Activities in Introductory Statistics Michael Sullivan Joliet Junior College msulliva@jjc.edu sullystats@gmail.com www.sullystats.com

StatCrunch HSSSCL-QUIPU-STEYR-DAYAN-MAINT-JOKES

StatCrunch Join the group “MAA Workshop at COD"

Retrieving Existing Data StatCrunch Data Sets b. StatCrunch This! Brooksbaseball.net or Payscale.com c. Data.world, Kaggle, and Quora d. https://www.dhs.gov/tsa-claims-data e. How to manipulate data. Go to Yahoo!Finance and download stock price change data. Create a rate of return column; create an up/down/no change column

Collect Your Own Data In StatCrunch, select MyStatCrunch > My Surveys Click Create a survey

Descriptive Statistics Mean versus Median Correlation Regression by Eye Influential observations Go to https://media.pearsoncmg.com/ph/esm/esm_sullivan_sst5e_17/applets/sst5e_studentactivity_applets.html Select the applet “Regression Influence” Open the classroom notes at sullystats.com Go through the regression influence activity.

Simpson’s Paradox Quantitative Data -- SAT Teacher data in GAISE Go to StatCrunch.com and open the data file “SAT versus Teacher Salaries”. Draw a scatter diagram of Salary versus SAT Find least-squares regression between Salary and SAT. Create a new variable entitled “low”, “med”, “high” using the following command: ifelse(Percent Students Taking SAT <= 22,low,ifelse(Percent Students Taking SAT <= 49,med,high)) Low Percent <= 22%; Med: 23 <= Percent <= 49; High >= 50 v. Draw a scatter diagram by Percent of Students Taking SAT

Simpson’s Paradox – Qualitative Data

Common Mistakes in Statistics Common Mistakes Webpage (Univ of Texas)

Sampling Distributions I. Sampling Distribution for the Mean Using Simulation i. StatCrunch “Home Run 2016” data Using Applet i. Applets > Sampling Distributions Sampling Distribution for the Proportion

2016 Home Runs Draw a histogram of the population data. Find the population mean and population standard deviation. Obtain 1000 random samples of size n = 9 from the population. For each sample, find the mean. Draw

2016 Home Runs Select Data > Sample

2016 Home Runs Now find the mean of each sample. Draw a histogram of the means. Find the mean and standard deviation of the sample means.

Student’s t-Distribution

“The Introductory Statistics Course: A Ptolemaic Curriculum?” by George W. Cobb Technology Innovations in Statistics Education Volume 1, Issue 1 2007 http://escholarship.org/uc/item/6hb3k0nz The Ptolemaic model of the universe survived as the model for the universe from the first century to the 1500s. The Earth is the center of the universe just as the normal model is the center of the inferential universe.

Those who know that the consensus of many centuries has sanctioned the conception that the earth remains at rest in the middle of the heavens as its center, would, I reflected, regard it as an insane pronouncement if I made the opposite assertion that the earth moves. Copernicus developed a better model, but it wasn’t immediately accepted. Those who know that the consensus of many centuries has sanctioned the conception that the earth remains at rest in the middle of the heavens as its center, would, I reflected, regard it as an insane pronouncement if I made the opposite assertion that the earth moves. Nicolaus Copernicus Read more: http://www.brainyquote.com/quotes/authors/n/nicolaus_copernicus_2.html#ixzz1d7NaRt00 For I am not so enamoured of my own opinions that I disregard what others may think of them. Nicolaus Copernicus Read more: http://www.brainyquote.com/quotes/authors/n/nicolaus_copernicus.html#ixzz1d7NiSjhL “Real knowledge is to know the extent of one's ignorance.”

