Take a Chance on Statistical Edutainment
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| Title: | Take a Chance on Statistical Edutainment |
|---|---|
| Language: | English |
| Authors: | Lesser, Lawrence M. (ORCID |
| Source: | Teaching Statistics: An International Journal for Teachers. Spr 2022 44(1):34-42. |
| Availability: | Wiley. Available from: John Wiley & Sons, Inc. 111 River Street, Hoboken, NJ 07030. Tel: 800-835-6770; e-mail: cs-journals@wiley.com; Web site: https://www.wiley.com/en-us |
| Peer Reviewed: | Y |
| Page Count: | 9 |
| Publication Date: | 2022 |
| Document Type: | Journal Articles Reports - Descriptive |
| Descriptors: | Statistics Education, Mathematical Concepts, Thinking Skills, Probability |
| DOI: | 10.1111/test.12293 |
| ISSN: | 0141-982X |
| Abstract: | Thinking probabilistically is an essential part of thinking statistically, and the probability learning objectives that this article focuses on are those that are important in the underpinning of statistics and statistical models. Like mathematical statistics, probability can be considered purely from a mathematical viewpoint, but the focus here is on understanding concepts. |
| Abstractor: | ERIC |
| Entry Date: | 2022 |
| Accession Number: | EJ1324860 |
| Database: | ERIC |
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| FullText | Links: – Type: pdflink Url: https://content.ebscohost.com/cds/retrieve?content=AQICAHj0k_4E0hTGH8RJwT4gCJyBsGNe_WN95AvKlDbXJGqwxwGCXp6uHkVTR_9yyYujQ24EAAAA4zCB4AYJKoZIhvcNAQcGoIHSMIHPAgEAMIHJBgkqhkiG9w0BBwEwHgYJYIZIAWUDBAEuMBEEDK07GQTAiyj26cI7SAIBEICBm7Kjd0gfQF4d7D0sZRtwEZodcCHOK3nq-ZiMa2oq9kNtndkfyFh32gjBmuk4EhuTYNs3vNbdCPo56_1CZ2Czf-h8xoYZxXxNlhaIvsKGcH4_YTDGPgiFre7bw4tMe7QTah0mkc8HsUhV6fitg0AaiB9-jvfjHTEJEZS97vInY10k1I7r3mTJh4zDfGbWtc6tXGivJ6HcMZ1CWvKa Text: Availability: 1 Value: <anid>AN0154795841;d8y01mar.22;2022Jan24.04:16;v2.2.500</anid> <title id="AN0154795841-1">Take a chance on statistical edutainment </title> <p>Chances are that edutainment fun items can engage students in discussing and learning key concepts about probability.</p> <p>Keywords: teaching; cartoon; joke; magic; probability; song; teaching statistics; video</p> <hd id="AN0154795841-2">INTRODUCTION</hd> <p>Maybe you are thinking about probability because chances are you are reading this article as a pdf! Thinking probabilistically is an essential part of thinking statistically, and the probability learning objectives we will focus on here are those that are important in the underpinning of statistics and statistical models. Like mathematical statistics, probability can be considered purely from a mathematical viewpoint, but the focus here is on understanding concepts. This also involves considerable attention to the language of chance and probability. So, let us take a chance and dive in!</p> <hd id="AN0154795841-3">Basics</hd> <p></p> <ulist> <item> Be able to interpret statements about the chance that an event may happen, recognizing that such statements must adhere to a specific set of rules including:</item> <p></p> <item> Any probability is a number from 0 to 1.</item> <p></p> <item> Probability must refer to a well‐defined situation that includes any possible outcome to be considered (ie, the sample space), and the probability of the situation is 1.</item> <p></p> <item> If two events have no outcomes in common, the probability that at least one occurs is the sum of their individual probabilities.</item> <p></p> <item> The probability that an event does not occur is 1 minus the probability that the event does occur.</item> <p></p> <item> The chance that two events both occur is the chance of one event times the chance of the other given or assuming the first occurs. The chance that an event occurs assuming another event occurs is called the conditional probability of the first event given the second.</item> <p></p> <item> There are a number of other basic results that follow from the above, including:</item> <p></p> <item> When events do have outcomes in common, then the chance of at least one occurring is the sum of probabilities reduced by the probability of those common outcomes (a special case of the general inclusion‐exclusion principle).</item> <p></p> <item> If knowing that one event has happened does not change the chances for a second event, then they are said to be independent. The chance that two independent events both happen can therefore be found by multiplying the chances of each event.</item> <p></p> <item> Avoid common misconceptions in interpreting statements about chance events</item> <p></p> <item> Recognize <emph>equiprobability bias</emph> —outcomes of a probability experiment are usually not equally likely.</item> <p></p> <item> Recognize the <emph>conjunction fallacy</emph> because if the ways in which one event can happen are a subset of the ways that a second event happens, then the first event cannot have a higher probability. Students should also see this in a variety of language settings, including if one event implies another has happened, or if an event cannot occur unless another has also happened.