Stats 250 Practice Exam 2 F23 SOLUTIONS

STATS 250 F23 – PRACTICE Exam 2 • Exam 2 is on Monday, December 11 th, starting at 7:30 PM • Exam 2 room assignment is based on lab section number, see Canvas. • IMPORTANT: Bring a Photo ID to the exam. Your Photo ID must be presented upon submission of your exam. Instructions: 1. Write your name and UMID number in the spaces above. 2. Points for each question are indicated in square brackets [ ]. 3. This exam is closed-notes and closed-book. 4. Some questions require that you clearly write your final answer in a designated box. 5. If a question instructs you to “show your work” or “provide an explanation”, no credit will be awarded for giving only the final answer without correct supporting work. 6. No calculators are required, but you are allowed to use one if you would like to. Numerical answers may be expressed as fractions, if you prefer (e.g., an answer of 0.5, 50%, and ½ are equally acceptable). 7. Final answers must be legible and clear. If multiple final answers are given, the worst will be graded. 8. Instructors will not answer any questions about exam content. Answer each question to the best of your ability and show your work. 9. When you are finished with your exam, make sure that your name and UMID number are clearly written on the front page. Walk to the front of the room, show your Photo ID to a member of the instructional team, submit your exam to the folder associated with your last name, and provide your initials to indicate your attendance today. Statement of Academic Honesty. For this exam , I make the following truthful statements : ● I have not received, I have not given, nor will I give or receive, any assistance to another student taking this exam , including discussing the exam with students enrolled in the course. ● I will not plagiarize someone else’s work and turn it in as my own. ● I understand that if I am suspected of academic dishonesty, the incident will be turned over to the Office of the Assistant Dean. ● I understand that acts of academic dishonesty may be penalized to the full extent allowed by the University of Michigan Student Conduct Code, including receiving a failing grade for the course. I recognize that I am responsible for understanding the provisions of the University of Michigan Student Conduct Code as they relate to this academic exercise. Student Signature: ____________________________________ Date: ____________________

