significant correlations by chance alone. Some time ago, the good people at FiveThirtyEight wrote an [article](http://fivethirtyeight.com/features/you-cant-trust-what-you-read-about-nutrition) on the shady practice of _p-hacking_, which refers the practice of applying data mining methods in order to obtain significant research findings. As an exploratory step to generate hypotheses, such an approach has its merits, but, more often than not, people present their findings without disclosing the process of how they were obtained or accounting for the multitude of hypothesis tests that were conducted. In this section, we'll get our own hands dirty trying to find spurious correlations in the FiveThirtyEight [data](http://fivethirtyeight.com/features/you-cant-trust-what-you-read-about-nutrition). As we'll discover, obtaining essentially meaningless results is relatively easy. To help mitigate the risk of spurious correlations, we'll discuss some remedies we can employ when faced with the task of running a large number of hypothesis tests. The hope is that our findings will serve as a reminder that we must **always** be cautious when dredging through massive amounts of data. `48:T423,md` For our analysis, we'll use two-sample [Student's _t_-tests](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/ttest2) to check whether respondents with different characteristics exhibit significant differences in terms of food intake. For example, we could check whether those respondents with diabetes consume more green salad than those respondents without diabetes. **Exercise 29**: split the nutrition dataset into two groups. One group should include those respondents with diabetes and the other group should include those respondents without diabetes. > **Hint**: a function to ["bifurcate"](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/utils/bifurcate-by) a collection according to a predicate function is available as [\`stdlib.bifurcateBy(collection, predicate)\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/utils/bifurcate-by) in this notebook. `49:T77e,md` One problem with the [Bonferroni correction](https://en.wikipedia.org/wiki/Bonferroni_correction) is that the adjusted significance level is overly conservative. Accordingly, its use often coincides with a large increase in the false negative rate. As an alternative to the [Bonferroni correction](https://en.wikipedia.org/wiki/Bonferroni_correction), we can use the the Benjamini–Hochberg procedure, which controls the [False Discovery Rate (FDR)](https://en.wikipedia.org/wiki/False_discovery_rate) at a signficiance level ${tex`\alpha`}. The Benjamini–Hochberg procedure is as follows: 1. Order the [_p_-values](https://en.wikipedia.org/wiki/P-value) in ascending order and denote each [_p_-value](https://en.wikipedia.org/wiki/P-value) as ${tex`P_{(1)} \ldots P_{(m)}`}. 2. For a given ${tex`\alpha`}, find the largest ${tex`k`} such that ${tex`P_{(k)} \leq {\frac {k}{m}}\alpha`}. 3. Reject the **null hypothesis** for the ${tex`k`} hypotheses corresponding to the ${tex`k`} lowest [_p_-values](https://en.wikipedia.org/wiki/P-value). Graphically, the procedure corresponds to the following: 1. Plot the ordered [p-values](https://en.wikipedia.org/wiki/P-value) ${tex`P_{(k)}`} against ${tex`k`}. 2. Draw a line through the origin with slope ${tex`\frac{\alpha}{m}`}. 3. Declare as significant all hypotheses that are to the left of where the points cross the line. **Exercise 34**: [plot](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/plot/ctor) the ordered [_p_-values](https://en.wikipedia.org/wiki/P-value) from your analysis of the nutrition data. How many significant results do you obtain when controlling the FDR for ${tex`\alpha = 0.05`}? A boilerplate [plot](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/plot/ctor) instance has already been provided below. `4a:T867d,male_first_names_en = 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Notebook: Statistical Inference (iJS 1a)","cells":[{"id":0,"value":"md`# Workshop Notebook: Statistical Inference (iJS 1a)`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":8,"value":"md`\nIn this workshop notebook, we will explore statistical inference.\n\n> _Statistical inference_ is defined as the development of procedures, methods, and theorems that allow for extraction of information and meaning from data generated by stochastic (i.e., random) processes.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":877,"value":"md`\n### Table of Contents\n\n- Populations vs Samples\n- Sampling Techniques\n - Bias\n - Parameter Estimation\n- Modeling\n - Probability Distributions\n- Hypothesis Testing\n - p-values\n - Confidence Intervals\n- p-hacking\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":14,"value":"md`\n* * *\n\n## Populations vs Samples\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":16,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":115,"value":"md`\n**Exercise 1**: in the following cell, create a list of _populations_, and, for each population, list one or more possible subsets of that population.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":27,"value":"md`\n| Population | Subsets |\n| ---------- | ------- |\n| animals | bipedal, mammals |\n| movies | has actor x, is a drama |\n| emails | emails written by x, emails written from 9-10 in the morning |\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":35,"value":"md`\n* * *\n\n## Sampling Techniques\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":37,"value":"md`\nNow that we have distinguished a _population_ from a _sample_, we can turn our attention to the topic of **sampling techniques**. _Sampling_ is the process of selecting a representative group from the _population_ under study (emphasis on the word **representative**).