</item> <p></p> <item> Recognize that probabilities of events in different situations—that is, under different conditions—should not be compared. A particular example of this is called the <emph>prosecutor's fallacy</emph> —the probability of evidence given an innocent explanation does not give information on its own about the probability of lack of an innocent explanation given the evidence. Both may be low at the same time (eg, in situations where the probability of the evidence is itself very low). In general, it is essential to pay careful attention to the conditions under which a probability statement applies in real‐world applications.</item> </ulist> <hd id="AN0154795841-4">Concepts for the underpinning of important aspects of statistical reasoning</hd> <p></p> <ulist> <item> Understand key concepts of the law of large numbers and expected value</item> <p></p> <item> As the number of independent repetitions of a random phenomenon increases, the relative frequency of a particular event tends to get closer and closer to its probability of occurring at each repetition. However, the variability of the number of times that the event occurs tends to grow, compared with what is expected (which is the number of repetitions times the probability of occurring at each repetition).</item> <p></p> <item> As the size of a random sample of independent observations on a variable increases, the (arithmetic) average of the observed values gets closer and closer to the expected value of the variable (the "population average"): The above result that the relative frequency of an event gets closer to its probability is a special case of this.</item> <p></p> <item> Recognize the <emph>gambler's fallacy</emph>. The law of large numbers does not work by compensation. Unusual runs of events will not change the probabilities about future events. When the repetitions of a random phenomenon are independent, the process has no memory.</item> <p></p> <item> Understand aspects of the behavior of sample statistics and the normal approximation</item> <p></p> <item> Sample statistics vary from sample to sample and the sampling distribution represents the distribution of the values the statistic might take over all possible samples.</item> <p></p> <item> As the number of independent repetitions of a random phenomenon increases, the distribution of the sample proportion (the relative frequency of a particular event) becomes approximately normal.</item> <p></p> <item> The above result for the sample proportion is a special case of the following: When a random sample of independent observations of a variable is used to collect data, then the sampling distribution of the sample mean is approximately normal as the sample size increases.</item> <p></p> <item> Understand that different approaches in statistical inference and data analysis may use different approaches and/or probability models, although all rules of probability must hold regardless of the perspective used</item> <p></p> <item> A non‐Bayesian approach views parameters as fixed and considers the probability of different data occurring given parameter values.</item> <p></p> <item> The Bayesian approach also considers the probability model for the data given the parameters, but adds an extra layer to this by also considering parameters as random variables having distributions and then updates the probability distribution of parameters given observed data using Bayes' rule. Knowledge of the situation is used for the initial (before data) distribution(s) of the parameter(s): this is called the prior distribution. Hence this approach is particularly useful in complex situations where there is some knowledge of the parameters. The updated distribution(s) of the parameters given the observed data are called the posterior distribution.</item> <p></p> <item> When the rules of a process are specified, probabilities may be estimated using simulation, usually with great accuracy as the number of simulated replicates of the process becomes large.</item> </ulist> <hd id="AN0154795841-5">BASICS</hd> <p></p> <hd id="AN0154795841-6">Probabilities are in [0, 1]</hd> <p>While students may hear a sports coach demanding "110% effort," it is thankfully rare to hear people actually refer to probabilities as numbers outside the interval from 0 to 1. One exception might involve confusion among decimals, fraction, and percent or between probabilities and odds (eg, a 50‐50 chance means the probability is 50%, not 50). Perhaps some humor can help our students internalize that a practical consequence of this simple rule is that any calculation they do yielding a value outside the [0,1] interval needs to be checked and redone. Here is a joke we wrote to help teachers reinforce that probabilities cannot be negative, while making sure students are used to the language conventions of medical test results and understand that "the probability of a negative test result" is NOT "a negative probability":</p> <p>I took a COVID test today. Chances are the test comes back positive, 'cause you can't have a negative probability.