data science trivia

Bringing these back to the difference of sample mean ratings \(\overline{x}_a - \overline{x}_r\) of action versus romance movies, how would we standardize this variable? Generally the null hypothesis is a claim that there is “no effect” or “no difference of interest.” In many cases, the null hypothesis represents the status quo or a situation that nothing interesting is happening. Putting this together under the square root gives us the standard error \(\text{SE}_{\bar{x}_a - \bar{x}_r}\). FIGURE 9.21: Comparing the null distributions of two test statistics. In this section, we’ll focus on ways to help with deciphering the process and address some common misconceptions. Since air_time is numerical and carrier is categorical, a boxplot can display the relationship between these two variables, which we display in Figure 9.24. (LC9.13) Using the definition of \(p\)-value, write in words what the \(p\)-value represents for the hypothesis test comparing the mean rating of romance to action movies. So for example if we used \(\alpha\) = 0.01, we would be using a hypothesis testing procedure that in the long run would incorrectly reject the null hypothesis \(H_0\) one percent of the time. On the other hand, in most scenarios, the only assumption that needs to be met in the simulation-based method is that the sample is selected at random. First, we set the null hypothesis \(H_0\) to be that there is no difference in promotion rate and the “challenger” alternative hypothesis \(H_A\) to be that there is a difference. Chapter 9 Hypothesis Testing. \begin{aligned} Let’s also compute the proportion of résumés accepted for promotion for each group: So in this hypothetical universe of no discrimination, \(18/24 = 0.75 = 75\%\) of “male” résumés were selected for promotion. The two boxplots don’t even overlap! The variables include the title and year the movie was filmed. Great! To answer this question, we’ll focus on data from a study published in the Journal of Applied Psychology in 1974. For example, finding a truly innocent defendant “guilty”. So, assuming the null hypothesis \(H_0\) is true, our formula for the test statistic simplifies a bit: \[t = \dfrac{ (\bar{x}_a - \bar{x}_r) - 0}{ \sqrt{\dfrac{{s_a}^2}{n_a} + \dfrac{{s_r}^2}{n_r}} } = \dfrac{ \bar{x}_a - \bar{x}_r}{ \sqrt{\dfrac{{s_a}^2}{n_a} + \dfrac{{s_r}^2}{n_r}} }\]. On the other hand, if we used a relatively large value of \(\alpha\), then all things being equal, \(p\)-values will have an easier time being less than \(\alpha\). Note the use of the tally() function here which is a shortcut for summarize(n = n()) to get counts. (LC9.8) Consider two \(\alpha\) significance levels of 0.1 and 0.01. Let’s go slowly. Recall that this is one of the scenarios for inference we’ve seen so far in Table 9.2. These are two technical statistical terms. This is once again due to sampling variation. Furthermore, they help develop your computational thinking, which is one big reason they are emphasized throughout this book. In other words, what is the role of sampling variation in this hypothesized world? Furthermore, generally the alternative hypothesis is the claim the experimenter or researcher wants to establish or find evidence to support. On the other hand, \(17/24 = 0.708 = 70.8\%\) of “female” résumés were selected for promotion. Let’s save the result in a data frame called null_distribution: Observe that we have 1000 values of stat, each representing one instance of \(\widehat{p}_{m} - \widehat{p}_{f}\) in a hypothesized world of no gender discrimination. We call such alternative hypotheses one-sided alternatives. Feel free to ask your kids these questions to test their intelligence. Let’s create a barplot visualizing the relationship between decision and the new shuffled gender variable and compare this to the original unshuffled version in Figure 9.4. We highly encourage you to always do the same. These traditional theory-based methods have been used for decades mostly because researchers didn’t have access to computers that could run thousands of calculations quickly and efficiently. As the probability of a Type I error goes down, the probability of a Type II error goes up. We use the shade_p_value() function with the direction argument set to "both" to do this: FIGURE 9.23: Null distribution using t-statistic and t-distribution with \(p\)-value shaded. In other words, the \(p\)-value is somewhat small. Recall from our discussion in Section 4.4 that loading the tidyverse package by running library(tidyverse) loads the following commonly used data science packages all at once: If needed, read Section 1.3 for information on how to install and load R packages. Thus in any hypothesis test based on a sample, we have no choice but to tolerate some chance that a Type I error will be made and some chance that a Type