STATISTICS HELP CARD Correlation and Regression Parameter Estimators ࠵? ! = ࠵? ࠵? " ࠵? # ࠵? $ = ࠵? ’ − ࠵? ! ࠵?̅ Residuals ࠵? = ࠵? − ࠵? , = ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵? ࠵? − ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵? ࠵? Chi-Square Tests Test for Goodness of Fit Test of Independence Expected Count ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵? = ࠵?࠵? %$ Expected Count ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵? = (࠵?࠵?࠵? ࠵?࠵?࠵?࠵?࠵?)(࠵?࠵?࠵?࠵?࠵?࠵? ࠵?࠵?࠵?࠵?࠵?) ࠵?࠵?࠵?࠵?࠵? ࠵? Test Statistic ࠵? & = ∑ (()*+,-+./+#0+12+.) ! +#0+12+. ࠵?࠵? = ࠵? − 1 Test Statistic ࠵? & = ∑ (()*+,-+./+#0+12+.) ! +#0+12+. ࠵?࠵? = (࠵? − 1)(࠵? − 1) Properties of a Chi-Square Distribution A ࠵? & random variable has mean = df and standard deviation = B2࠵?࠵? One Proportion One Mean Mean Difference Difference in Two Means Parameter p Parameter ࠵? Parameter ࠵? . Parameter ࠵? ! − ࠵? & Statistic ࠵?̂ Statistic ࠵?̅ Statistic ࠵? ̅ Statistic ࠵?̅ ! − ࠵?̅ & Standard Error ࠵?. ࠵?. (࠵?̂) = G ࠵?̂(1 − ࠵?̂) ࠵? Standard Error ࠵?. ࠵?. (࠵?̅) = ࠵? √࠵? Standard Error ࠵?. ࠵?. I࠵? ̅ J = ࠵? . √࠵? Standard Error ࠵?. ࠵?. (࠵?̅ ! − ࠵?̅ & ) = G ࠵? ! & ࠵? ! + ࠵? & & ࠵? & Confidence Interval ࠵?̂ ± ࠵? ∗ ࠵?. ࠵?. (࠵?̂) Confidence Interval ࠵?̅ ± ࠵? ∗ ࠵?. ࠵?. (࠵?̅) Confidence Interval ࠵? ̅ ± ࠵? ∗ ࠵?. ࠵?. I࠵? ̅ J Confidence Interval (࠵?̅ ! − ࠵?̅ & ) ± ࠵? ∗ ࠵?. ࠵?. (࠵?̅ ! − ࠵?̅ & ) Hypothesis Test ࠵? = ࠵?̂ − ࠵? $ N ࠵? $ (1 − ࠵? $ ) ࠵? Hypothesis Test ࠵? = ࠵?̅ − ࠵? ( ࠵?. ࠵?. (࠵?̅) Hypothesis Test ࠵? = ࠵? ̅ − 0 ࠵?. ࠵?. (࠵? ̅ ) Hypothesis Test ࠵? = (࠵?̅ ! − ࠵?̅ & ) − 0 ࠵?. ࠵?. (࠵?̅ ! − ࠵?̅ & ) If the p- value is: Greater than 0.10 (࠵? > ࠵?. ࠵?࠵?) Between 0.05 and 0.10 (࠵?. ࠵?࠵? < ࠵? ≤ ࠵?. ࠵?࠵?) Between 0.01 and 0.05 (࠵?. ࠵?࠵? < ࠵? ≤ ࠵?. ࠵?࠵?) Less than 0.01 (࠵? ≤ ࠵?. ࠵?࠵?) we will say we have: little evidence some evidence strong evidence very strong evidence Est. Effect size Interpretation | ࠵?࠵?࠵?࠵?࠵?࠵? ࠵?࠵?࠵?࠵? ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?| ≤ 0.2 Small 0.2 < |࠵?࠵?࠵?࠵?࠵?࠵? ࠵?࠵?࠵?࠵? ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?| ≤ 0.5 Small-to-moderate 0.5 < |࠵?࠵?࠵?࠵?࠵?࠵? ࠵?࠵?࠵?࠵? ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?| ≤ 0.8 Moderate-to-large |࠵?࠵?࠵?࠵?࠵?࠵? ࠵?࠵?࠵?࠵? ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?| > 0.8 large

1. Better Coffee ~ Some Stats 250 students were having a discussion about whether Starbucks or RoosRoast has better coffee. After conducting a large study, they concluded that a majority of all UM student’s prefer RoosRoast coffee over Starbucks. a. Provide the appropriate statistical notation for the parameter that would be found in the null and alternative hypotheses and define the parameter in context. Statistical Notation:_____ p __________ b. The resulting test statistic is 2.3 and the corresponding p-value is 0.0107. Provide an interpretation of the p-value in context. Assuming the population proportion of all UM Students who prefer RoosRoast coffee over Starbucks is equal to 0.5, the probability of obtaining a z-test statistic of 2.3 or larger is 0.0107. OR Assuming the population proportion of all UM Students who prefer RoosRoast coffee over Starbucks is equal to 0.5, and if we were to repeat the sampling process many times, we would expect to obtain a z-test statistic of 2.3 or larger in about 1.07% of the repetitions. 2. Window Color Study ~ A study investigates whether car window tinting affects drivers’ ability to quickly detect low-contrast targets, potentially impacting road safety. A large random sample of US drivers was selected to participate in the study. Participants were randomly assigned to drive a car with a predetermined degree of tinting on the windows ( TINT : "no window tinting" or "some window tinting") and the INSPECTION TIME (in milliseconds) was recorded. The aim of this study is to understand the average effect of tinting on inspection times for US drivers. to investigate the effect of window tinting on inspection time, on average, for all U.S. drivers. a. For the variable TINT , bubble in the correct role and variable type ROLE o Explanatory variable o Confounding variable o Response variable TYPE o Nominal o Ordinal o Discrete o Continuous b. For the variable INSPECTION TIME , bubble in the correct role and variable type. o Explanatory variable o Confounding variable o Response variable o Nominal o Ordinal o Discrete o Continuous