Let’s use StatCrunch to simulate the building of this normal model Let’s use StatCrunch to simulate the building of this normal model. Assume we are sampling from a population with mean 100 and standard deviation 15. Let’s take simple random samples of size n = 9. In a typical introductory statistics course, students are exposed to the sampling distribution of the sample mean. The student is taught that the distribution of the sample mean (under certain conditions) is at least normally distributed with mean mu and standard deviation sigma over the square root of the sample size. This model is meant to describe the distribution of the sample mean if we were able to obtain many, many samples of size n and compute the sample mean. Of course, the conditions mentioned above are that either the distribution from which the sample is drawn is normal, or the sample size is sufficiently large. Do the simulation. Find the sample mean for each sample. Draw a normal curve with mean = 100, standard deviation = 15/sqrt(9) = 5. Center the histogram at 100 (try start bin at 82, bin width = 4) Notice that the sample mean rarely deviates from the population mean by more than 10. So, perhaps we can use this “margin of error” with a sample mean. Sounds awesome, but there is one problem…we don’t know the “margin of error” without knowing the sampling distribution, and we don’t know the sampling distribution without knowing the distribution of the parent population. Of course, if we knew the parent population, we wouldn’t need to estimate a parameter in the first place!!!

Problem: How can we estimate the “margin of error” when we only have sample data? Enter Bradley Efron (in 1979). He suggests sampling with replacement from the sample data many, many times to find a proxy for the sampling distribution of the sample statistic.

Verb: "to bootstrap is to help (oneself) without the aid of others” Adjective: "relying entirely on one's efforts and resources". In 1979, Bradley Efron presented an alternative approach to constructing confidence intervals using a normal model, or Student's t-distribution. The method is called Bootstrapping. The general idea behind this method is to use the power of the computer to build the distribution of the sample mean from sample data - the only resource you have to describe the distribution of the sample mean. Remember, the normal model is used because it represents a good fit to what would happen if we were allowed (able) to obtain many, many random samples from a population and determine the sample statistic. For example, the earlier slide shows what would happen if we took many, many samples of size n = 9 from a normal population with mean 100 and standard deviation 15.

Now, suppose instead we only had n = 10 observations, but did not have the ability to obtain many, many samples from the population. Could we still build a distribution of sample means from these n = 10 observations and use this distribution as an approximation of the sampling distribution of the statistic (say, the sample mean)? Efron says "yes"! The idea is that we sample from the n = 10 observations with replacement over and over; and then compute the sample statistic that interests us. The thinking behind this is that the 10 observations are “lots and lots of copies” of the original population.

Two Worthwhile Sources on Bootstraps This document shows the capture rate of confidence intervals when sampling from different populations for different methods in constructing the interval. https://docs.google.com/spreadsheets/d/1MNOCwOo7oPKrDB1FMwDzsYzvLoK-IBqoxhKrOsN1M2A/edit#gid=0 Tim Hesterberg: http://amstat.tandfonline.com/doi/full/10.1080/00031 305.2015.1089789

Using the Bootstrap Method to Construct a Confidence Interval The following data represent the price per square foot of a random sample of recently sold condominiums in Miami Beach, FL. 275.24 271.77 274.81 283.03 287.07 275.78 271.16 270.14 280.95 277.30 Source: www.zillow.com Discuss “narrowness” bias (Hesterberg). Bootstrap standard errors tend to be too narrow by a factor of sqrt[(n-1)/n]. For example, the sample standard deviation for the Miami Beach data is s = 5.483, so s/sqrt(n) = 5.483/sqrt(10) = 1.734 I get a standard error for 5000 runs of the Miami data equal to 1.61. Computing 1.61*sqrt(10/9) = 1.697 Construct a 95% confidence interval for the mean price per square foot of a condominium in Miami Beach, FL using a bootstrap sample.

Do your students understand what a P-value measures? If you asked your students to define a P­-value, what type of response would you get? Do you believe your students have a solid grasp of what a P­-value represents? Do you believe that students truly understand what the models we use to obtain a P-value actually represent? Is there anything we could do to increase our students' conceptual understanding of a P-value? Yes!! Let’s go through an example where we use randomization to approximate a P-value. From this type of simulation, students will get a real sense as to what the P-value is measuring and why it can be used to judge the statements made in the null and alternative hypothesis. Finally, these randomization techniques will allow students to see why models such as the normal model might be useful in finding P-values.

