MONTGOMERY COLLEGE - TAKOMA PARK CAMPUS MATH 117 - ELEMENTS OF STATISTICS FINAL EXAMINATION Given Date: 08/08/2016 at 6PM EST Prof: Dr. J. Wulu Due Date: 8/10/2016 at 11:59PM EST Max Point: 500 Submit pdf file via MyMC Class Blackboard email to the course professor only! Name______________________________ Date______________________ Score_______ Instructions: This is a Take-Home Exam. You may refer to your text and other course materials as you work on the exam, and you may use a calculator or a computer. Record your answers and work in this document. This examination has three parts. Observe points assigned to each part. Answer all questions (or problems). Make sure your answers are as complete as possible. Show all of your work and reasoning. In particular, when there are calculations involved, you must show how you come up with your answers with critical work and/or necessary tables. Answers that come straight from programs or software packages will not be accepted. If you need to use software (for example, MS Excel, and StatKey) and/or online or hand-held calculators to aid in your calculation, please cite the source and explain how you get the results. When requested, show all work and write all answers in the spaces allotted on the following pages. You may hand-write your work and submit you completed work in class on time. In either case, show work neatly and correctly, following standard statistical conventions. Each step should follow clearly and completely from the previous step. If necessary, you may attach extra pages. You must complete the test individually. Neither collaboration nor consultation with others is allowed. Your test will receive a zero grade unless you complete the following honor statement. Please sign your name below the following honor statement: I promise that I did not discuss any aspect of this test with anyone other than my instructor. I further promise that I neither gave nor received any unauthorized assistance on this test, and that the work presented herein is entirely my own. Signed Name _____________________ Date___________________ Page 1 of 17 PART I: Choose and circle the correct/best/appropriate options (Multiple-Choice Questions). (160 points--5 pts each) 1-4: Scores on an exam (out of 100 points) given in a large introductory statistics course are displayed in the provided histogram. 1. Which best describes the shape of the distribution of exam scores? A) Approximately symmetric B) Skewed to the left C) Skewed to the right D) None 2. Based on the histogram, which value is likely the mean exam score? A) 82 B) 88 C) 76 D) 92 3. Based on the histogram of exam scores, which value is likely the median exam score? A) 92 B) 88 C) 82 D) 72 4. Based on the histogram, the standard deviation of the exam scores is likely closest to which of these values? A) 0.5 B) 10 C) 5 D) 1 Page 2 of 17 5. A sample of size 46 with a mean of 13.6 is to be used to construct a confidence interval for . A bootstrap distribution based on 1,000 samples is created. Where will the bootstrap distribution be centered? A) 46 B) 13.6 C) D) 1,000 6. The average SAT-Critical Reading score for college bound students taking the exam in the 2009-2010 academic year was 501. A highly selective university wants to know if their 2010 incoming class had an average SAT-Critical Reading score that was higher than the national average. Which of the following possible samples provides the most evidence for this claim? A) Sample A B) Sample B C) Sample C D) Sample D 7. Using a significance level of 5%, the appropriate conclusion for a test with a p-value of 0.0421 would be: A) Reject Ho B) Do not Reject HO C) Indeterminate 8. It is believed that about 37% of college students binge drink (5 or more drinks for men, and 4 or more drinks for women, in two hours). Administrators at a small university of 6,000 students want to do a study to determine if the proportion of their students who binge drink differs from 37%. They select a sample of 98 students enrolled at the university to survey about their drinking behavior. When generating the randomization distribution for this test, how large should each individual randomization sample be? A) B) C) D) 98 because that is the size of the original sample 1,000 to get an accurate randomization distribution 6,000 because that is the size of the university 2,220 because that is 37% of the students at the university Page 3 of 17 9. Which of the following samples provides the most evidence that the amount of time spent studying for an exam and the grade on the exam are positively