c. The researcher uses R and provides the following R output to assess if, on average, the inspection time is different among all US drivers with no window tinting (group 1) versus all US drivers with some window tinting (group 2). Bubble in the most plausible 90% confidence interval for μ no tint -μ some tint . o (-13.96, 12.45) o (-12.56, 11.72) o (-14.61, -11.05) o (11.28, 13.94) Provide an explanation of how you arrived at this interval. Because the p-value of 0.01915 is between 1% and 5%, we have strong evidence to support Ha, which suggests that μ 1 -μ 2 ≠0, hence the 95% confidence interval constructed using this same data, would not contain the value 0 and would contain only positive because the test statistic value is positive indicating that ࠵?̅ ! ࠵?࠵? ࠵?࠵?࠵?࠵?࠵?࠵?࠵? ࠵?ℎ࠵?࠵? ࠵?̅ " . d. The researcher conducts three different follow-up studies. Study A Study B Study C Test results ࠵? 2.74 ࠵? 0.007 ࠵? / 0.037 Test results ࠵? 2.15 ࠵? 0.034 ࠵? / 0.312 Test results ࠵? 0.681 ࠵? 0.498 ࠵? / 0.013 Which of the three follow-up studies do you expect was based on the largest overall sample size? Provide a 2-3 sentence justification of your choice. Study A had the most participants, which is why its test statistic is higher compared to other studies. A large number of participants can make it easier to obtain a small p-value. Think of the large sample size like zooming in with a microscope: it helps us see tiny details that we couldn't with the naked eye. In this case, those tiny details are the small effects we're trying to measure. Even though these effects don't make a big difference, the large number of observations gives us the precision to detect them and suggest that our initial assumption (the null hypothesis) might be off and we should support the alternative claim.

4. Pokémon Go players in Ann Arbor ~ A researcher plans to survey a random sample of Pokémon Go players in Ann Arbor to estimate the population proportion who are young. a. The researcher takes a large random sample of Ann Arbor Pokémon Go players and provides a 98% confidence interval that ranges between 60% and 68%. i. Provide an interpretation of the 98% confidence level in context. If we were to repeat this sampling process many times, we expect about 98% of the resulting confidence intervals to contain the unknown population proportion of all Ann Arbor Pokémon Go players who are young. ii. Suppose that, using the same data, we construct a 90% confidence interval instead of the original 98% confidence interval. Bubble in the correct symbol and center of the 90% confidence interval. o ࠵?̂ = 0.63 o ࠵? = 0.63 o ࠵?̂ = 0.64 o ࠵? = 0.64 o ࠵?̂ = 0.65 o ࠵? = 0.65 Bubble in a reasonable value for the 90% margin of error. o 0.04 o 0.028 o 0.044 o 0.05 Bubble in the distribution that was used to find the 90% multiplier. o ࠵?࠵?࠵?࠵?࠵?࠵? Distribution o ࠵? -distribution o ࠵? " distribution o ࠵? distribution 5. Name That Scenario ~ One important aspect of statistics is to understand which statistical methods or procedures are appropriate to use to address the research problem or question of interest. For each description of a research question below, clearly select and write the letter corresponding to the statistical analysis technique most appropriate for addressing that research question. [ Note : you may use each procedure more than once.]