Articles on P-values ASA’s Statement on P-values http://www.tandfonline.com/doi/full/10.1080/00031305.2016.1154108 Scientific Method: Statistical Errors http://www.nature.com/news/scientific-method-statistical-errors-1.14700 P-value Precision and Reproducibility http://www.tandfonline.com/doi/abs/10.1198/tas.2011.10129 Statistical Significance and P-values https://www.sciencedaily.com/releases/2016/03/160307092305.htm An interesting article on the media and reproducibility http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0172650#pone.0172650.s002

Introduction to P-values The proportion of the human population that is left-handed is 0.12. Mensa is an organization of high-IQ individuals (to qualify to be a member of Mensa one’s IQ must be at the 98th percentile). In a random sample of 20 members of Mensa, it was found that 4 were left-handed. Does this result suggest that a higher proportion of Mensa members are left-handed that the general population? What is the variable of interest in this study? Is it qualitative or quantitative? What are the null and alternative hypotheses? What can’t a fair coin be used to simulate the outcomes of this study? Use the Urn applet in StatCrunch to build a null model that could be used to simulate the outcomes of the study. Hint: There are approximately 120,000 members of Mensa worldwide. Under the assumption the null hypothesis from part (b) is true, how many of the 120,000 Mensa members would be left-handed? How many would not be left-handed? Use the Urn applet built in part (d) to simulate 1 repetition of this study. Explain what the results represent. Use the Urn applet built in part (d) to simulate 5000 more repetitions of this study. What does the result suggest? Use the binomial probability distribution function to find P(X < 4) where n = 20 and p = 0.12. (d) Need 120,000(0.12) = 14,400 red balls for the left-handers; need 120,000(0.88) = 105,600 blue balls for the right-handers.

The Role of Sample Size The proportion of the human population that is left-handed is 0.12. Mensa is an organization of high-IQ individuals (to qualify to be a member of Mensa one’s IQ must be at the 98th percentile). In a random sample of 120 members of Mensa, it was found that 24 were left-handed. Does this result suggest that a higher proportion of Mensa members are left-handed that the general population? Build the Urn applet again and run 5000 simulations. Be sure to select proportions. Click Analyze. Find the mean and standard deviation of the 2000 sample proportions. Determine the mean and standard deviation of the sample proportion assuming p = 0.12. Compare to part (b). Use the normal model to find the probability of observing 24 or more left-handers from a population whose proportion is 0.12. (c) Mean = 0.12; Standard Deviation = 0.0297

Professors Honey Kirk and Diane Lerma of Palo Alto College developed a “learning community curriculum that blended the developmental mathematics and the reading curriculum with a structured emphasis on study skills.” In a typical developmental mathematics course at Palo Alto College, 50% of the students complete the course with a letter grade of A, B, or C. In the experimental course, of the 16 students enrolled, 11 completed the course with a letter grade of A, B, or C. Source: Kirk, Honey & Lerma, Diane, “Reading Your Way to Success in Mathematics: A Paired Course of Developmental Mathematics and Reading” MathAMATYC Educator; Vol. 1. No. 2 Feb, 2010. (a) What proportion of the students enrolled in the experimental course passed with an A, B, or C?

(b) Describe how a coin might be used to simulate the outcome of this experiment to gauge whether the results are unusual. In testing hypotheses, we assume the statement in the null hypothesis is true. Here, we are assuming the proportion of students in the experimental section who will pass is no different from the traditional class. So, we assume p = 0.5. We can treat each of 16 coins as a student and let heads represent pass, tails represent fail. Flip the 16 coins and count the number of heads. This would represent the outcome of the experiment if half the students pass the experimental course.