\n\nLet's return to the example of the alien race above. Suppose for a moment that we do not know much about this alien race. In particular, we are unaware as to the planet's origin story and its peculiar naming practice. And suppose we would like to make a guess as to the ratio of males and females based on the entire set of inhabitant names. However, we're lazy and cannot be bothered to go through the entire list of _N_ names.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":119,"value":"md`\n**Exercise 2**: what are some ways we could \"sample\" the list of names to make an educated guess as to the ratio? In the cell below, write down your ideas. We've already provided one way to help get you started.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":52,"value":"md`\n- find the first 4 names longer than 5 letters and compute the ratio of male to female names\n- take the first ${tex`n`} names in the list\n- group the names into ${tex`n`} groups and take 1 name from each group\n- ...\n\n\n \n \n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":55,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":73,"value":"md`\n### Bias\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":77,"value":"md`\nWhenever a _sample_ is not representative of a _population_, we say that the sample is _biased_. As an example, let's consider the following dataset.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":104,"value":"sample_dataset_emails_per_day","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":113,"value":"md`\nEach item in this dataset consists of a first name, a gender, and the average number of emails a person receives per day.\n\n**Exercise 3**: in one or more cells below, generate random samples where _n = 10_, and, based only on that sample, calculate the average number of emails any given person receives per day. The formula for the sample mean is\n\n${tex.block`\\bar{x} = \\frac{1}{n} \\sum_{i=0}^{n-1} x_i`}\n\n \n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":170,"value":"md`\n> **Hint**: if a sampling method involves generating [pseudorandom numbers](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/random/base), you may find it useful when debugging to use a seeded [pseudorandom number generator](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/random/base) in order to always generate the same sequence of pseudorandom numbers.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1060,"value":"function sampleMean( x ) {\n var sum = 0;\n for ( var i = 0; i < x.length; i++ ) {\n sum += x[ i ]; \n }\n return sum/x.length;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":131,"value":"function e3_sample( x, n ) {\n var out = new Array( n );\n var idx;\n for ( var i = 0; i < n; i++ ) {\n idx = stdlib.base.random.discreteUniform( 0, x.length-1 );\n out[ i ] = x[ idx ][ 2 ];\n }\n return out;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1121,"value":"sampleMean( e3_sample( sample_dataset_emails_per_day.slice( 1 ), 10 ) )","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":129,"value":"md`\n**Exercise 4**: repeat Exercise 3, but this time using a different sampling technique from Exercise 2.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":135,"value":"{\n // Pick a random starting index and take 10 consecutive entries...\n var idx = stdlib.base.random.discreteUniform( 0, sample_dataset_emails_per_day.length-11 );\n return sampleMean( stdlib.pluck( sample_dataset_emails_per_day.slice( idx, idx+10 ), 2 ) ); \n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":147,"value":"md`\n> **Question**: assuming \"random\" samples, does the sample mean vary much from one run to the next? What about between different sampling techniques? Why or why not? (feel free to create a cell below containing your thoughts)\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":151,"value":"md`\n**Exercise 5**: for one or more of the sampling techniques from the previous two exercises, generate a [plot](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/plot/ctor) which shows the relationship between the sample mean and _n_. To help get you started with [plotting](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/plot/ctor), a boilerplate function call is provided.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":166,"value":"e5_x = {\n // Return a set of sample sizes...\n return stdlib.linspace( 1, 1000, 1000 );\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1071,"value":"e5_samples = {\n // For each sample size, generate a new random sample...\n var out = new Array( e5_x.length );\n var d = sample_dataset_emails_per_day.slice( 1 ) ;\n for ( var i = 0; i < out.length; i++ ) {\n out[ i ] = e3_sample( d, e5_x[i] );\n }\n return out;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1077,"value":"e5_means = {\n // For each sample, compute the sample mean...\n var out = new Array( e5_samples.length );\n for ( var i = 0; i < out.length; i++ ) {\n out[ i ] = sampleMean( e5_samples[ i ] ); \n }\n return out;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":154,"value":"plt5 = stdlib.plot({\n 'x': [ e5_x ], // e.g., 'x': [ n ]\n 'y': [ e5_means ], // e.g., 'y': [ mu ]\n 'xLabel': 'n',\n 'yLabel': 'sample mean',\n 'yMin': 0.0,\n 'width': 600\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":158,"value":"html`${plt5.render('html')}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":144,"value":"md`\n**Challenge 1**: create an interactive widget immediately following this cell which, for a predetermined range for _n_, plots the sample mean for two or more sampling techniques. The widget should include a slider for _n_, which, when updated, highlights the sample mean values for the respective techniques.\n\n> **Hint**: for interactive HTML elements, see the documentation on [tagged template literals](https://beta.observablehq.com/@mbostock/standard-library) in Observable.\n\n> **Hint**: because the range of possible _n_ values is known beforehand, you can precompute all sample means and only need to update the highlighted values.\n\n> **Hint**: shown below is snapshot of an interactive widget displaying the results of one sampling technique. Your output should be similar.\n\n