</p> <p>Students in a course covering the negative binomial distribution may appreciate this quip:</p> <p>A judge thought probabilities can be negative when he found out the number of trials before the third acquittal was a negative binomial probability.</p> <hd id="AN0154795841-7">Probability of sample space = 1</hd> <p>This rule forces students to understand what it means to identify the required situation, that is, lay out a sample space with "all possible outcomes" and avoid mistakes where statements about probabilities in differing sample spaces or conditional spaces are mixed together as if they were parts of the same whole. For example, have students critique this joke that is more than "some" of its parts:</p> <p>When you pick an adult at random, there's a 38% chance they take a conservative position on the issue, a 47% chance they take a liberal position, and a 41% chance they are undecided about how percentages work!</p> <hd id="AN0154795841-8">Equiprobability bias</hd> <p>Students of all ages commonly have this bias, in part because most textbook examples of dice, cards, coins, spinners, etc., involve symmetric outcomes that <emph>are</emph> equally likely. We cannot leave it to chance that they will outgrow this bias without intervention. For example, teachers can bring in asymmetric predecessors of dice such as astragalus (knuckle) bones (see pictures of such objects from many cultures at https://en.wikipedia.org/wiki/Knucklebones) or the pig‐shaped plastic dice of the game Pass the Pigs [<reflink idref="bib11" id="ref1">11</reflink>]. Another example is the (non‐50%) probability of a Hershey's Kiss® candy landing on its base, and hands‐on activities to explore this are given in Reference [<reflink idref="bib33" id="ref2">33</reflink>]. Such examples involving food for explorations in statistics class tend to be especially engaging for students. Other types of items allowing discussion of equiprobability bias include poetry ("50‐50" in Reference [<reflink idref="bib17" id="ref3">17</reflink>]) and song ("1 in 2" in Reference [<reflink idref="bib21" id="ref4">21</reflink>], which includes lesson guidance). The character in the latter item talks about having a 50% chance of winning the lottery because "either I hit jackpot or I don't," a conclusion so blatantly unrealistic that it can help a student identify the erroneous reasoning and more readily avoid it in less blatant contexts.</p> <hd id="AN0154795841-9">Probability of unions</hd> <p>The formula P(A or B) = P(A) + P(B)−P(A and B) is made intuitive with a simple Venn diagram of two overlapping circles, a diagram popularized in the 1880s by English logician John Venn. Students can see that the overlap area gets counted twice and so subtracting the overlap corrects for that (as a special case of the inclusion‐exclusion rule for multiple events). Venn diagrams also play a role in humor because, for example, puns can be represented as the intersection of two sets [<reflink idref="bib31" id="ref5">31</reflink>]. As a fun activity here, an instructor can mention a survey where randomly selected people are asked if they are an "early‐bird" or a "night owl" and whether their favorite analog radio station is an AM or FM broadcast. The teacher would then reveal the fun "Venn Diagram" in the left of Figure 1 below and ask if it captured the relationship you would see in results from this survey. Discussion should include how the "AM" is taking two meanings, the idea that shapes in a Venn diagram represent events (not variables), and perhaps discussing the ambiguity in the descriptions in relation to the survey. Finally, the instructor would reveal a proper Venn diagram like the one on the right side of Figure 1 and also ask what is represented by area within the black rectangle (sample space) but outside the circles (ie, a night owl who listens to FM radio). Slides to implement this activity are in Appendix S1.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/D8Y/01mar22/test12293-fig-0001.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="test12293-fig-0001.jpg" title="1 Venn diagram activity including meme (left side adapted from https://www.reddit.com/r/memes/comments/ac1py4/) and proper Venn diagram on right side" /> </p> <p></p> <hd id="AN0154795841-11">Disjoint or not?