(c) Use MINITAB, StatCrunch, or some other statistical spreadsheet to simulate 1000 repetitions of this experiment assuming the probability a randomly selected student passes the course is 0.5. The histogram below, which represents the results of 1000 repetitions of the experiment. Use your results or the results below to gauge the likelihood of 11 or more students passing the course if the true pass rate is 0.5. That is, determine the approximate P-value. What does this tell you? Note the shape of the distribution.

Shape!!

(d) Use the binomial probability distribution to determine an exact P-value. P(X > 11) = 1 – 0.8949 = 0.1051

(e) Now suppose that the actual study was conducted on 48 students and 33 passed the course with an A, B, or C. This would be a study that has ten times as many subjects. What is the proportion of students who passed in this experiment? How does the result compare with part (a)? This demonstrates the impact sample size can have on the outcome of a hypothesis test. Notice the proportion of students who pass the experimental section is the same, but now the result is statistically significant. This shows that small sample sizes require an incredible amount of evidence against the statement in the null hypothesis before one can reject the null.

Testing a Hypothesis about a Population Mean: Simulation   Coors Field is home to the Colorado Rockies baseball team and is located in Denver, Colorado. Denver is approximately one mile above sea level where the air is thinner. Therefore, baseballs are thought to travel farther in this stadium. Does the evidence support this belief? In a random sample of 15 homeruns hit in Coors Field, the mean distance the ball traveled was 417.6 feet. Does this represent evidence to suggest that the ball travels farther in Coors Field than it does in the other Major League ballparks?

Testing a Hypothesis about a Population Mean: Simulation

Testing a Hypothesis about a Population Mean: Simulation To do this, we will obtain 1000 simple random samples of size n = 15 from the population. Data > Sample Select the variable “Distance”. Enter 15 for the sample size. Check the radio button “Stacked with sample ID”. Open in new data table. Compute the mean of each sample by selecting Stat > Summary Stats > Columns Select Distance(Sample) for the variable; group by Sample. Only compute the mean.

Why is the P-value so much larger using Student’s t-distribution?

The Bootstrap Adjust the data so that the mean of the sample data equals the mean stated in the null hypothesis.

Obtain a 10,000 bootstrap samples Obtain a 10,000 bootstrap samples. Click Analyze to export the results to the spreadsheet.

Compute the mean and standard deviation of the bootstrapped means Compute the mean and standard deviation of the bootstrapped means. Compare the standard deviation of the bootstrapped means to s/sqrt(n) where n = 15. They should be close. Remember, bootstrap standard errors tend to be smaller (narrowness bias).

Repeat the Simulation with New Sample Data

Bootstrap of New Data

Testing a Hypothesis about Two Independent Population Proportions Joliet Junior College implemented a redesign of its Intermediate Algebra course. The Math Redesign Program (MRP) is a mastery-based learning model in which students must work at a minimally prescribed pace and each module must be mastered prior to moving to the next module. The goal of the research was to determine if the MRP course results in a higher pass rate than the traditional lecture course. During the Fall, 2015 semester, Professor Egner recruited 48 volunteer students who were going to enroll in Intermediate Algebra in the Spring, 2016 semester. She randomly assigned the students to either the MRP course or traditional course. Professor Egner taught both sections. At the end of the semester, she determined whether a student passed, or failed, the course. The table summarizes the results of the study in a contingency table. Does this evidence suggest the MRP course is superior to the traditional course as measured by pass rates?   MRP Course Traditional Course Total Pass 19 14 33 Fail 5 10 15 24 48

Math Redesign Program at JJC What is the response variable in the study? What is the explanatory variable in the study? What type of experimental design is this? What role does randomization play in this study? What or who are the experimental units in the study? Are any variables controlled? Is there the potential for confounding in this study? Explain.