\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":310,"value":"md`\n### Parameter Estimation\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":313,"value":"$35","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1085,"value":"e6_x = {\n // Generate a linearly-spaced array whose elements correspond to the \"indices\" of the simulated numbers...\n return stdlib.linspace( 1, 10000, 10000 );\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":324,"value":"e6_y = {\n // For each element in `x`, generate a normallyd-distributed pseudorandom number:\n return stdlib.inmap( new Array( e6_x.length ), stdlib.base.random.normal.factory( 10.0, 3.0 ) );\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":503,"value":"plt6 = stdlib.plot({\n 'x': [ e6_x ], // e.g., 'x': [ stdlib.linspace(1,10000,10000) ]\n 'y': [ e6_y ], // e.g., 'y': [ rvs ]\n 'xLabel': 'x',\n 'yLabel': 'y',\n 'yMin': -10.0,\n 'yMax': 30.0,\n 'width': 600\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":506,"value":"html`${plt6.render('html')}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":327,"value":"$36","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":350,"value":"e7_y = stdlib.inmap( e6_y.slice(), stdlib.incrmean() )","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":354,"value":"md`\n**Exercise 8**: generate a [plot](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/plot/ctor) showing the relationship between _n_ (the _sample_ size) and the estimate of the sample mean. A boilerplate plot invocation has been provided below.\n\n> **Hint**: to generate a linearly spaced array for the \\`x\\` values, see [linspace](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/math/utils/linspace) (available as [\\`stdlib.linspace(a, b, n)\\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/math/utils/linspace) in this notebook).\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":357,"value":"plt8 = stdlib.plot({\n 'x': [ stdlib.linspace( 1, e7_y.length, e7_y.length ) ], // e.g., 'x': [ n ]\n 'y': [ e7_y ], // e.g., 'y': [ mu ]\n 'xLabel': 'n',\n 'yLabel': 'sample mean',\n 'width': 600,\n 'yMin': 9.0,\n 'yMax': 11.0\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":360,"value":"html`${plt8.render('html')}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":370,"value":"md`\n**Challenge 2**: update the above plot to also display the true mean ${tex`\\mu`}.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":373,"value":"md`\n* * *\n\n## Modeling\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":375,"value":"$37","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":383,"value":"md`\n### Probability Distributions\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":385,"value":"$38","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":415,"value":"viewof mu = html``","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1138,"value":"viewof sigma = html``","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1141,"value":"e9_x = stdlib.linspace( -10.0, 10.0, 1000 )","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1147,"value":"e9_y = stdlib.inmap( e9_x.slice(), stdlib.base.dists.normal.pdf.factory( mu, sigma ) )","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":409,"value":"plt9 = stdlib.plot({\n 'x': [ e9_x ], // e.g., 'x': [ x ]\n 'y': [ e9_y ], // e.g., 'y': [ pdf ]\n 'xLabel': 'x',\n 'yLabel': 'pdf',\n 'yMin': 0.0,\n 'yMax': 0.5,\n 'width': 600\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":413,"value":"html`${plt9.render('html')}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":426,"value":"md`\n**Exercise 10**: repeat the previous exercise for the [Cauchy Distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/cauchy/pdf) with parameters ${tex`x_0`} and ${tex`\\gamma`}.\n\n${tex.block`f(x|\\gamma,x_0)=\\frac{1}{\\pi\\gamma\\,\\left[1 + \\left(\\frac{x-x_0}{\\gamma}\\right)^2\\right]}\\!`}\n\n> **Hint**: the function to evaluate the [probability density function](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/cauchy/pdf) for a [Cauchy distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/cauchy) is available as [\\`stdlib.base.dists.cauchy.pdf(\\`${tex`x`}, ${tex`x_0`}, ${tex`\\gamma`}\\`)\\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/cauchy/pdf) in this notebook.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":430,"value":"''","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":433,"value":"md`\n**Exercise 11**: repeat the previous exercise for the [Student's _t_-distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/t/pdf) with degrees of freedom ${tex`\\nu`}.\n\n${tex.block`f(x|\\nu) = \\frac{1} {\\sqrt{\\nu}\\,B\\left( \\tfrac{1}{2}, \\tfrac{\\nu}{2} \\right )} \\left(1+\\frac{x^2}{\\nu} \\right)^{-\\frac{\\nu+1}{2}}`}\n\n> **Hint**: the function to evaluate the [probability density function](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/t/pdf) for a [Student's _t_-distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/t) is available as [\\`stdlib.base.dists.t.pdf(x, \\` ${tex`\\nu`}\\`)\\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/t/pdf) in this notebook.