</hd> <p>Disjoint refers to events themselves, so this question can often involve careful consideration of language and identification of events. Most students have seen many examples of simple disjoint events, so there can be lively discussions about situations such as weather, for example, what is disjoint with "sunny"? Some other examples are: is a parcel arriving late disjoint with the event it does not arrive at all? What is disjoint with the event "she bought no more than two bottles"? The importance of making explicit the fact that students must think about whether events are disjoint is reinforced by this pair of limericks from Reference [<reflink idref="bib3" id="ref6">3</reflink>]:</p> <p>Now, here is an essential point:</p> <p>If A is from B disjoint</p> <p>then to give you, I'm glad</p> <p>permission to add</p> <p>at the risk of seeming flamboyant.</p> <p>and</p> <p>And now here's another cool fact</p> <p>on which you might soon need to act:</p> <p>You still, my comrade,</p> <p>have permission to add</p> <p>and orders to also subtract.</p> <hd id="AN0154795841-12">Probability of complementary events</hd> <p>The "complementary events" rule P(A) = 1−P(not A) is important because it is often much simpler to calculate the probability of the complement of an event than the probability of the original event, especially events involving something happening at least once, as in the famous question "What is probability there is at least one birthday match among people in the room?" Of course, we could also ask "if we make 10 attempts at humor in this column, what is the chance at least one of them is funny?" The U.S. secondary school teacher Mary McLellan's song parody [<reflink idref="bib27" id="ref7">27</reflink>] about the probability of something happening at least once includes this memorable couplet:</p> <p>Probability of at least 1</p> <p>Listing all the combinations is not fun</p> <p>Probability of at least 1</p> <p>Easier with 1 minus the probability of none.</p> <p>Beyond canned probability exercises, it is important to note that this commonly‐used rule connects to teaching about important statistics topics such as multiple comparisons, which is not always covered in a first course. In that case it is of great interest to know that the probability of at least one statistically significant result can be expressed as the complement of the event that there are no statistically significant results. Another application example is testing a pooled community sample of wastewater for RNA from SARS‐CoV‐2 or the risk of having at least one infected person in a gathering of a specific size (see https://covid19risk.biosci.gatech.edu for an app showing estimates of this risk in different geographical settings).</p> <hd id="AN0154795841-13">Independence</hd> <p>The P(A and B) = P(A)P(B) rule for independent events has arithmetic that is intuitive in equally‐likely random situations (eg, a 6×2 table for the possible outcomes of rolling a fair six‐sided die and flipping a fair coin makes clear that the intersection of "heads" and "3" is 1/12, which is the product of the independent events: 1/2 × 1/6). However, students often struggle with the concept of independence itself. This is exacerbated by its everyday meaning as "separate" (eg, independent nations) when disjoint (ie, mutually exclusive) nonempty events can actually <emph>never</emph> be independent. To reinforce that, see if students try to calculate an answer to this question: "If the nations of Germany and Brazil are estimated to have 10% and 5% chances, respectively, of winning the next World Cup, what is the probability that they both win?" (Logically, there is no sporting chance this can happen, of course).</p> <p>To make students distinguish between disjoint and independent events (as well as think about sample space in the process), teachers can assign an event relationships activity [<reflink idref="bib1" id="ref8">1</reflink>], in which students classify pairs of events as disjoint, independent, both, or neither. Students will notice at the end of the activity that one cell in the table of possibilities remains empty and can be asked for an explanation, even if in hindsight, of why two events cannot be disjoint and independent at the same time.</p> <p>It may also help to introduce some jokes that force students to think about the independence assumption itself. For example, this joke adapted from an online post by David McElroy:</p> <p>The hitchhiker I picked up asked, "Why did you pick me up? How do you know I'm not a serial killer?" I replied that the chances would be astronomical for one car to have <emph>two</emph> serial killers.</p> <p>The multiplication rule can also be used to set up Hen Fetsch's 1954 magic trick "Mental Epic," as described in Reference [<reflink idref="bib22" id="ref9">22</reflink>]. While a teacher could play a performance video [<reflink idref="bib10" id="ref10">10</reflink>] and ask prompting questions before, during, or after the video, I (the first author) enjoy performing a low‐tech version adapted from and explained in Reference [<reflink idref="bib6" id="ref11">6</reflink>], where I bring to class typical equiprobability objects—52‐card deck, 6‐sided die, 5‐region spinner, and a coin—and invite everyone to make predictions of the results of a single outcome of each. As we go through predicting, then observing, the four events one at a time, I incorporate expected value into the discussion by asking, "About how many people in the room would we expect to get <emph>this</emph> part correct?" And finally, I ask, "If these four events are independent, what is the probability of getting all four predictions correct?" and because it is small (1/3120), students are shocked that my predictions (and no one else's) were all correct.