Math Redesign Program at JJC

Math Redesign Program at JJC The difference of 0.209 seems significant. That is, it sure seems like the MRP course has a higher pass rate. That said, is it possible that the MRP course had a higher pass rate due to random chance? Put another way, is it possible the 33 students who passed would have passed regardless of which course they were enrolled in and they just happened to end up in the MRP course? So, there are two possibilities here: (1) The MRP course is not effective and the higher pass rate in the MRP course was due to random chance. That is, the proportion of students who pass the MRP course equals the proportion who pass the traditional course. (2) The MRP course is effective and this explains the difference in pass rates. That is, the proportion of students who pass the MRP course is greater than the proportion of students who pass the traditional course.

Math Redesign Program at JJC

Math Redesign Program at JJC To develop a conceptual understanding for building the null model, we use an urn. Let 33 green balls represent the 33 students who passed the course and let 15 red balls represent the 15 students who did not pass. Mix the 48 balls in the urn and randomly choose 24 balls. These 24 balls will represent the 24 students who were assigned to the MRP course. Notice that this random assignment is done under the assumption the statement in the null hypothesis is true because each ball has an equally likely chance of going to the MRP course or the traditional course. Note: We are not saying that passing versus not passing is equally likely, just that the likelihood of a “passing” student going to the MRP course is the same as that student going to the traditional course.

Math Redesign Program at JJC Test Statistic

Math Redesign Program at JJC Be sure to point out that the distribution of outcomes is centered at 0. Why? We conducted this test under the assumption there is no difference in the effectiveness of the courses. In other words, when the null hypothesis is true, we would expect the MRP program to have a higher proportion of passers half the time and the traditional course to have a higher proportion of passers about half the time. What would happen if we increased the sample size ten-fold, but had the same proportion of success in each treatment group?

P-value using Fisher’s Exact Test The P-value would be the probability of obtaining 19 or more successes from the redesign group from the 33 successes. Side note: Rumor has it that Fisher developed this test after Dr Muriel Bristol claimed to be able to determine whether milk was added to tea first or second. See “The Lady Tasting Tea”.

Math Redesign Program at JJC In the 2000 repetitions of the experiment (through the applet), 217 resulted in a sample difference of 0.209 or higher. We estimate the P-value to be 217/2000 = 0.1085. The probability of observing a sample difference in proportions of 0.209 or higher under the assumption the difference in proportions is zero is 0.1085.

Math Redesign Program at JJC What would happen if we increased the sample size five- fold? So, we now have 95 out of 120 MRP students passing and 70 out of 120 Traditional Students passing.

Math Redesign Progam at JJC

Math Redesign Program at JJC

vs. A study was conducted by researchers designed to determine if application of duct tape is more effective than cryotherapy (liquid nitrogen applied to the wart for 10 seconds every 2 to 3 weeks) in the treatment of common warts." The researchers randomly divided 40* patients into two groups. The 20 patients in Group 1 had their warts treated by applying duct tape to the wart for 6.5 days and then removing the tape for 12 hours, at which point the cycle was repeated, for a maximum of 2 months. The 20 patients in Group 2 had their warts treated by cryotherapy for a maximum of six treatments. Once the treatments were complete, it was determined that 17 patients in Group 1 (duct tape) and 12 patients in Group 2 (cryotherapy) had complete resolution of their warts. Does the evidence suggest duct tape is a more effective treatment? I recommend doing the following activity with students through both a tactile simulation as well as through the use of applets prior to the introduction of the traditional models we use. Source: Dean R. Focht III, Carole Spicer, Mary P. Fairchok. "The Efficacy of Duct Tape vs. Cryotherapy in the Treatment of Verruca Vulgaris (the Common Wart)," Archives of Pediatrics and Adolescent Medicine, 156(10), 2002. *These numbers were adjusted slightly for convenience.

Basketball Predictions Two famous college basketball analysts are Jay Bilas and Dick Vitale. In 116 college basketball games, Bilas correctly predicted the outcome of 77 of them; in 116 games, Vitale correctly predicted the outcome of 69 of them. Is Bilas better at predicting games?