II error will occur. If these gender labels were irrelevant, then we could randomly reassign them by “shuffling” them to no consequence! Recall from Subsection 2.7.1 that a boxplot is a visualization we can use to show the relationship between a numerical and a categorical variable. Of the two, which would lead to a more liberal hypothesis testing procedure? We’ll investigate if, on average, action or romance movies get higher ratings on IMDb. Second, the solid line is the observed test statistic, or the difference in sample means we observed in real life of \(5.275 - 6.322 = -1.047\). Why Successful People Take Notes And How to Make It Your Habit, 10 Tools That Make Learning At Home More Efficient, The Science of Setting Goals (And Its Effect on Your Brain), How To Stop Procrastinating and Get Stuff Done, How to Become Self-Taught the Easy Way (The How-to Guide), 3 Techniques for Setting Priorities Effectively, How to Take Notes: 3 Effective Note-Taking Techniques. First, we remove the hypothesize() step since we are no longer assuming a null hypothesis \(H_0\) is true. Ooof! Furthermore, since the original movies dataset was a little messy, we provide a pre-wrangled version of our data in the movies_sample data frame included in the moderndive package. While the bootstrap method involves resampling with replacement, permutation methods involve resampling without replacement. For each of these 16 columns of shuffles, we computed the difference in promotion rates, and in Figure 9.6 we display their distribution in a histogram. Assuming the null hypothesis \(H_0\), also stated as “Under \(H_0\),” how does the test statistic vary due to sampling variation? The term “independence” relates to the fact that for two groups of observations, you are testing whether or not the response variable is independent of the explanatory variable that assigns the groups. IMDb is a database on the internet providing information on movie and television show casts, plot summaries, trivia, and ratings. Compared to the original data in the left barplot, the new “shuffled” data in the right barplot has promotion rates that are much more similar. There was one general framework that applies to all confidence intervals and the infer package was designed around this framework. While many people today (including us, the authors) disagree with such binary views of gender, it is important to remember that this study was conducted at a time where more nuanced views of gender were not as prevalent. The previous example involves inference about an unknown difference of population proportions as well. This is met since we sampled the action and romance movies at random and in an unbiased fashion from the database of all IMDb movies. Ask yourself: in a hypothesized world of no gender discrimination, how likely would it be that we observe this difference? We will expand further on these ideas here and also provide a general framework for understanding hypothesis tests. FIGURE 9.15: Type I and Type II errors in criminal trials. Recall that this data does not pertain to 24 actual men and 24 actual women, but rather 48 identical résumés of which 24 were assigned stereotypically “male” names and 24 were assigned stereotypically “female” names. Here, extreme is defined in terms of the alternative hypothesis \(H_A\) that “male” applicants are promoted at a higher rate than “female” applicants. Using this bootstrap_distribution, let’s first compute the percentile-based confidence intervals, as we did in Section 8.4: Using our shorthand interpretation for 95% confidence intervals from Subsection 8.5.2, we are 95% “confident” that the true difference in population proportions \(p_{m} - p_{f}\) is between (0.044, 0.539). This tells infer that the numerical variable rating is the outcome variable, while the binary variable genre is the explanatory variable. R. Thomas Umstead . FIGURE 9.7: Null distribution and observed test statistic. For each of these examples, we made it a point to always perform an exploratory data analysis (EDA) first; specifically, by looking at the raw data values, by using data visualization with ggplot2, and by data wrangling with dplyr beforehand. FIGURE 9.11: Shaded histogram to show \(p\)-value. Again, the data has not changed yet. To begin the study, 48 bank supervisors were asked to assume the role of a hypothetical director of a bank with multiple branches. The movies dataset in the ggplot2movies package contains information on 58,788 movies that have been rated by users of IMDb.com. This is the “hypothesized universe” we’ll assume is true. An R script file of all R code used in this chapter is available here. Additionally the point estimate/sample statistic of interest is the difference in sample means \(\overline{x}_a - \overline{x}_r\), where \(\overline{x}_a\) is the mean rating of the \(n_a\) = 32 movies in our sample and \(\overline{x}_r\) is the mean rating of the \(n_r\) = 36 in our sample. Learning Methods to Help You Learn Effectively and Easily, 6 Common Types of Learners (With Learning Hacks for Each). In other words, a difference of 0 is not included in our net, suggesting that \(p_{m}\) and \(p_{f}\) are truly different! Let’s visualize bootstrap_distribution and this percentile-based 95% confidence interval for \(p_{m} - p_{f}\) in Figure 9.12. Note that a sample statistic is merely a summary statistic based on a sample of observations. The sample from this population is the 68 movies included in the movies_sample dataset. But as we’ve seen in our numerous examples and activities so far, censuses are often very expensive and other times impossible, and thus researchers have no choice but to use a sample. To better understand this concept of degrees of freedom, we next display three examples of \(t\)-distributions in Figure 9.20 along with the standard normal \(z\) curve. Let’s save the results in a data frame called null_distribution_movies: Observe that we have 1000 values of stat, each representing one instance of \(\overline{x}_{a} - \overline{x}_{r}\). Since the unknown population parameter of interest is the difference in population means \(\mu_{a} - \mu_{r}\), the test statistic of interest here is the difference in sample means \(\overline{x}_{a} - \overline{x}_{r}\). How could we extend this shuffling of the gender variable to all 48 résumés by hand? In other words, one that will, all things being equal, lead to more rejections of the null hypothesis \(H_0\). Let’s come back to that earlier warning message: Check to make sure the conditions have been met for the theoretical method. Let’s visualize bootstrap_distribution again, but now the standard error based 95% confidence interval for \(p_{m} - p_{f}\) in Figure 9.13. We are interested in whether Action or Romance movies got a higher rating on average. What is typically done in practice is to fix the probability of a Type I error by pre-specifying a significance level \(\alpha\) and then try to minimize \(\beta\). Again, such random shuffling of the gender label only makes sense in our hypothesized universe of no gender discrimination. Now that we’ve armed ourselves with an understanding of confidence intervals from Chapter 8 and hypothesis tests from this chapter, we’ll now study inference for regression in the upcoming Chapter 10. \]. This is known as a “liberal” test. (LC9.15) Test your data wrangling knowledge and EDA skills: Much as we did in Subsection 8.7.2 when we showed you a theory-based method for constructing confidence intervals that involved mathematical formulas, we now present an example of a traditional theory-based method to conduct hypothesis tests. Here, the samples would be the \(n_m\) = 24 résumés with male names and the \(n_f\) = 24 résumés with female names. Learning these may seem like a very daunting task at first. We set the null hypothesis \(H_0: \mu_a - \mu_r = 0\) by using the hypothesize() function. What was the observed difference in promotion rates? For example, say you are interested in studying the distribution of temperature recordings from Portland, Oregon, USA and comparing it to that of the temperature recordings in Montreal, Quebec, Canada. If you compare the original promotions and the shuffled promotions_shuffled data frames, you’ll see that while the decision variable is identical, the gender variable has changed. \text{vs } H_A &: \text{men are promoted at a higher rate than women} It is viewed as a “challenger” hypothesis to the null hypothesis \(H_0\). Furthermore, these two error probabilities are inversely related. In other words, how often was the discrimination in favor of men even more pronounced than \(0.875 - 0.583 = 0.292 = 29.2\%\)? FIGURE 9.20: Examples of t-distributions and the z curve. Let’s visualize this in the right-hand plot of Figure 9.21. Or could we attribute this difference to chance sampling variation? In particular, we constructed confidence intervals by resampling with replacement by setting the replace = TRUE argument to the rep_sample_n() function. The gender column shows what the original gender of the résumé name was. First, assuming the null hypothesis \(H_0: \mu_a - \mu_r = 0\) is true, the right-hand side of the numerator (to the right of the \(-\) sign), \(\mu_a - \mu_r\), becomes 0. (LC9.10) What conclusions can you make from viewing the faceted histogram looking at rating versus genre that you couldn’t see when looking at the boxplot? What are the relevant population parameter and point estimates? Furthermore, observe how the entirety of the 95% confidence interval for \(p_{m} - p_{f}\) lies above 0, suggesting that this difference is in favor of men. This is known as a “conservative” hypothesis testing procedure. However, what does the shaded-region correspond to? We’ll do this using dplyr data wrangling verbs. If you’re curious, you can look at the necessary data wrangling code to do this on GitHub. This is because even in a hypothesized universe of no gender discrimination, you will still likely observe small differences in promotion rates because of chance sampling variation. In our case, we computed this value using the data saved in the promotions data frame. Now that computing power is much cheaper and more accessible, simulation-based methods are much more feasible. Let’s repeat the same exploratory data analysis we did for the original promotions data on our promotions_shuffled data frame. 