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":435,"value":"viewof nu = html``","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1157,"value":"e11_x = stdlib.linspace( -10.0, 10.0, 500 )","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1160,"value":"e11_y = stdlib.inmap( e11_x.slice(), stdlib.base.dists.t.pdf.factory( nu ) )","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1163,"value":"plt11 = stdlib.plot({\n 'x': [ e11_x ], // e.g., 'x': [ x ]\n 'y': [ e11_y ], // e.g., 'y': [ pdf ]\n 'xLabel': 'x',\n 'yLabel': 'pdf',\n 'yMin': 0.0,\n 'yMax': 0.5,\n 'width': 600\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1166,"value":"html`${plt11.render('html')}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":444,"value":"md`\n**Exercise 12**: repeat the previous exercise for the [F distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/f/pdf) with parameters ${tex`d_1`} and ${tex`d_2`}.\n\n${tex.block`f(x| d_1,d_2) = \\frac{\\sqrt{\\frac{(d_1\\,x)^{d_1}\\,\\,d_2^{d_2}} {(d_1\\,x+d_2)^{d_1+d_2}}}} {x\\,\\mathrm{B}\\!\\left(\\frac{d_1}{2},\\frac{d_2}{2}\\right)}`}\n\n> **Hint**: the function to evaluate the [probability density function](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/f/pdf) for an [F distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/f) is available as [\\`stdlib.base.dists.f.pdf(\\`${tex`x`}, ${tex`d_1`}, ${tex`d_2`}\\`)\\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/f/pdf) in this notebook.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":439,"value":"''","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":456,"value":"md`\n**Exercise 13**: repeat the previous exercise for the [${tex`\\chi^2`} distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/chisquare/pdf) with degrees of freedom ${tex`k`}.\n\n${tex.block`f(x|k) = {\\frac{1}{2^{\\frac {k}{2}}\\Gamma\\left({\\frac{k}{2}}\\right)}}\\;x^{{\\frac {k}{2}}-1}e^{-{\\frac {x}{2}}}`}\n\n> **Hint**: the function to evaluate the [probability density function](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/chisquare/pdf) for a [${tex`\\chi^2`} distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/chisquare) is available as [\\`stdlib.base.dists.chisquare.pdf(\\`${tex`x`}, ${tex`k`}\\`)\\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/chisquare/pdf) in this notebook.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":460,"value":"''","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":463,"value":"md`\n**Exercise 14**: repeat the previous exercise for the [gamma distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/gamma/pdf) with parameters ${tex`\\alpha`} and ${tex`\\beta`}.\n\n${tex.block`f(x|\\alpha,\\beta)=\\frac{\\beta^\\alpha}{\\Gamma(\\alpha)} x^{\\alpha \\,-\\, 1} e^{- \\beta x }`}\n\n> **Hint**: the function to evaluate the [probability density function](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/gamma/pdf) for a [gamma distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/gamma) is available as [\\`stdlib.base.dists.gamma.pdf(\\`${tex`x`}, ${tex`\\alpha`}, ${tex`\\beta`}\\`)\\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/gamma/pdf) in this notebook.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":466,"value":"''","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":474,"value":"$39","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":476,"value":"''","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":479,"value":"$3a","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":492,"value":"''","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":517,"value":"md`\n**Exercise 17**: repeat the previous exercise for the [exponential distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/exponential/pdf) with parameter ${tex`\\lambda`}.\n\n${tex.block`f(x|\\lambda) = \\begin{cases} \\lambda e^{-\\lambda x} & x \\ge 0 \\\\ 0 & x < 0 \\end{cases}`}\n\n> **Hint**: the function to evaluate the [probability density function](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/exponential/pdf) for an [exponential distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/exponential) is available as [\\`stdlib.base.dists.exponential.pdf(\\`${tex`x`}, ${tex`\\lambda`}\\`)\\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/exponential/pdf) in this notebook.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":519,"value":"''","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":523,"value":"$3b","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":525,"value":"''","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":530,"value":"md`\n* * *\n\n## Hypothesis Testing\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":532,"value":"$3c","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":544,"value":"dataset_page1 = [\n 1, 0, 0, 0, 0, 0, 1, 0, 0, 1,\n 0, 0, 0, 1, 0, 0, 0, 1, 0, 0,\n 0, 1, 1, 0, 0, 0, 0, 0, 0, 0\n]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":548,"value":"dataset_page2 = [\n 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n 0, 0, 0, 0, 0, 0, 1, 0, 0, 0,\n 0, 0, 0, 0, 0, 0, 0, 1, 0, 0\n]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":552,"value":"md`\nA cursory look at the data suggests that the first page is more effective at converting users. This observation may, in fact, be correct; however, we'd undoubtedly prefer a means for statistical verification and certainly a means which does not rely on us having to manually inspect relative proportions. To this end, let's see if we can statistically confirm our belief that the **higher proportion** of users who signed up on the first page compared to the second page is **not** due to chance alone.\n\n**Exercise 19**: calculate the proportion of users who signed up per page.