Basketball

Basketball

Randomization Test for Means We can also use random assignment to compare two independent means. The data in this scenario is collected from a completely randomized design with two levels of treatment and a quantitative response variable. Or, the data is collected using an observational study in which there are two distinct groups and the variable of interest is quantitative. For example, suppose Professor Sullivan suspects that the female students in his Elementary Algebra course spend more time on homework than the male students. To confirm his suspicion, he randomly selects 12 male and 12 female students and records the amount of time (in minutes) on task for a particular section based on MyMathLab reports.

The sample mean time spent on homework for the 12 males is 80 The sample mean time spent on homework for the 12 males is 80.6 minutes while the sample mean time spent on homework for the 12 females is 99.8 minutes. The dot plot and sample means both suggest that females are spending more time on homework, on average, but is the difference significant? Or, is it possible, simply due to random chance that we obtained a sample mean difference of 99.8 – 80.6 = 19.2 minutes and the two groups study the same amount of time, on average. The value of 19.2 is the test statistic for this study. Is the difference in sample means of 19.2 minutes statistically significant, suggesting that females spend more time on homework than males? Or, is it possible that the difference is due to random chance and there is no difference in the study times in the two groups?” There are two possibilities:   There is no association between gender and study times and the difference in study times shown in Table 1 is due to random chance. That is, study time is not related to gender. There is an association between gender and study times. Namely, females, on average, spend more time studying and this explains the difference in sample means. That is, study time is related to gender.

Enter the gender of each student and his or her study time into the StatCrunch spreadsheet. Now, select Data, then highlight Sample. Select the column “Gender”, make the sample size 24 (for the 24 students). Let “Number of samples:” equal 1. Click Compute!.

The sample mean time for males is 89 The sample mean time for males is 89.8 minutes and the sample mean time spent for females is 90.5 minutes. The sample mean difference (females minus males) is 90.5 – 89.8 = 0.7 minute. This sample mean difference would also become part of the null model.

Designing a study Traditional underwriting to determine the risks associated with lending include credit scores, income, and employment history. The online lender ZestFinance wanted to determine whether people who fill out loan applications using all capital letters versus those who use all lower case letters versus those who fill out the application using upper and lowercase letters accurately default at different rates. Explain how to obtain and analyze data in order to determine whether the method used to fill out loan applications results in different default rates.

The Challenge in Polling One of the challenges in polling for elections is deciding who to include in your frame. (a) Suppose you were asked to conduct a poll for a senatorial election. Explain how you might design your poll. In your explanation include a discussion of the difference between "registered voters" and "likely" registered voters. What role would stratification play in your poll? (b) Voter turnout is different for presidential election cycles (2012, 2016, 2020, and so on) versus non-presidential election cycles (2014, 2018, 2022, and so on). Explain the role election cycle plays in voter turnout and explain how this may affect your polling model. (c) During the 2014 election, Nate Silver of FiveThirtyEight said  "the pre-election polling averages (not the FiveThirtyEight forecasts, which also account for other factors) in the 10 most competitive Senate races had a 6-percentage point Democratic bias as compared to the votes counted in each state so far." Explain what this means and explain how this would have impacted polling results compared with actual results.

The Law of Large Numbers Suppose you live in a town with two hospitals - one large and the other small. On a given day in one of the hospitals, 60% of the babies who were born were girls. Which one do you think it is? Or, is it impossible to tell. Support your decision.