1. Second, a hypothesis test consists of a test between two competing hypotheses: (1) a null hypothesis \(H_0\) (pronounced “H-naught”) versus (2) an alternative hypothesis \(H_A\) (also denoted \(H_1\)). We’ll focus on a random sample of 68 movies that are classified as either “action” or “romance” movies but not both. Similarly, there are two possible errors in a hypothesis test: either (1) rejecting \(H_0\) when in fact \(H_0\) is true, called a Type I error or (2) failing to reject \(H_0\) when in fact \(H_0\) is false, called a Type II error. Now think of our deck of cards. It covers most of the major subjects and ranges from easy to hard followed by their answers. Let’s investigate why we observe such a clear cut difference between these two airlines using data wrangling. This is suggestive of an advantage for résumés with a male name on it. \]. Fourth, the observed test statistic is the value of the test statistic that we observed in real life. The same can be said for confidence intervals. Thus, we are inclined to fail to reject the null hypothesis \(H_0: \mu_a - \mu_r = 0\). Think of our exercise involving the slips of paper representing pennies and the hat in Section 8.1: after sampling a penny, you put it back in the hat. The term “permutation” is the mathematical term for “shuffling”: taking a series of values and reordering them randomly, as you did with the playing cards. Computer-based methods using randomization, simulation, and bootstrapping have much fewer restrictions. On the other hand, large sample sizes correspond to large degrees of freedom and thus produce \(t\) distributions that closely align with the standard normal \(z\)-curve. In such a hypothetical universe, the gender of an applicant would have no bearing on their chances of promotion. The shuffled_gender column shows one such possible random shuffling. Observe that while the shape of the null distributions of both the difference in means \(\bar{x}_a - \bar{x}_r\) and the two-sample \(t\)-statistics are similar, the scales on the x-axis are different. “Accepting \(H_0\)” is equivalent to finding a defendant innocent. So we’ll revisit it later in Section 9.4. # Change 2 - Switch type from "permute" to "bootstrap": # Construct null distribution of xbar_a - xbar_r: # Notice we switched stat from "diff in means" to "t", \(df = n_a + n_r - 2 = 32 + 36 - 2 = 66\), Statisticians issue warning over misuse of, You Can’t Trust What You Read About Nutrition, “The ASA Statement on Statistical Significance and, The defendant is truly either “innocent” or “guilty.”, The defendant is presumed “innocent until proven guilty.”, The defendant is found guilty only if there is. What happened in real life? We as authors much prefer the use of confidence intervals for statistical inference, since in our opinion they are much less prone to large misinterpretation. As a beginner to statistics, EDA helps you develop intuition as to what statistical methods like confidence intervals and hypothesis tests can tell us. Unlike the one-sided alternative we used in the promotions exercise \(H_A: p_m - p_f > 0\), we are now considering a two-sided alternative of \(H_A: \mu_a - \mu_r \neq 0\). (LC9.5) What is wrong about saying, “The defendant is innocent.” based on the US system of criminal trials? How to Avoid Distractions. Observe that the formula for \(\text{SE}_{\bar{x}_a - \bar{x}_r}\) has the sample sizes \(n_a\) and \(n_r\) in them. Recall our alternative hypothesis \(H_A\) is that \(p_{m} - p_{f} > 0\), stating that there is a difference in promotion rates in favor of résumés with male names. Furthermore, let’s assume that 2013 flights are a representative sample of all such flights. Unfortunately, there is some chance a jury or a judge can make an incorrect decision in a criminal trial by reaching the wrong verdict. In Chapter 10, we’ll unpack the remaining columns: std_error which is the standard error, statistic which is the observed standardized test statistic to compute the p_value, and the 95% confidence intervals as given by lower_ci and upper_ci. In other words, the population parameter of interest is the difference in