correlated? A) Sample A B) Sample B C) Sample C D) Sample D 10-11: A study recently described in Attention, Perception, and Psychophysics investigated the impacts of mult-tasking on people who play video games and those who don't. Participants in the study were asked to perform three visually demanding tasks with (dual-task) and without (singletask) answering unrelated questions over the phone. One of the tasks involved tracking multiple circles moving around on a computer monitor. At the 5% significance level, the authors of the study concluded "tracking accuracy was significantly worse in the dual-task condition" for both people who play video games and those who do not. 10. What conclusion would the authors have made at the 10% significance level? A) Tracking accuracy was significantly worse in the dual-task condition. B) Tracking accuracy was not significantly worse in the dual-task condition. C) Not enough information 11. What conclusion would the authors have made at the 1% significance level? A) Tracking accuracy was significantly worse in the dual-task condition. B) Tracking accuracy was not significantly worse in the dual-task condition. C) Not enough information Page 4 of 17 12-15: Find the z* values based on a standard normal distribution for each of the following. 12. An 86% confidence interval for a proportion A) 1.080 B) 1.476 C) 0.994 D) 1.96 13. An 88% confidence interval for a correlation. A) 2.575 B) 1.175 C) 2.326 D) 1.555 14. A 78% confidence interval for a mean. A) 0.772 B) 1.227 C) 1.514 D) 1.126 15. A 66% confidence interval for a slope. A) 0.954 B) 0.412 C) 0.754 D) 1.016 16-18: A set of hypotheses, some information from one or more samples, and a standard error from a randomization distribution are provided. Find the value of the standardized z-test statistic. 16. Test Ho: p = 0.75 versus Ha: p > 0.75 when the sample has n = 20, p = 0.78, and SE = 0.015. A) 0.03 B) 2 C) -2 D) 1.5 17. Test H o : = 26 versus H a : 26 when the sample has n = 75, x = 25.5, s = 5.4, and SE = 0.6. A) -1.5 B) 0.278 C) 2.5 D) -0.833 18. Test H o : 1 = 2 and H a : 1 2 when the samples have n1 = 35, x1 = 78, s1 = 15, n2 = 40, x2 = 72, and s2 = 12. The standard error of x1 x2 from the randomization distribution is 3.2. A) 1.875 B) 6 C) 0.4 D) 0.5 19-21: Find the p-value based on a standard normal distribution for the standardized test statistic and provided alternative hypothesis. 19. z = -1.86 for H a : p < 0.5 A) 0.031 B) 0.969 C) 0.062 D) 0.937 Page 5 of 17 20. z = 2.36 for H a : > 86 A) 0.982 B) 0.0182 C) 0.991 D) 0.0091 21. z = 1.75 for H a : p1 p2 A) 0.960 B) 0.040 C) 0.080 D) 0.920 22-26: Computer output from a regression analysis is provided. The regression equation is Y = 72.9 - 0.519 X Predictor Constant X Coef 72.909 -0.5195 SE Coef 2.037 0.1946 T 35.79 -2.67 P 0.000 0.008 22. What is the sample slope for this model? A) 72.909 B) 2.037 C) -0.5195 D) 0.1946 23. What is the sample intercept for this model? A) 72.909 B) 2.037 C) -0.5195 D) 0.1946 24. What is the standard error of the sample slope? A) 72.909 B) 2.037 C) -0.519 D) 0.1946 25. What is the p-value for testing if the slope in the population is different from zero? D) 0.008 A) 0.5195 B) 0.1946 C) p < 0.001 26. The sample size in this situation is n = 157. What are the degrees of freedom for constructing a confidence interval for, or performing a test about, the population slope? A) 157 B) 156 C) 155 D) 153 27-29: Consider taking random samples of size 50 from Population A with proportion 0.45 and random samples of size 40 from Population B with proportion 0.38. 27. Find the mean of the distribution of differences in sample proportions, p A p B . A) 0.45 B) 0.38 C) 0.07 D) -0.07 28. Find the standard error of the distribution of differences in sample proportions, p A p B . A) 0.0108 B) 0.0704 C) 0.0768 D) 0.1041 Page 6 of 17 29. Are the sample sizes for both groups large enough for the Central Limit Theorem to apply so that the differences in sample proportions follow a normal distribution? A) Yes B) No 30-32: Observed counts from a sample are provided in the following table. The expected counts from a null hypothesis are given in parentheses. Category Observed (Expected) A 42 (45.33) B 38 (45.33) C 56 (45.33) 30. What is the 2 - test statistic? A) 3.941 B) 3.711 C) 4.315 D) 2.983 31. How many degrees of freedom are there? A) 1 B) 2 C) 3 D) 4 32. Based on the expected counts, which of the following is most likely the null hypothesis? A) H o : p A = pB = pC = 1 C) H o : p A = 0.2, pB = 0.4, pC = 0.4 3 B) H o : p A = 0.25, pB = 0.25, pC = 0.5 D) H o : p A = 0.5, pB = 0.2, pC = 0.3 PART II: Provide relatively short but concise answers. (150 points--6 pts each) 1-4: State whether the data are best described as a population or a sample. 