\n\n> **Hint**: for a binary random variable (i.e., a random variable which can assume only one of two values; e.g., \\`0\\` and \\`1\\`), calculating the **mean** gives you the proportion of \"\\`1\\`\" observations. This is the **only** instance in which calculating a numerical [summary statistic](https://en.wikipedia.org/wiki/Summary_statistics) for [categorical data](https://en.wikipedia.org/wiki/Categorical_variable) is appropriate.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":565,"value":"e19_p1 = sampleMean( dataset_page1 )","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1170,"value":"e19_p2 = sampleMean( dataset_page2 )","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":570,"value":"md`\n> **Question**: are the proportions the same or different? If the proportions are different, how much larger is one proportion than the other (e.g., \"100x larger\")? And if different, do you feel more or less confident, based on the relative difference, that the first page is \"superior\" to the second with regard to user conversion? Why? (feel free to create a cell below containing your thoughts)\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":578,"value":"$3d","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":608,"value":"stdlib.ttest2( dataset_page1, dataset_page2 ).print()","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":615,"value":"md`\n> **Question**: based on the test results, do we conclude that the first page or the second page is more effective at converting visitors?\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":620,"value":"md`\nWhen all test assumptions are satisfied, the significance level of a test, commonly denoted by the Greek letter ${tex`\\alpha`}, informs you as to the probability that you reject the **null hypothesis** even though the **null hypothesis** is true. Hypothesis testing gives us only this guarantee and does **not** provide any guarantees when we fail to reject a false **null hypothesis**.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":617,"value":"md`\n> **Question**: observe that the test decision has an inherent uncertainty; i.e., a small probability exists that you might have reached a false conclusion. How large is this probability?\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":624,"value":"$3e","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":629,"value":"stdlib.ttest2( dataset_page1, dataset_page2, {'alternative': 'less'} ).print()","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1175,"value":"stdlib.ttest2( dataset_page1, dataset_page2, {'alternative': 'greater'} ).print()","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":634,"value":"md`\n> **Question**: when A/B testing websites, which is preferred: a two-sided or a one-sided statistical hypothesis test? Why? (feel free to create a cell below containing your thoughts)\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":686,"value":"$3f","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":692,"value":"viewof p1 = html``","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1186,"value":"viewof n1 = html``","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1179,"value":"viewof p2 = html``","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1188,"value":"viewof n2 = html``","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1182,"value":"function randomBinarySequence( n, p ) {\n var out = new Array( n );\n var i;\n for ( i = 0; i < n; i++ ) {\n if ( stdlib.base.random.randu() < p ) {\n out[ i ] = 1;\n } else {\n out[ i ] = 0;\n }\n }\n return out;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1192,"value":"function c3_sort_descending( a, b ) {\n if ( a > b ) {\n return -1;\n }\n if ( a === b ) {\n return 0;\n }\n return 1;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":697,"value":"pltc3 = stdlib.plot({\n 'x': [ stdlib.linspace( 0, 1, n1 ), stdlib.linspace( 0, 1, n2 ) ], // e.g., 'x': [ stdlib.linspace(0,1,n1), stdlib.linspace(0,1,n2) ]\n 'y': [ randomBinarySequence( n1, p1 ).sort( c3_sort_descending ), randomBinarySequence( n2, p2 ).sort( c3_sort_descending ) ], // e.g., 'y': [ sorted_dataset1, sorted_dataset2 ]\n 'xLabel': 'proportion',\n 'yLabel': 'converted',\n 'yMin': -0.1,\n 'yMax': 1.1,\n 'width': 600\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":699,"value":"html`${pltc3.render('html')}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":714,"value":"$40","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":744,"value":"c4_p1 = 0.1","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1199,"value":"c4_p2 = stdlib.linspace( c4_p1+0.1, 1.0, 100 )","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1203,"value":"c4_ns = {\n var maxN = 1000;\n var out;\n var res;\n var n;\n var i;\n \n out = new Array( c4_p2.length );\n for ( i = 0; i < c4_p2.length; i++ ) {\n n = 1;\n while ( true ) {\n res = stdlib.ttest2( randomBinarySequence( n, c4_p1 ), randomBinarySequence( n, c4_p2[ i ] ) );\n if ( res.rejected ) {\n out[ i ] = n;\n break;\n }\n n += 1;\n if ( n >= maxN ) {\n break;\n }\n }\n }\n return out;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1212,"value":"c4_dp = {\n var out = new Array( c4_p2.length );\n var i;\n for ( i = 0; i < out.length; i++ ) {\n out[ i ] = c4_p2[ i ] - c4_p1;\n }\n return out;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":737,"value":"pltc4 = stdlib.plot({\n 'x': [ c4_dp ], // e.g., 'x': [ delta_p ]\n 'y': [ c4_ns ], // e.g., 'y': [ n ]\n 'xLabel': 'Δp',\n 'yLabel': 'n',\n 'xMin': 0.0,\n 'xMax': 1.0,\n 'yMin': 0,\n 'width': 600,\n 'lineStyle': [ 'none' ],\n 'symbols': [ 'open-circle' ]\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":739,"value":"html`${pltc4.render('html')}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":754,"value":"md`\n> **Question**: what conclusions can you draw regarding the relationship between ${tex`n`} and ${tex`\\Delta p`}?