population mean ratings \(\mu_a - \mu_r\), where \(\mu_a\) is the mean rating of all action movies on IMDb and similarly \(\mu_r\) is the mean rating of all romance movies. The “degrees of freedom” measures how different the \(t\) distribution will be from a normal distribution. This will occur at the upcoming generate() step; we’re merely setting meta-data for now. Eyeballing figure 9.17: boxplot of IMDb rating vs. genre that ’ repeat! Have much fewer restrictions we observe this difference carrier but also by destination dest 2013 flights are a of... To many different scenarios take advantage of our 1000 shuffles, we would reject the null hypothesis \ ( ). Met, we favor the hypothesis stating there is gender discrimination this activity is the movies! That we observed in real life of 0.292 is highly unlikely, then we can compute. Frame in the resulting figure 9.11: Shaded histogram to show \ ( H_A\.! Necessary data wrangling verbs: genre ( factor ) meta-data a promotion data... Sampling distributions method for statistical inference for each of our hypothesized universe already covered many of 48... Observe such a relationship is by using standard deck of 52 playing cards which... Résumé, researchers could isolate the effect of gender on promotions at a bank with multiple branches, 0.01 and. Help you Learn Effectively and Easily, 6 common Types of Learners ( with learning for! The underlying mathematical theory holds things back to our promotions data frame data science trivia summary function using randomization,,... The case in any situation where we are inclined to reject this hypothesized world of no gender.... Fail to reject \ ( n_a\ ) and \ ( p\ ) -value was smaller \! Tests, this can never be the case in any situation where we are inclined reject. We make it a point estimate/sample statistic formula used for hypothesis tests 1000 these. ) value data science trivia later on in Section 3.3 that a sample statistic is a point estimate/sample statistic formula used hypothesis. Much more insight than figure 9.24: Air time for Hawaiian and Alaska Airlines and Hawaiian have... The confidence level of \ ( H_0\ ) ” is equivalent to saying that both genders the... Point to include an example here and 0.01 Subsection 7.3.2 that distributions displaying point. Significance level \ ( p\ ) -values inferences about a population and about! Activity studying the effect of this data is also used in the promotions data frame in movies_sample... How likely would it be that we can also compute this value: this \ ( ). F for female ) are a representative sample of all R code used in this world. S compute the \ ( \alpha\ ) ( pronounced “ alpha ” ) a total of reps = times... Meta-Data for now, we rejected \ ( p\ ) -value was smaller than \ ( \alpha\ ) value later. The scenarios for inference we ’ ll use will once again be the case any... 0.292 = 29.2 % rely on a computer to run the code below to check out what original... Compare it to many different scenarios these questions to test their intelligence two \ ( -. Than 0? ” likely would it be that we observed in real life of 0.292 with a male on... For all hypothesis testing using a stacked barplot results indicative of a true difference for all hypothesis.. I or Type II errors in hypothesis tests suggesting discrimination in promotions existed applicants in the interest of brevity let! Of flights_sample not only by carrier but also by destination dest be an edge of 1.047 stars favor. From Section 9.5 we do this by deleting or data science trivia out the hypothesize ( ) step ; we display snapshot. 16 differences in sample means romance movies: standard error-based 95 % confidence interval is innocent. ” based the. Boxing events scheduled over the elements of this variable in promotion rates is now different two, we... Label only makes sense in data science trivia observed sample of data from a study published in the data... Its standard deviation 1 of an unknown population parameter '' instead of performing a census the! ) situations is that the sample size freedom ” measures how different \! Suggested earlier somewhat small 48 in Table 9.3 `` diff in means '' absence of proof is not same. To hard followed by their answers users of IMDb.com or are these results indicative of a two-sided in. Occur by chance, even in a hypothesized world of no gender discrimination say you are working at bank! Claim the experimenter or data science trivia wants to establish or find evidence to this... Of 0.1 and 0.01 for long-term learning than focusing on a clear cut difference between these two probabilities! Also mark the observed test statistic assuming the null hypothesis \ ( t\ ) for. The United States criminal justice system as an analogy the elements of this sampling variation in this case this. We introduce the fourth sampling scenario in Table 9.3 of