</p> <p>In general, perceived low probabilities in magic motivate hypothesis testing in statistics. As noted in Reference [<reflink idref="bib22" id="ref12">22</reflink>, p. 265], "Magic catches our attention, and therefore has the potential to engage the mind of a probability/statistics student precisely because (it appears) an unlikely event has happened that seems difficult to explain by chance alone."</p> <p>Teaching tips in Reference [<reflink idref="bib23" id="ref13">23</reflink>] show how a 2.2‐minute film by Samuel Rapien [<reflink idref="bib32" id="ref14">32</reflink>] can be used as a vehicle to discuss probability rules and concepts, such as applying the multiplication rule for independent events to assess how unusual are specific streaks of coin flips, die rolls, and card draws. Teachers can efficiently and memorably recap most of the basic rules with the 1‐minute song "Probability Rules Rap" in finished or interactive form [<reflink idref="bib20" id="ref15">20</reflink>].</p> <hd id="AN0154795841-14">Conditional probability</hd> <p>Independence is a special case of the general rule P(A and B) = P(A)P(B|A), and in general, there is a great need for activities and learning experiences to significantly increase students thinking about conditioning language and conditional probability. A simple way to start the discussion is to share this excerpt from a U.S. work of young adult fiction [<reflink idref="bib34" id="ref16">34</reflink>]: "Did you know that people who meet at least three different times within a 24‐hour period are 98% more likely to meet again?" The meaning of the phrase "98% more likely" is good fodder for class discussion as students might confuse its interpretation between a 1% chance of becoming 99% or becoming 1.98%. Follow this up by asking "How can the statement in the story be interpreted to seem plausible?" Conditional probability is also involved in the 1958 magic trick "Mental Image" by Dr. (Stanley) Jaks (described in Reference [<reflink idref="bib22" id="ref17">22</reflink>]). And, the lower right cartoon in Figure 2 gives students the chance to recognize that P(struck by lightning) is much lower than P(struck by lightning | standing outside in the storm)! The cartoon in the lower left of Figure 2 may be used to help instructors emphasize the point that assumptions of independence cannot just be declared without careful attention to checking their validity.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/D8Y/01mar22/test12293-fig-0002.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="test12293-fig-0002.jpg" title="2 Cartoons [13–15,29] (clockwise from upper left) on simulation, law of large numbers, conditional risk, and conditional probability" /> </p> <p></p> <hd id="AN0154795841-16">Conjunction fallacy</hd> <p>Jessica Utts [<reflink idref="bib37" id="ref18">37</reflink>] refers to Kahneman and Tversky [<reflink idref="bib12" id="ref19">12</reflink>] giving background about a person who majored in philosophy and was active in social justice causes and then asking which of these two statements is more likely: "Linda is a bank teller" vs "Linda is a bank teller and is active in the feminist movement." A statistics edutainment version of this could be to ask: What is more likely: "I am an entertainer" or "I am a statistics instructor and an entertainer"?</p> <hd id="AN0154795841-17">CONSEQUENCES</hd> <p>To help his kids understand the consequences of various probability rules, a Spanish‐speaking father might teach them probabiliDAD. And teachers may appreciate knowing about collections such as <ulink href="http://www.eduteka.org/MI/master/interactivate">http://www.eduteka.org/MI/master/interactivate</ulink> that offer probability apps in Spanish. Of course, the simple probability rules discussed in Section 2 have important consequences in statistics such as the law of large numbers, the central limit theorem (CLT), Bayes' rule, and the underpinning for the use of simulation.