a. A study examined the effect of Intraaortic balloon counterpulsation (IABP) on the length of hospital stay post-treatment in heart attack patients with cardiogenic shock. In the study, 300 patients received IABP, with an average hospital stay of 10 days, while the control group of 298 patients, who did not receive IABP, had an average hospital stay of 12 days. The researchers are investigating whether, on average, IABP influences the length of hospital stay among heart attack patients suffering from cardiogenic shock. Procedure: _________ D __________ PROCEDURES A. One-sample z-test for one population proportion B. One-sample t-test for one population mean C. One-sample t-test for the population mean of differences (paired data) D. Two-sample t-test for the difference in two population means E. ANOVA F. One-sample t-test for the population slope of a regression line G. ࠵? ! test of goodness of fit H. ࠵? ! test of independence b. In 2012, Dr. Meian He of the Harvard School of Public Health published a study about the relationship between blood type and heart disease. When discussing his results, he concluded that a person’s blood type (Type A, B, AB, or O) has an association with their risk of developing heart disease (Yes = heart disease, No = no heart disease). What procedure might he have used to arrive at this conclusion? Procedure: ______ H ____________ c. Birds of many species retain the same breeding partner year after year. In some species, male and female partners migrate separately and spend the winter in different places, often thousands of miles apart. A recent study recorded the spring arrival dates (measured as number of days since March 31 st ) of male and female bird-couples on the breeding grounds in the year after they were observed breeding together. The researchers hope to use the collected data as evidence of a relationship between the arrival dates of coupled male and female birds. Procedure: ________ F __________ d. Japan’s Automobile Manufacturer’s Association provides data on exported vehicles. In 2005, cars, trucks, and buses constituted 86.4%, 12.1%, and 1.5% of exports, respectively. Researchers plan to collect data at random from automobile manufacturers regarding their exports this year to see if these proportions have changed. Procedure: _________ G _________ e. Researchers concerned with waste produced by grocery bags surveyed a random sample of ࠵? = 500 shoppers at a local grocery store and asked them whether they preferred paper bags over plastic ones. They hope to use their results to assess whether customers generally prefer one material over the other. Procedure: ________ A __________ f. Ornithologists (zoologists who specialize in the study of birds) were interested in investigating whether birds that migrate long distances tend to have longer wingspans than similar, nonmigratory species. A recent study compared the wing lengths (in millimeters) of random samples of two subspecies of dark-eyed juncos, only one of which migrates each winter.

7. Living Arrangements ~ A survey was conducted on a random sample of 316 college students. Two of the survey questions were: 1) What is your current living arrangement? 2) What is your current geographical location? The results of these two categorical questions are provided in the table below. Asia/Pacific Europe Africa Americas Total Extended Family 28 (20.25) 8 (16.70) 26 (15.2) 18 (27.85) 80 Parents & Siblings 34 (34.94) 30 (28.82) 28 (26.2) 46 (48.04) 138 Significant Other/ Roommate 10 (13.67) 16 (11.28) 4 (10.25) 24 (18.80) 54 Solo 8 (11.14) 12 (9.20) 2 (8.35) 22 (15.31) 44 Total 80 66 60 110 316 a. Which chi-square test would be appropriate for assessing if there is a significant relationship between Living Arrangements and Geographical Location for the population of all college students? o Chi-square Test of Goodness of Fit o Chi-square Test of Independence b. The researcher is interested in using these results to compute a few proportions; i. Provide an estimate of the population proportion of college students who live by themselves (solo) and reside in Europe. o ࠵? = !" %!& o ࠵? = !" && o ࠵? = !" ’’ o ࠵?̂ = !" %!& o ࠵?̂ = !" && o ࠵?̂ = !" ’’ ii. Provide an estimate of the population proportion of college students in Africa who reside with their extended family. o ࠵? = "& &# o ࠵? = "& %!& o ࠵? = "& (# o ࠵?̂ = "& &# o ࠵?̂ = "& %!& o ࠵?̂ = "& (#