\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":638,"value":"md`\n### Simulation Study\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":640,"value":"$41","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":662,"value":"''","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":666,"value":"plt22 = stdlib.plot({\n 'x': [ /* insert your `x` values */ ], // e.g., 'x': [ stdlib.linspace(-10,10,n) ]\n 'y': [ /* insert your pseudorandom numbers */ ], // e.g., 'y': [ rvs ]\n 'xLabel': 'x',\n 'yLabel': 'y',\n 'width': 600\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":668,"value":"html`${plt22.render('html')}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":674,"value":"md`\n**Exercise 23**: using the same approach as in the previous exercise, generate \\`M\\` simulated datasets. For each simulated dataset, perform a one-sample [Student's _t_-test](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/ttest), which tests the hypothesis that the true mean of the distribution from which we drew pseudorandom numbers is equal to zero. Store the [_p_-value](https://en.wikipedia.org/wiki/P-value) of each test in an array of length \\`M\\`. Calculate the proportion of [_p_-values](https://en.wikipedia.org/wiki/P-value) which are less than ${tex`\\alpha`}. \n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":676,"value":"''","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":683,"value":"md`\n> **Question**: is the proportion of [_p_-values](https://en.wikipedia.org/wiki/P-value) reasonably close to ${tex`\\alpha`}, as we would expect?\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":759,"value":"$42","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":780,"value":"''","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":783,"value":"pltc5 = stdlib.plot({\n 'x': [ /* insert your `n` values */ ], // e.g., 'x': [ stdlib.linspace(1,n,n) ]\n 'y': [ /* insert your proportions */ ], // e.g., 'y': [ y ]\n 'xLabel': 'n',\n 'yLabel': 'proportion',\n 'width': 600,\n 'yMin': 0.0,\n 'yMax': 1.1,\n 'lineStyle': [ 'none' ],\n 'symbols': [ 'open-circle' ]\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":786,"value":"html`${pltc5.render('html')}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":788,"value":"md`\n> **Question**: what conclusions can you draw regarding the relationship between ${tex`n`} and the proportion of [_p_-values](https://en.wikipedia.org/wiki/P-value) which are less than ${tex`\\alpha`}?\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":791,"value":"$43","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":804,"value":"''","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":809,"value":"md`\n> **Question**: do the [_p_-values](https://en.wikipedia.org/wiki/P-value) follow a [uniform distribution](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/base/dists/uniform/pdf)? How does varying \\`M\\` affect your conclusions?\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":814,"value":"$44","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":818,"value":"''","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":821,"value":"plt25 = stdlib.plot({\n 'x': [ /* insert your `x` values */ ], // e.g., 'x': [ stdlib.linspace(-10,10,n) ]\n 'y': [ /* insert your pseudorandom numbers */ ], // e.g., 'y': [ rvs ]\n 'xLabel': 'x',\n 'yLabel': 'y',\n 'width': 600\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":823,"value":"html`${plt25.render('html')}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":828,"value":"$45","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":833,"value":"''","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":836,"value":"md`\n> **Question**: is the proportion of [_p_-values](https://en.wikipedia.org/wiki/P-value) reasonably close to the specified significance level, as we would expect? Why or why not?\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":846,"value":"md`\n> **Question**: what happens as you vary dataset length?\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":848,"value":"md`\n### Confidence Intervals\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":850,"value":"$46","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":857,"value":"''","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":861,"value":"md`\n> **Question**: are you able to verify that the confidence intervals contain the value ${tex`\\mu`} around ${tex`1-\\alpha`} percent of the time?\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":873,"value":"md`\n> **Question**: does varying dataset length affect your conclusions? Why or why not?\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":869,"value":"md`\n> **Aside**: while in reality we are not going to repeat a specific test multiple times, we are likely to conduct many different tests over a long period of time. If we conduct them all at a significance level ${tex`\\alpha`}, we can be certain that ${tex`1-\\alpha`} of them contain the true value (provided all test assumptions are valid).\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":885,"value":"md`\n* * *\n\n## p-hacking\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":887,"value":"$47","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":919,"value":"md`\n**Exercise 28**: as a first step, take some time to familiarize yourself with the FiveThirtyEight dataset, which is available in this notebook as \\`nutrition_data\\`. The data originates from a [survey](https://en.wikipedia.org/wiki/Food_frequency_questionnaire) (referred to as the \"[food frequency questionnaire](https://en.wikipedia.org/wiki/Food_frequency_questionnaire)\") completed by respondents regarding their eating habits and personal characteristics.