1. A student purchases a bag of Milk Chocolate M&M's from the vending machine. The makers of M&M's state that when they package their candies they thoroughly mix the colored candies together and randomly put them into packages _____________________________________________ 2. A professor wants to schedule a review session for an exam. He asks all students enrolled in the course their preferred time, and they all respond. _____________________________________________ Page 7 of 17 3. A researcher has identified a beach with a substantial number of driftwood logs. She randomly chooses 30 logs and takes core samples from those logs. _____________________________________________ 4. A football fan recorded the number of rushing yards for all NFL running backs who played last season. _____________________________________________ 5. In elementary school (Grades 1 through 6) there is a strong association between a child's height and reading ability. What is a possible confounding variable that would help explain this relationship? Explain briefly. _____________________________________________ _____________________________________________ 6-9: A university's Admissions staff sends one of four different representatives to work at college fairs. A study was conducted to evaluate the relative effectiveness of the four representatives. For each college fair over the course of the year, the number of inquiries from students, the type of fair (large or small), the representative who worked at that fair, and the percent of inquiries that resulted in applications were recorded. It was found that one of the representatives was far more effective at getting lots of inquiries. 6. What are the cases in this study? _____________________________________________ 7. What are the variables recorded in this study? List them and identify each as either categorical or quantitative. _____________________________________________ 8. Is this an observational study or an experiment? Briefly justify your answer. _____________________________________________ Page 8 of 17 9. Can we conclude that sending the most effective representative to more college fairs will increase the number of inquiries from those college fairs? Why or why not? _____________________________________________ _____________________________________________ 10-16: Trying to determine the number of students to accept is a tricky task for universities. The Admissions staff at a small private college wants to use data from the past few years to predict the number of students enrolling in the university from those who are accepted by the university. The data are provided in the following table. Number Accepted 2,440 2,800 2,720 2,360 2,660 2,620 Number Enrolled 611 708 637 584 614 625 10. What is the explanatory (X) variable? _____________________________________________ 11. What is the response (Y) variable? _____________________________________________ 12. Find the correlation between the number of students accepted and enrolled. Use two decimal places in your answer. _____________________________________________ 13. Find the least squares regression line for predicting the number enrolled from the number accepted. _____________________________________________ 14. Interpret the slope in context. _____________________________________________ Page 9 of 17 15. Interpret the intercept of the line in context. Does the interpretation make sense? _____________________________________________ _____________________________________________ 16. Suppose Admissions has announced that 2,575 students have been accepted this year. Use your regression equation to predict the number of students that will enroll. _____________________________________________ _____________________________________________ 17- 20: In an August 2012 Gallup survey of 1,012 randomly selected U.S. adults (age 18 and over), 53% said that they were dissatisfied with the quality of education students receive in kindergarten through grade 12. They also report that the "margin of sampling error is plus or minus 4%." 