</p> <hd id="AN0154795841-18">Law of large numbers and expected value</hd> <p>Throughout an introductory statistics course, students see a common theme about the relationship between sample size and the consistency of an average around its expected value: the margin of error is smaller for larger sample sizes; the average of multiple measurements is more reliable than a single measurement; more repetitions of a simulation converge to the probabilities at hand, bigger sample sizes give you more power to detect differences, and so forth. A 42‐second musical statement of the law of large numbers encapsulating this theme can be played from McLellan [<reflink idref="bib28" id="ref20">28</reflink>]. The cartoon [<reflink idref="bib15" id="ref21">15</reflink>] in the upper right of Figure 2 provides a way to introduce the slightly broader idea of the long‐run behavior of chance processes so that the convergence of averages and proportions can be contrasted with the divergence of sums and counts. The Shiny app [<reflink idref="bib7" id="ref22">7</reflink>] and the associated lesson guidance for using the app in Reference [<reflink idref="bib38" id="ref23">38</reflink>] provide a quickly digested way to explore this contrast on one screen.</p> <p>A misconception about independence associated with the law of large numbers shows up when students consider events over time, as if events have "memory" of the past, as in this joke adapted from https://<ulink href="http://www.onlinemathlearning.com/math-jokes-statistics.html:">www.onlinemathlearning.com/math-jokes-statistics.html:</ulink></p> <p>A man decides to walk home from a bar during a storm.</p> <p>'Aren't you afraid of being struck by lightning?' his friend asks.</p> <p>'Nope! In our county, statistics show one person per year gets struck by lightning, and that person died in the hospital 3 weeks ago'.</p> <p>In case it seems there is no chance someone in real life would use that type of reasoning, students can critique an actual quotation from the media. An example accompanied by discussion questions [<reflink idref="bib16" id="ref24">16</reflink>] concerns a city in the United States, which suffered in 2006 what was described as a 500‐year‐flood, prompting a city council representative to claim there was no danger of another one for almost 500 years.</p> <p>And then there are people who play a state or national lottery (or advertise lottery "strategies") as if drawings are somehow not independent, as if knowing past winning numbers can increase one's chances of picking future winning numbers. Such misconceptions are debunked in Larry Lesser's song "The Gambler" [<reflink idref="bib18" id="ref25">18</reflink>]. The quote attributed to American comedian Jay Leno gives a thought‐provoking perspective: "How come you never read a headline like 'Psychic Wins Lottery'?" An instructor might ask students how predictions about the lottery that defy probability rules when numbers are drawn independently can be tested in practice.</p> <p>Two common student misconceptions must be addressed in teaching about the law of large numbers (LLN). First, the LLN does not work by compensation (see above); unusual runs of events will not change the probabilities about future events. To address this issue, along with the resources related to independence mentioned above, an instructor might point to a quote from <emph>The Simpsons</emph>, a decades‐running animated show broadcast in over 70 countries, where character Lisa Simpson declares that "hot streaks are a statistical illusion!" Secondly, students often wrongly perceive the idea of convergence to imply that hitting the expected value <emph>exactly</emph> becomes more likely with more trials. Using the Shiny app mentioned above and discussing extreme cases (clearly getting exactly one head in two tosses is much more common than getting exactly 500000000 heads in a billion tosses) are helpful in dispelling this notion. Finally, low‐stakes assessment questions that can be used with a student response system are provided in Appendix S2 to help instructors quickly gauge student understandings of the LLN.</p> <hd id="AN0154795841-19">The normal approximation</hd> <p>Contrasting the idea of the distribution over individual values of the variable of interest (the "theoretical population") with the distribution of all possible values that a statistic might take (the sampling distribution) is important for understanding the logic of statistical inference; regardless of what inference methods are used (simulation, randomization, normal approximations, other distributions, Bayesian). Introductory classes often concentrate on sample averages or proportions as a common statistic arising in large sample surveys where the CLT justifies using the normal approximation. An interactive song [<reflink idref="bib4" id="ref26">4</reflink>] "Central Limit Theorem" now has accompanying interactive videos in which advanced student inputs will yield one of three possible versions of the song (and video). For instance, if students pick "heights at a parent‐child campout" as their example of a non‐normal distribution, then the animation in the video illustrating the CLT shows the bimodal distribution of heights that would form the population and how average heights of samples picked from that population would have a bell‐shaped histogram. Since different students will have chosen different examples of non‐normal populations, an instructor can ask the class to share their results and highlight how different examples