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":921,"value":"nutrition_data","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":924,"value":"md`\n> **Question**: how many respondents are there? Is this data quantitative, qualitative, or a mixture of both?\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":932,"value":"md`\n> **Question**: based on this dataset, what sorts of hypotheses might you want to test? (feel free to create a list in a new cell below)\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":935,"value":"$48","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":942,"value":"e29_groups = stdlib.bifurcateBy( nutrition_data, ( v ) => v.diabetes === 'Yes' )","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":945,"value":"md`\n**Exercise 30**: for each group, extract the frequency of green salad consumption (key: \\`GREENSALADFREQ\\`).\n\n> **Hint**: be sure to convert the frequencies to integer values.\n\n> **Hint**: extracting frequencies can be done with two one-liners via [inmap](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/utils/inmap) (available as [\\`stdlib.inmap(collection, fcn)\\`](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/utils/inmap) in this notebook). Just be sure to slice (i.e., copy) the raw (bifurcated) data **before** invocation!\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":949,"value":"e30_g1 = stdlib.inmap( e29_groups[ 0 ].slice(), ( v ) => parseInt( v.GREENSALADFREQ ) )","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1224,"value":"e30_g2 = stdlib.inmap( e29_groups[ 1 ].slice(), ( v ) => parseInt( v.GREENSALADFREQ ) )","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":953,"value":"md`\n**Exercise 31**: perform a two-sample two-sided [Student's _t_-test](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/ttest2) on the two frequency collections.\n\n> **Hint**: for pretty-printed output, use the \\`print()\\` method, which will print the exact definition of the **alternative hypothesis**.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":955,"value":"stdlib.ttest2( e30_g1, e30_g2 ).print()","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":958,"value":"md`\n> **Question**: is green salad consumption significantly different between the two groups?\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":961,"value":"md`\n> **Aside**: a few points to remember when _p_-hacking:\n\n> - Even if no real association exists, when using a significance level of 5% (as is standard), you expect to detect an average of five significant results per 100 tests, thus corresponding to a false positive rate of 5%.\n> - Even if you were to detect a true effect, the question still persists as to whether that effect is of any practical significance.\n> - As we explored above, as sample sizes become larger, hypothesis tests become better at detecting even very small differences across groups.\n> - Finally, we should keep in mind that correlation does **not** imply causation, as a confounding factor may exist that affects both variables and is thus responsible for the observed difference!\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":967,"value":"md`\n**Exercise 32**: as we did above for green salad consumption frequency, iterate over all 1066 frequency and quantity responses for those respondents with and without diabetes. For each set of responses, perform a two-sample [Student's _t_-test](https://github.com/stdlib-js/stdlib/tree/a5fc046e1a13d5938d45b8cb70b45fe14fbbe5ed/lib/node_modules/%40stdlib/stats/ttest2). Record whether the test result is significant. Upon analysis completion, print a list of statistically significant results (e.g., \"those with diabetes are less likely to eat green salad\").\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":970,"value":"{\n var keys = Object.keys( nutrition_data[ 0 ] ).slice( 27 );\n var res;\n var out;\n var g2;\n var g1;\n var k;\n var i;\n \n out = [];\n for ( i = 0; i < keys.length; i++ ) {\n k = keys[ i ];\n g1 = stdlib.inmap( e29_groups[ 0 ].slice(), ( v ) => parseInt( v[ k ] ) );\n g2 = stdlib.inmap( e29_groups[ 1 ].slice(), ( v ) => parseInt( v[ k ] ) );\n res = stdlib.ttest2( g1, g2 );\n if ( res.rejected ) {\n if ( res.xmean < res.ymean ) {\n out.push( 'those with diabetes are less likely: '+k );\n } else {\n out.push( 'those with diabetes are more likely: '+k );\n }\n }\n }\n return out;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":973,"value":"md`\n> **Question**: did you find any interesting statistically significant differences between the two groups? If so, which differences are most surprising?\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":980,"value":"md`\n**Challenge 6**: repeat the previous exercise for the remaining personal characteristics (e.g., whether someone is left-handed, has a particular belly button, owns a dog, etc).\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":982,"value":"''","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":985,"value":"md`\n> **Question**: did you find any interesting statistically significant differences between groups? If so, which differences are most surprising?\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":990,"value":"md`\n**Challenge 7**: repeat the previous exercise for all personal characteristic combinations (e.g., has diabetes vs owns a dog).\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":992,"value":"''","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":997,"value":"md`\n> **Question**: did you find any interesting statistically significant differences between groups? If so, which differences are most surprising?