17. What is the population of interest? _____________________________________________ 18. What is the sample being used? _____________________________________________ 19. What is the population parameter of interest, and what is the correct notation for this parameter? What is the relevant statistic? _____________________________________________ _____________________________________________ 20. Find an interval estimate for the parameter of interest. Interpret it in terms of dissatisfaction in the quality of education students receive. Use two decimal places in your answer. _____________________________________________ _____________________________________________ Page 10 of 17 21-25: An Internet provider contacts a random sample of 300 customers and asks how many hours per week the customers use the Internet. It found the average amount of time spent on the Internet per week to be about 7.2 hours. 21. Define the parameter of interest, using the proper notation. _____________________________________________ 22. Use the information from the sample to give the best estimate of the population parameter. _____________________________________________ 23. Describe how to use the data to select one bootstrap sample. What statistic is recorded from this sample? _____________________________________________ _____________________________________________ 24. The standard error is about 0.458. Find and interpret a 95% confidence interval for the parameter. Round the margin of error to two decimal places. _____________________________________________ _____________________________________________ 25. Percentiles of the bootstrap distribution are provided. Use the percentiles to report a 95% confidence interval for the parameter. 1% 2.5% 5% 10% 25% 50% 75% 90% 95% 97.5% 99% 6.174 6.322 6.438 6.593 6.866 7.17 7.481 7.78 7.947 8.082 8.304 _____________________________________________ _____________________________________________ Page 11 of 17 PART III: Solve each of the following problems. (190 points --10 pts each) 1-4: A sample of 148 college students at a large university reports getting an average of 6.85 hours of sleep last night with a standard deviation of 2.12 hours. 1.Verify that it is reasonable to use the t-distribution to construct a confidence interval for the average amount of sleep students at this university got last night. ________________________________________ 2. Construct a 98% confidence interval for the average amount of sleep students at this university got last night. Use two decimal places in your margin of error. ________________________________________ 3. Provide an interpretation of your interval in the context of this data situation. ________________________________________ 4. Suppose you want to conduct a similar study at your university. Assuming that the standard deviation of this sample is a reasonable estimate of the standard deviation of sleep time at your university, how many students do you need to survey to estimate the mean sleep time of students at your university with 95% confidence and a margin of error of 0.5 hours? ________________________________________ 5-10: Final grades in Professor Albert's large calculus class are approximately normally distributed with a mean of 76 (%) and standard deviation of 8 (%). 5. Draw a sketch of this normal distribution and label at least three points on the horizontal axis. ________________________________________ Page 12 of 17 6. In Professor Albert's course, students who earn less than a 60% in the class are assigned a failing grade (F). What proportion of the students earned F's? Report your answer with three decimal places. ________________________________________ 7. In Professor Albert's course, students who earn above a 94% are assigned an "A". What proportion of students earned A's? Report your answer with three decimal places. ________________________________________ 8. What proportion of students earn between an 82% and 88% in this class? Report your answer with three decimal places. ________________________________________ 9. What is the 25th percentile in this course? Report your answer with one decimal place. ________________________________________ 10. The top 30% of students earned scores above what value? Report your answer with one decimal place. ________________________________________ 11. Are all colors equally likely for Dark Chocolate M&M's? Data collected from a randomly selected bag of Dark Chocolate M&M's are provided. Blue Brown Green Orange Red Yellow 77 74 57 81 62 84 Test, at the 5% level, if this sample provides evidence that not all colors are equally likely for Dark Chocolate M&M's. Include all details of the test. (Use the p-value approach - Show 5 steps) Step 1: (Hypotheses) Step 2: (Standardized Test Statistic) Step 3: (P-value) Step 4: (Decision) Step 5: (Conclusion) Page 13 of 17 12-13: Penalties in ice hockey occur