all led to the same shape for the sampling distribution of the average. (We note that the nonstandard capitalization of "normal" when referring to the probability distribution is pedagogically intentional, in light of how the word is used differently in an everyday sense (see Reference [<reflink idref="bib19" id="ref27">19</reflink>] for an activity using a comic strip).) This song‐based lesson uses approximately 15 minutes to complete and can then be paired with the use of the CLT Shiny app [<reflink idref="bib39" id="ref28">39</reflink>], where users can directly control both the sample size and the shape of the theoretical distribution in multiple examples about averages and proportions. A student handout in Appendix S3 provides lesson guidance for using the app in class when tablets or computers are available.</p> <hd id="AN0154795841-20">Multiple interpretations and Bayes' rule</hd> <p>Bayes' rule, P(A|B) = P(B|A)*P(A)/P(B), developed by 18th‐century English statistician Reverend Thomas Bayes, can be explored with the Shiny app [<reflink idref="bib26" id="ref29">26</reflink>]. This app and the interactive Tom Toce song "It Might Not Be That Bad" [<reflink idref="bib36" id="ref30">36</reflink>] both explore Bayes' theorem using the classic example of seeking the probability that a person has a disease given that they test positive when you have information about the probability of testing positive when you have the disease (test sensitivity), about the probability that the test is negative when you do not have the disease (test specificity), and about the prevalence of the disease in the population. Teachers can have students examine how the results given in the lyrics align with the results they see in engaging with the app. The short story "Ladies' Night" set in a casino setting by Canadian mathematician Robert Dawson [<reflink idref="bib5" id="ref31">5</reflink>] can also be used in an out‐of‐class assignment in association with the study of probability rules, Bayes' theorem, and expectations as they relate to games of chance such as the Monty Hall "3 doors" problem. For example, students might verify the probability statements made by the character on the top of page 301 in the story.</p> <p>Probability models in inference and data analysis describe the distribution of data observed or collected on variables with assumed or known distribution(s) and the parameters that describe this distribution(s) (ie, called the likelihood when considered as a function of the parameters). A Bayesian approach also assumes a distribution for the parameter(s), and the inferential aims in this approach lie in understanding the uncertainty in our knowledge of parameters for a given set of data:</p> <p>GRAPH</p> <p>Bayes' rule thus gives the underpinning for relating these two. By providing your current best estimate of the distribution of a parameter, Bayes' rule takes that <emph>prior</emph> information and uses the model for the data given the parameter to yield the <emph>posterior</emph> distribution of the parameter given the data. This is celebrated in the song "Prior", a parody by Mark Glickman [<reflink idref="bib8" id="ref32">8</reflink>] of the 1969 hit "Venus" by the Dutch band Shocking Blue. Teachers can ask students to relate the themes in the song to their counterparts in Bayes' rule (ie, "extra information" as new data/evidence and how a "prior" is required to use Bayes' rule).</p> <p>After introducing the concept of prior and posterior distributions, an instructor teaching statistical inference might choose to go further and introduce students to the concept of Bayes factors (which compares probabilities between models) to compare with the <emph>P</emph> value used in hypothesis testing (which looks at probabilities of obtaining the data or more extreme under the null hypothesis, without assuming distributions over parameter values). A fun method of teaching about Bayes factors uses the card trick described in Glickman [<reflink idref="bib9" id="ref33">9</reflink>].</p> <hd id="AN0154795841-21">Simulation</hd> <p>Using simulation to study the distribution of results arising from probability models allows the quick exploration of the effects of changing parameters and other characteristics of the model.</p> <p>This allows students to concentrate on the key principles of the behavior of models rather than the mathematical procedures involved in manipulating them. Thinking about the key components of a probability model that are needed to give software a recipe for simulation helps students to think algorithmically: an important learning objective in its own right [<reflink idref="bib30" id="ref34">30</reflink>]. The cartoon in the upper left of Figure 2 can be used to launch a class discussion on these issues.