\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":999,"value":"md`\n> **Question**: can you think of any reasons why the previous analysis might be flawed? If so, what are those flaws? (feel free to write your thoughts in a new cell below)\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1009,"value":"md`\n### Bonferroni Correction\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1011,"value":"md`\nOne solution for the problem of multiple comparisons is the [Bonferroni correction](https://en.wikipedia.org/wiki/Bonferroni_correction). The procedure is straightforward: instead of testing each individual hypothesis at a significance level of ${tex`\\alpha`}, test each at ${tex`\\tfrac{\\alpha}{m}`}, where ${tex`m`} is the number of hypotheses. One can show that the family-wise error rate (FWER; i.e., the probability of rejecting **at least** one true **null hypothesis**) is less than ${tex`\\alpha`}.\n\n**Exercise 33**: had you used a [Bonferroni correction](https://en.wikipedia.org/wiki/Bonferroni_correction) in previous exercises, at what significance level should you have performed your tests?\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1019,"value":"0.05/1066","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1022,"value":"md`\n### Controlling the False Discovery Rate (FDR)\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1024,"value":"$49","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1032,"value":"''","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1035,"value":"plt34 = stdlib.plot({\n 'x': [ /* insert your `x` values */ ], // e.g., 'x': [ stdlib.linspace(1,1066,1066), stdlib.linspace(1,1066,1066) ]\n 'y': [ /* insert your y values and p-values */ ], // e.g., 'y': [ line_with_slope_alpha_over_m, pvals ]\n 'xLabel': 'x',\n 'yLabel': 'P_k',\n 'width': 600\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1038,"value":"html`${plt34.render('html')}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1003,"value":"md`\n* * *\n\n## Conclusions\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1005,"value":"md`\nIn this notebook, you achieved the following:\n\n- Learned about random sampling.\n- Explored how sample size affects estimate reliability.\n- Performed simulations, gaining a sense as to why simulation is important.\n- Created plots both for visualizing results and for verifying data shape and simulation behavior.\n- Learned about statistical significance and quantifying uncertainty.\n- Gained an intuition as to best practice and as to why one should always regard correlations with suspicion.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":420,"value":"md`\n* * *\n\n## Notes\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":422,"value":"md`\n- This notebook uses [stdlib](https://github.com/stdlib-js/stdlib), a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing applications. [stdlib](https://github.com/stdlib-js/stdlib) contains over 2,000 functions for mathematics, statistics, data processing, streams, and more. For more information, see the GitHub [repository](https://github.com/stdlib-js/stdlib).\n- This notebook is part of a series of notebooks used as part of a workshop on data science in JavaScript held at the [International JavaScript Conference](https://javascript-conference.com/) (iJS), held in London, UK, 11-13 April 2018.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":88,"value":"md`\n* * *\n\n## Appendix\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":299,"value":"md`\nAn appendix containing functionality used in running this notebook.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":91,"value":"stdlib = require( \"https://unpkg.com/@stdlib/stdlib@0.0.35/dist/stdlib-flat.min.js\" )","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":97,"value":"$4a","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":99,"value":"$4b","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":93,"value":"sample_dataset_emails_per_day = {\n var rpareto1 = stdlib.base.random.pareto1.factory( 1.0, 3.0, { 'seed': 1234 } );\n var names = male_first_names_en.concat( female_first_names_en );\n var len = names.length;\n var out;\n var tmp;\n var i;\n \n out = [];\n\n // Header line:\n out.push( [ 'name', 'gender', 'emails_per_day' ] );\n\n // Data:\n for ( i = 0; i < len; i++ ) {\n tmp = new Array( 3 );\n\t tmp[ 0 ] = names[ i ];\n\t if ( i < male_first_names_en.length ) {\n\t\t tmp[ 1 ] = 'male';\n\t } else {\n\t\t tmp[ 1 ] = 'female';\n\t }\n\t tmp[ 2 ] = stdlib.base.ceil( rpareto1() );\n out.push( tmp );\n }\n return out;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":912,"value":"nutrition_data = {\n var d3 = await require('https://d3js.org/d3.v5.min.js');\n return d3.csv('https://raw.githubusercontent.com/fivethirtyeight/data/master/nutrition-studies/raw_anonymized_data.csv');\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":107,"value":"md`\n* * *\n\n## License\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":109,"value":"md`\nCopyright (c) 2018 The Stdlib Authors.\n\nLicensed under the Apache License, Version 2.0 (the \"License\");\nyou may not use this file except in compliance with the License.\nYou may obtain a copy of the License at\n\n http://www.apache.org/licenses/LICENSE-2.0\n\nUnless required by applicable law or agreed to in writing, software\ndistributed under the License is distributed on an \"AS IS\" BASIS,\nWITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\nSee the License for the specific language governing permissions and\nlimitations under the License.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W4c","stdlib":"1","titling":"auto"}} 32:"$31:body"