when a player breaks one of the rules of the game. In most cases, when a penalty occurs, the offending player is placed in the penalty box (the length of time spent in the penalty box depends on the severity of the penalty), and the team has to play with fewer people on the ice, which can result in an advantage for the opposing team. The number of penalties per game for several randomly selected games are displayed for three college men's ice hockey teams. Team A B C Overall 9 7 3 Penalties 9 5 11 3 5 1 7 2 4 9 5 8 n 5 5 5 15 x 8.6 4.2 4.8 5.87 s 2.191 2.280 2.588 2.973 12. Use the summary information to compute the three sums of squares needed for using ANOVA to test for a difference in mean number of penalties among these three teams. Round each to two decimal places. Between Groups = SSG = _____________________ Within Groups = SSE = ______________________ Total = SSTotal = _________________________ 13. Construct the ANOVA table and test, at the 5% significance level, for a difference in mean number of penalties among these three hockey teams. Use two decimal places when rounding decimal values. Include the details of your test. Source df SS Groups Error Total (Use the p-value approach - Show 5 steps) Step 1: (Hypotheses) MS F Step 2: (Standardized Test Statistic) Step 3: (P-value) Step 4: (Decision) Step 5: (Conclusion) Page 14 of 17 14-16: Students in a large lecture class want to know who has, on average, more Facebook friends, male or female students. The data for the students are displayed in the provided dotplots and summary statistics are available in the provided table. Gender Male Female n 68 91 x 678.9 616.1 s 317.6 330.6 14. Is it reasonable to use a t-distribution for inference about the difference in mean number of Facebook friends for male and female students at this university? ________________________________________ ________________________________________ 15. Construct a 95% confidence interval for the difference in mean number of Facebook friends for male and female students at this university. Use two decimal places in your margin of error. ________________________________________ 16. Test, at the 5% level, if this sample provides evidence of a significant difference in the mean number of Facebook friends for male and female students at this university. Include all of the details of the test. (Use the p-value approach - Show 5 steps) Step 1: (Hypotheses) Step 2: (Standardized Test Statistic) Step 3: (P-value) Step 4: (Decision) Step 5: (Conclusion) Page 15 of 17 17-19: "Black Friday," which occurs annually the day after Thanksgiving, is one of the biggest shopping days of the year. During the 2012 holiday season, many stores created controversy by starting their mega-sales on Thanksgiving itself. In a random sample of 25 individuals who shopped during the Black Friday four-day weekend (Thursday Sunday), the average amount spent was $399.40 with standard deviation $171.10. The data are displayed in the provided dotplot. 17. Construct and interpret a 95% confidence interval for the average amount spent by individuals who shopped over the 2012 Black Friday weekend. Use two decimal places in your margin of error. ________________________________________ ________________________________________ 18. Suppose we know that in a random sample of n = 22 individuals who shopped over Black Friday weekend in 2011 the average amount spent was $381.30 with standard deviation $119.80. Construct and interpret a 95% confidence interval for the difference in the mean amount spent between the 2012 and 2011 Black Friday weekends. Round the margin of error to two decimal places. Recall that for the 2012 sample of 25 individuals, the average amount spent was $399.40 with standard deviation $171.10. Dotplots of both samples are provided. _______________________________________ ________________________________________ Page 16 of 17 19. Suppose we know that in a random sample of n = 22 individuals who shopped over Black Friday weekend in 2011 the average amount spent was $381.30 with standard deviation $119.80. Recall that for the 2012 sample of 25 individuals, the average amount spent was $399.40 with standard deviation $171.10. Dotplots of both samples are provided. Test, at the 5% level, if the samples provide evidence that Black Friday shoppers spent more, on average, in 2012 than they did in 2011. Include all of the details of the test. (Use the p-value approach - Show 5 steps) Step 1: (Hypotheses) Step 2: (Standardized Test Statistic) Step 3: (P-value) Step 4: (Decision) Step 5: (Conclusion) Page 17 of 17