The Role of Randomness One measure of successful investing is being able to "beat the market". To beat the market in any given year, an investor must earn a rate of return greater than the rate of return of some market basket of stocks, such as the Dow Jones Industrial Average (DJIA) or Standard and Poor's 500 (S&P500). Suppose in any given year, there is a probability of 0.5 that a particular investment advisor beats the market for his/her clients. (a) If there are 5000 investment advisors across the country, how many would be expected to beat the market in any given year? 2500 (b) Assuming beating the market in one year is independent of beating the market in any other year, what is the probability that a randomly selected investment advisor beats the market in two consecutive years? Based on this result, how many of 5000 investment advisors would be expected to beat the market for two consecutive years? 0.25; 1250 (c) Assuming beating the market in one year is independent of beating the market in any other year, what is the probability that a randomly selected investment advisor beats the market in five consecutive years? Based on this result, how many of 5000 investment advisors would be expected to beat the market for five consecutive years? 0.03125; 156.25 (d) Assuming beating the market in one year is independent of beating the market in any other year, what is the probability that a randomly selected investment advisor beats the market in ten consecutive years? Based on this result, how many of 5000 investment advisors would be expected to beat the market for ten consecutive years? 0.00098; 4.9

The Role of Randomness (e) Assume a randomly selected investment advisor can beat the market with probability 0.5 and investment results from year to year are independent. Suppose we randomly select 5000 investment advisors and determine the number x who have beaten the market the past ten years. Explain why this is a binomial experiment (assuming there are tens of thousands of investment advisors in the population) and clearly state what a success represents. (f) Use the results of part (e) to determine the probability of identifying at least six investment advisors who will beat the market for ten consecutive years. Interpret this result. Is it unusual to identify at least six investment advisor who consistently beats the market even though his/her underlying ability to beat the market is 0.5? Explain. 0.3635

The Black Swan Open the data file “Daily Percentage Change in S&P500” in StatCrunch. Draw a histogram of the variable “Percentage Change.” Describe the shape of the distribution. Compute the mean and standard deviation percentage change.

The Black Swan (c) Determine the percentage change value for Use the normal model to determine the proportion of observations that should be less than three standard deviations below the mean. Redraw the histogram with a lower class limit of the first class of -10 and a class width of 0.5. In StatCrunch, there is an option called “Dividers” in the histogram dialogue box. Select the Percent radio button. Click Compute! Double-click on the left-most divider and enter the value found in part (c). What proportion of observations are less than this value? Is this close to the value given by the normal model? Ans: -3.71

The Black Swan (f) In his book The Black Swan, author Nassim Nicholas Taleb warns of using the normal model for financial markets. According to the normal model, what proportion of the days should the percentage change of the S&P 500 be less than -7.0%? Ans: 0.0000000073 (g) The number of days one should expect to wait before observing an event equals 1/p where p is the probability of observing the event. Use the result of part (f) to determine the number of days one should expect to wait before observing a percentage change in the S&P500 of -7.0 or less. Ans: 136,986,301 (h) There were 4023 trading days from 11/7/00 to 11/3/2016. How many of these days had a percentage change in the S&P500 of -7.0 or less? What does this result suggest?

Deciding on an Inferential Method It is well-documented that watching TV, working on a computer, or any other activity involving artificial light can be harmful to sleep patterns. Researchers wanted to determine if the artificial light from e-Readers also disrupted sleep. In the study, 12 young adults were given either an iPad or printed book for four hours before bedtime. The following evening, they switched reading devices. Whether the individual received the iPad or book first was determined randomly. Bedtime was 10 PM and the time to fall asleep was measured each evening. It was found that participants took an average of 10 minutes longer to fall asleep after reading on an iPad. The P-value for the test was 0.009. Source: Anne-Marie Chang, et.al. “Evening Use of Light-Emittting eReaders Negatively Affects Sleep, Circarian Timing, and Next-Morning Alertness” PNAS 2015 112(4) 1232-1277, doi:10.1073/pnas.1418490112. (a) What was the research objective? (b) What is the response variable? Is it quantitative or qualitative? (c) What is the treatment? (d) Is this a designed experiment or observational study? What type? (e) Interpret the P-value.