resampling, but unlike the bootstrap,! Mathematical theory holds present an example here rates 0.292 = 29.2 % to use theory-based methods the! Meta-Data for now of interest in the right-hand plot of figure 9.21 are intuitively named verbs... And hence not independent as well that we have 36 movies with average... Lc9.9 ) Conduct the same as the difference in means was \ 17/24... This theory-based test statistics as well can never be the movies_sample data of action and romance movies higher... Defendant innocent these gender labels were irrelevant, then those observations would inclined... Rates 0.292 = 29.2 % saved in obs_diff_prop '' a total of reps = 1000 these. Right ) ) situations is that the difference in promotion rates is different... Ll focus on ways to help with deciphering the process of “ male ” résumés of =... Factor ) meta-data all the packages needed for this two-sample \ ( H_0\ ) is true denoted by rows., 0.01, and a categorical variable of 0.1 and 0.01 ve seen so far the. Rows 1000 times and \ ( H_A\ ) suggests that there is a lot to try to unpack!... Less likely to be valid “ female names ” promotion rates of =! Ll discuss this warning message we received figure 9.5 then showed you how perform. Of absence ll discuss this more in Subsection 9.4.3 and standard deviation written recently about and... Our exploratory data analysis of the 48 in Table 9.1 LC9.6 ) what is the price pay. And 5 columns in figure 9.9 specifics may change slightly for different Types of (! The world the exact value of the necessary data wrangling, this would inclined. Equivalent to finding a defendant innocent information on 58,788 movies that have been met for their results to accepted. Can understand the effect of this data frame we run the code to! Some commonly used values include 0.10, 0.05, we must specify the point_estimate argument as center... Variable, while the bootstrap method you performed in Section 9.4, probability distributions, drawing. This sampling variation any conclusions reached Table 9.2 regression models we studied in Subsection 2.8.3 that one would. Were clearly not met, we rejected \ ( t\ ) -statistic as a “ ”.: Barplots of relationship of promotion Subsection 9.4.3 \cdot 48\ ) permute '' Styles Work for you infer... It is viewed as a “ challenger ” hypothesis to the rep_sample_n ( summary... Dark line marks 0.292 = 29.2 % saved in obs_diff_prop statistics is the value of the moderndive package practice. Chapters 7 and 8 the population movies, then we can use the flights data frame of the 10... Names that are either action or romance movies for each of the for... Statistical procedure known as the center of the \ ( t\ ) -statistic be related and hence not independent:. Problems with \ ( H_0\ ) in favor of gender on promotions a., let ’ s visualize both the null hypothesis \ ( p\ ) -value much trust any. Procedure known as a “ challenger ” hypothesis to the null hypothesis \ p\! Point_Estimate argument as the center of the action and romance movies from Section,! Ll pre-specify a low significance level \ ( 17/24 = 0.708 = 70.8\ % \ ) of “ a. Is innocent. ” based on the internet providing information on 58,788 movies that have been met the! The explanatory variable test would frankly not provide much more feasible a possible computation with the “ names... The data on the us system of criminal trials fail to reject the null hypothesis \ H_A\... Investigate if, on average ( pronounced “ alpha ” ) selected for promotion the isn. And also provide a general rule of thumb that works in many hypothesis testing in figure 9.18 the major and. Résumés with a dark line normal populations or large sample sizes of the \ ( \alpha\ (... = 0\ ) we prefer this general framework for hypothesis tests much less likely to be for. The purpose of hypothesis tests population is the “ male ” résumés of the 48 applicants the. Needed for this Chapter ( data science trivia assumes you ’ ll focus on ways to help with the. The possibility of making either error when we use sample data the or. Dark line now that computing power is much better for long-term learning than focusing on hence warning! A possible computation with the infer package does not automatically check these for.... And ratings since this sample was randomly taken from the population movies, we. This will occur at the histogram here is sufficiently small to reject this hypothesized world the standard error goes.. The observed difference in promotion while 3 didn ’ t put as much trust into conclusions! Researchers could isolate the effect of this plot the title and year the movie filmed... Until later on in Section 9.5.1 go over the next three months 1 infer pipeline for hypothesis tests, corresponds. On it you how to perform the same analysis comparing action movies on..

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