c. Alongside the bolded observed counts, the table above also shows the expected counts in italics. Which cell contributed the largest amount to the overall test statistic ࠵? " = 36.671 ? Bubble only one. o European students living with extended family o European students living with parents and siblings o African students living with extended family o African students living with parents and siblings d. Assuming that there is not a significant relationship between Living Arrangement and Geographical Location for the population of all college students, what is the expected test statistic value? The expected test statistic is the degrees of freedom. For a chi-squared test of independence, the degrees of freedom are (r-1)(c-1) =(4-1)(4-1) =9 8. Tree Growth ~ Trees grow faster due to global warming. Does rapid growth mean that trees will not survive as long? To help address this question with data, the tree rings that are formed as a tree grows were examined for 46 randomly selected dead trees of a particular species of pines. Tree ring width in the first 25 years tells us how fast the tree grows in its younger years. The survival age of trees observed in the data ranged from 47 to 190 years. The variables of interest are: ࠵? = TRW = Tree ring width in the first 25 years measured in millimeters (mm) ࠵? = AGE = Survival age of trees in years Results of the associated regression analysis are shown below: Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 4.84207 0.08015 60.41 < 2e-16 *** AGE -0.0026 0.00067 -3.92 0.00031 *** a. Use the results to report the estimated OLS regression equation summarizing the relationship between the observed variables ࠵? and ࠵? . Use correct notation . EQUATION: ____________ ࠵?࠵?࠵? / = 4.84207 − 0.0026 (࠵?࠵?࠵?) _______________________

d. Based on the data, among all trees represented by the sample, does there appear to be a significant negative linear relationship between tree ring width in the first 25 years and the survival age of the tree? o YES o NO Provide a 2-3 sentence justification of your choice. Please include appropriate numerical support: Since the p -value = 0.00031/2 = 0.000125 is much less 1%, there is very strong support for a significant negative linear relationship between TRW and Age, among the population of trees represented by the sample. e. Consider the following three correct 95% confidence intervals: Interval A : 4.545 ± 0.3545 Interval B : −0.0026 ± 0.00135 Interval C : 4.545 ± 0.0515 These intervals were computed to summarize these regression results. However, we forgot to include their labels. Based on the statement provided, identify the correct interval. a. With 95% confidence, we estimate that, on average, each additional year of a tree’s life results in a decrease in Total Ring Width (TRW) for all trees. Interval:_____ B _____ b. With 95% confidence, we estimate the average Total Ring Width (TRW) for all trees whose survival age is 113 years Interval:____ C ______ c. With 95% confidence, we estimate the Total Ring Width (TRW) for a tree whose survival age is 113 years. Interval:_____ A _____

9. Covid Vaccines ~ When vaccines were first made available, a number of Michiganders were getting their Covid vaccines in Ohio. Let us investigate if the vaccine doses in four Midwest states (Michigan and three bordering states) were distributed according to population size. The table below shows the location of vaccines administered from a random sample of vaccine doses and the population distribution for these four Midwest states. Michigan Ohio Illinois Indiana Total Administered vaccines 136 166 139 84 525 Population distribution for the four Midwest states 24% 29% 31% 16% 100% Using the information in the table, we wish to assess whether, for four Midwest states, the distribution of administered vaccine doses matches the distribution of the population. a. State the appropriate null hypothesis. ࠵? " : _______ H 0 : p Michigan = 0.24, p Ohio = 0.29, p Illinois = 0.31, p Indiana = 0.16 . __________________ b. Bubble in the value that correctly completes the statement below: Assuming the null hypothesis is true, approximately half of the values of the test statistic would be equal to _________________ or lower. o 2.366 o 4.108 o 5.317 c. The test statistic value is 5.501 and the ࠵? -value is 0.1386. What distribution was used to find the ࠵? - value? ࠵? " ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵? ࠵?࠵?࠵?ℎ 3 ࠵?࠵?࠵?࠵?࠵?࠵?࠵? ࠵?࠵? ࠵?࠵?࠵?࠵?࠵?࠵?࠵? d. Select the appropriate conclusion for the test. o Based on the data, we do not have enough evidence to suggest that, for the four Midwestern states, the distribution of administered vaccine doses matches the distribution of the population from the four Midwestern states. o Based on the data, we do not have enough evidence to suggest that, for the four Midwestern states, the distribution of administered vaccine doses does not match the distribution of the population from the four Midwestern states. We are looking for the median value of a ࠵? " (3) . Since a ࠵? " distribution is skewed to the right, the median has to be less then the mean=df=3.