Deciding on an Inferential Method In clinical trials of the allergy medicine Clarinex (5 mg), 3307 allergy sufferers were randomly assigned to either a Clarinex group or a placebo group. It was reported that 50 out of 1655 individuals in the Clarinex group and 31 out of 1652 individuals in the placebo group experienced dry mouth as a side effect of their respective treatments. Source: www.clarinex.com What type of experimental design is this? What is the response variable? Is it qualitative or quantitative? What is the explanatory variable? How many levels does the treatment have? The clinical trial was double-blind. What does this mean? Why is it important to have a placebo group? Does the sample evidence suggest that a side effect of Clarinex may be dry mouth? Do you think the results are practically significant? Why is a large sample size needed for this study?

Deciding on an Inferential Method For each study, explain which statistical procedure (estimating a single proportion; estimating a single mean; hypothesis test for a single proportion; hypothesis test for a single mean; hypothesis test or estimation of two proportions, dependent or independent; hypothesis test or estimation of two means, dependent or independent) would most likely be used for the research objective given. Assume all model requirements for conducting the appropriate procedure have been satisfied. Researchers measured regular testosterone levels in a random sample of athletes and then measured testosterone levels prior to an athletic event. They wanted to know whether testosterone levels increase prior to athletic events.

Deciding on an inferential method For each study, explain which statistical procedure (estimating a single proportion; estimating a single mean; hypothesis test for a single proportion; hypothesis test for a single mean; hypothesis test or estimation of two proportions, dependent or independent; hypothesis test or estimation of two means, dependent or independent) would most likely be used for the research objective given. Assume all model requirements for conducting the appropriate procedure have been satisfied. Do adult males who take a single aspirin daily experience a lower rate of heart attacks than adult males who do not take aspirin daily?

Deciding on an Inferential Method For each study, explain which statistical procedure (estimating a single proportion; estimating a single mean; hypothesis test for a single proportion; hypothesis test for a single mean; hypothesis test or estimation of two proportions, dependent or independent; hypothesis test or estimation of two means, dependent or independent) would most likely be used for the research objective given. Assume all model requirements for conducting the appropriate procedure have been satisfied. Does turmeric (a antioxidant that can be added to foods) help with depression? Researchers randomly assigned 200 adult women who were clinically depressed to two groups. Group 1 had turmeric added to their regular diet for one week; Group 2 had no additives in their diet. At the end of one week, the change in their scores on the Beck Depression Inventory was compared.

Deciding on an Inferential Method For each study, explain which statistical procedure (estimating a single proportion; estimating a single mean; hypothesis test for a single proportion; hypothesis test for a single mean; hypothesis test or estimation of two proportions, dependent or independent; hypothesis test or estimation of two means, dependent or independent) would most likely be used for the research objective given. Assume all model requirements for conducting the appropriate procedure have been satisfied. While exercising by climbing stairs, is it better to take one stair, or two stairs, at a time? Researchers identified 30 volunteers who were asked to climb stairs for two different 15-minute intervals taking both one stair and two stairs at a time. Whether the volunteer did one stair or two stairs first was determined randomly. The goal of the research was to determine if energy expenditure for each exercise routine was different.

Deciding on an Inferential Method For each study, explain which statistical procedure (estimating a single proportion; estimating a single mean; hypothesis test for a single proportion; hypothesis test for a single mean; hypothesis test or estimation of two proportions, dependent or independent; hypothesis test or estimation of two means, dependent or independent) would most likely be used for the research objective given. Assume all model requirements for conducting the appropriate procedure have been satisfied. What is the typical amount of time 20- to 24-year-old males spend brushing their teeth (each time they brush)?

Deciding on an Inferential Method For each study, explain which statistical procedure (estimating a single proportion; estimating a single mean; hypothesis test for a single proportion; hypothesis test for a single mean; hypothesis test or estimation of two proportions, dependent or independent; hypothesis test or estimation of two means, dependent or independent) would most likely be used for the research objective given. Assume all model requirements for conducting the appropriate procedure have been satisfied. Are a majority of registered voters in favor of a tax increase to reduce the federal debt?