</p> <p>The use of simulation in inference allows students to better focus on the core logic of inference [<reflink idref="bib2" id="ref35">2</reflink>] and is implemented in a variety of new curricula for introductory statistics [<reflink idref="bib25" id="ref36">25</reflink>,<reflink idref="bib35" id="ref37">35</reflink>]. A song to introduce the basic idea of using simulation to calculate a <emph>P</emph> value for a randomization test (by simulating lots of group assignments and seeing what proportion of them give more extreme test statistics than observed with the actual group assignments) can be found in Lesser and Pearl [<reflink idref="bib24" id="ref38">24</reflink>].</p> <hd id="AN0154795841-22">DISCUSSION</hd> <p>It is important to be able to teach probability in a way that links with statistical thinking, incorporates language, includes real‐world examples, and develops probabilistic thinking in both modeling and statistical investigations. It is also important to teach in a way that helps students distinguish a model from the true process it is intended to mimic, and to identify and understand the assumptions of a model. Probability as a purely mathematical topic may suggest investigation of interesting results, but this is different from probability as an integral part of statistics.</p> <hd id="AN0154795841-23">ACKNOWLEDGEMENT</hd> <p>The interactive songs mentioned in this work were supported by Division of Undergraduate Education Project SMILES, NSF/EHR/DUE 1544426 (PSU), 1544237 (UTEP).</p> <p>GRAPH: Appendix S1. Supporting Information.</p> <p>GRAPH: Appendix S2. Supporting Information.</p> <p>GRAPH: Appendix S3. Supporting Information.</p> <ref id="AN0154795841-24"> <title> Footnotes </title> <blist> <bibl id="bib1" idref="ref8" type="bt">1</bibl> <bibtext> Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.</bibtext> </blist> <blist> <bibl id="bib2" idref="ref35" type="bt">2</bibl> <bibtext> Funding information Division of Undergraduate Education, Grant/Award Numbers: 1544237, Project SMILES, NSF/EHR/DUE 1544426</bibtext> </blist> </ref> <ref id="AN0154795841-25"> <title> REFERENCES </title> <blist> <bibtext> R. P. Carey, Event relationships, 2021, available at https://<ulink href="http://www.causeweb.org/cause/resources/fun/games/event-relationships">www.causeweb.org/cause/resources/fun/games/event-relationships</ulink>.</bibtext> </blist> <blist> <bibtext> G. W. 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Toce, It might not be that bad, 2015, available at https://<ulink href="http://www.CAUSEweb.org/cause/resources/fun/songs/it-might-not-be-bad">www.CAUSEweb.org/cause/resources/fun/songs/it-might-not-be-bad</ulink> Interactive version at https://<ulink href="http://www.CAUSEweb.org/smiles/songs/not%5fthat%5fbad">www.CAUSEweb.org/smiles/songs/not%5fthat%5fbad</ulink>.</bibtext> </blist> <blist> <bibtext> J. M. Utts, Seeing through statistics, 4th ed., Cengage, Stamford, CT, 2015.</bibtext> </blist> <blist> <bibtext> S. L. Wang, A. Y. Zhang, S. Messer, A. Wiesner, and D. K. Pearl, Student‐developed Shiny applications for teaching statistics, J Stat Data Sci Educ 29 (2021), 1 – 10. https://doi.org/10.1080/26939169.2021.1995545.</bibtext> </blist> <blist> <bibtext> Y. Wang, Central limit theorem [R Shiny app], 2020, available at https://psu-eberly.shinyapps.io/Central_Limit_Theorem.</bibtext> </blist> </ref> <aug> <p>By Lawrence M. Lesser and Dennis K. Pearl</p> <p>Reported by Author; Author</p> </aug> <nolink nlid="nl1" bibid="bib11" firstref="ref1"></nolink> <nolink nlid="nl2" bibid="bib33" firstref="ref2"></nolink> <nolink nlid="nl3" bibid="bib17" firstref="ref3"></nolink> <nolink nlid="nl4" bibid="bib21" firstref="ref4"></nolink> <nolink nlid="nl5" bibid="bib31" firstref="ref5"></nolink> <nolink nlid="nl6" bibid="bib27" firstref="ref7"></nolink> <nolink nlid="nl7" bibid="bib22" firstref="ref9"></nolink> <nolink nlid="nl8" bibid="bib10" firstref="ref10"></nolink> <nolink nlid="nl9" bibid="bib23" firstref="ref13"></nolink> <nolink nlid="nl10" bibid="bib32" firstref="ref14"></nolink> <nolink nlid="nl11" bibid="bib20" firstref="ref15"></nolink> <nolink nlid="nl12" bibid="bib34" firstref="ref16"></nolink> <nolink nlid="nl13" bibid="bib37" firstref="ref18"></nolink> <nolink nlid="nl14" bibid="bib12" firstref="ref19"></nolink> <nolink nlid="nl15" bibid="bib28" firstref="ref20"></nolink> <nolink nlid="nl16" bibid="bib15" firstref="ref21"></nolink> <nolink nlid="nl17" bibid="bib38" firstref="ref23"></nolink> <nolink nlid="nl18" bibid="bib16" firstref="ref24"></nolink> <nolink nlid="nl19" bibid="bib18" firstref="ref25"></nolink> <nolink nlid="nl20" bibid="bib19" firstref="ref27"></nolink> <nolink nlid="nl21" bibid="bib39" firstref="ref28"></nolink> <nolink nlid="nl22" bibid="bib26" firstref="ref29"></nolink> <nolink nlid="nl23" bibid="bib36" firstref="ref30"></nolink> <nolink nlid="nl24" bibid="bib30" firstref="ref34"></nolink> <nolink nlid="nl25" bibid="bib25" firstref="ref36"></nolink> <nolink nlid="nl26" bibid="bib35" firstref="ref37"></nolink> <nolink nlid="nl27" bibid="bib24" firstref="ref38"></nolink> |
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