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Stat 1000 Fall 2019 Homework 11 (To be turned in via CourseWeb) 1. M&M Mars Company has varied the mix of colors for M&M's Plain Chocolate Candies over the years. These changes in color blends are the result of consumer preference tests. Most recently, the color distribution is reported to be 13% brown, 14% yellow, 13% red, 20% orange, 24% blue, and 16% green. You open up a 14-ounce bag of M&M's and find: 61 brown, 59 yellow, 49 red, 77 orange, 141 blue, and 88 green. Use a goodness of fit test to examine how well this bag fits the percents stated by the M&M Mars Company. Step 1: Give the expected counts for the different colors: (report to 2 decimal places) Brown Yellow Red Orange Blue Green 61.75 66.5 61.75 95 114 76 Step 2: Find the Chi-Square Statistic (report to 2 decimal places; I recommend doing all of this in one calculation to remove errors from rounding) 15.19 What are the degrees of freedom? 5 2. For the above Chi-square statistic, what is the P-value? a. 0.1 3. Can you reject the null hypothesis? YES NO 4. Does your sample bag of M&M's fit the percentages given by M&M Mars Company? YES NO

5. Students were asked in study whether or not they were harassed in person and whether or not they were harassed online. Here are the data for boys: Harassed Online Harassed in person Yes No Yes 183 154 No 48 578 Use a 2-sample proportion test to see whether there is a difference in the proportions of boys harassed online versus those harassed in person. HINT: Use marginal proportions to determine the proportions. Sample proportion of boys harassed online (report to 2 decimal places): 0.54 Sample proportion of boys harassed in person (report to 2 decimal places): 0.79 Find the z-statistic (report to 2 decimal places): -6.09388

6. Find the p-value for the above. Can you reject the null hypothesis? YES NO 7. Is there evidence that the proportion of boys harassed online is different from the proportion of boys harassed in person? YES NO 8. Now examine the previous example to see whether or not there is a relationship between the two variables. What are the expected counts (report to 2 decimal places)? Harassed Online Harassed in person Yes No Total Yes 80.84 256.16 337 No 150.16 475.84 626 Total 231 732 963 What are the degrees of freedom? 1 9. Calculate the Chi-square and find the p-value. Is there evidence that there is a relationship between the two variables?

YES NO 10. Computer software generated 500 random numbers that should look as if they are from a uniform distribution on the interval 0 to 1. They are categorized into five groups: less than or equal to 0.2 greater than 0.2 and less than or equal to 0.4 greater than 0.4 and less than or equal to 0.6 greater than 0.6 and less than or equal to 0.8 greater than 0.8. The counts in the five groups are, respectively: 114, 92, 108, 101, and 85. The probabilities of the five intervals are all the same. What is the probability? (report to one decimal place): 0.2 What is the expected count for each interval? What is the degree of freedom? 4

100

11. Find the Chi-square statistic and p-value for the previous question. Is there evidence that the sample is different from the random probabilities? YES NO 12. We would like to determine if there is a difference in mean credits taken by students in various years (1, 2, 3, 4, or other) at college. ANOVA was carried out on survey data from several hundred students at a certain university. Analysis of Variance for Credits Source DF SS Year 4 63.84 Error 440 1763.97 Total 444 1827.81 Level 1 2 3 4 Other

N 35 257 102 37 14

Mean 15.286 15.389 15.225 15.108 13.214

Pooled StDev = 2.002 The total sample size is: The number of groups is:

445 5

MS 15.96 4.01

F 3.98

StDev 1.824 1.901 1.934 1.912 3.580

P 0.003

95% CI 14.62, 15.95 15.14, 15.63 14.84, 15.61 14.44, 15.78 12.17, 14.26

13. Refer to question 12. As far as samples are concerned, students in this year averaged the least credits: a. 1 b. 2 c. 3 d. 4 e. Other 14. The size of the above p-value suggests which TWO of the following conclusions? a. There is evidence of a relationship in general between year and number of credits taken. b. There is no compelling evidence of a relationship in general between year and number of credits taken. c. Population mean number of credits is different for at least two of the five year levels. d. Population mean credits taken may be the same for all five year levels. 15. Which one of these would have helped to produce MORE evidence of a difference among credits? a. If similar means and standard deviations had arisen from smaller samples of students b. If the sample means had been closer together. c. If the sampled credits had been spread out less around the five group means. d. None of the above. 16. Many studies have suggested that there is a link between exercise and healthy bones. One study examined the effect of jumping on the bone density of growing rats. There were three treatments: a control with no jumping, a low-jump condition, and a high-jump condition (the jump height was 60 cm). After eight weeks of 10 jumps per day five days a week, the bone density of the rats was measured. Here are the data: Contr ol 611 621 614 593 593 653 600 554 603 569

Low Jump 635 605 638 594 599 632 631 588 607 596

High Jump 650 622 626 626 631 622 643 674 643 650

Make a table giving the sample size, mean, and standard deviation for each group of rats. (YOU CAN USE SOFTWARE SUCH AS MINITAB.)

Control Low Jump High Jump

n (report as integer) 10 10 10

Mean (1 decimal) 601.1 612.5 638.7

s (2 decimals) 27.36 19.33 16.59

17. For the above, is it reasonable to calculate the pooled variance? (can we assume HOV?) YES NO 18. For question 16, what is: DFG: 2 DFE: 29 19. For questions 16-18, find the F-statistic and p-value (OK to use software). Can you reject the null hypothesis? YES NO 20. Use the Bonferroni or another multiple-comparisons procedure to determine which pairs of means differ significantly. Which pair(s) differ significantly? You may select more than one. Control and High Jump Control and Low Jump High Jump and Low Jump None of them 21. Given that BMI = 29.578 - 0.655 PA, where PA is a measure for Physical Activity given by thousands of steps per day: What is the predicted BMI for a woman who averages 9500 steps per day? (report to one decimal place) 23.4 22. Given what you found in the previous question 21: If an observed BMI at x=9.5 were 24.3, what would be the residual? (report to 1 decimal place) 0.9 23. Given that: n =20 y = 28.5 + 1.4x SEb1 = 0.65 What is the test statistic? (report to 2 decimal places) 2.15

24. Given the t-statistic you found in question 23, would you reject the null hypothesis at the 5% level using a two-sided alternative? (you will need to find your p-value!) YES NO

25. Given that: n = 30 y = 30.8 + 2.1x SEb1 = 1.05 What is the test statistic? (report to 2 decimal places) 2 26. Given the t-statistic you found in question 25, would you reject the null hypothesis at the 5% level using a two-sided alternative? (you will need to find your p-value!) YES NO 27. Given that: n = 100 y = 29.3 + 2.1x SEb1 = 1.05 What is the test statistic? (report to 2 decimal places) 2 28. Given the t-statistic you found in question 27, would you reject the null hypothesis at the 5% level using a two-sided alternative? (you will need to find your p-value!) YES NO 29. The average undergraduate GPA for American colleges and universities was estimated based on a sample of institutions that published this information. Here are the data for the public schools in that report: Year 1992 1996 2002 2007 GPA 2.85 2.90 2.97 3.01 Make a scatterplot that shows the increase in GPA over time. Does a linear increase appear reasonable? YES NO 30. For the previous problem 29: Find the equation of the least-squares regression line for predicting GPA from year: GPA = b0 + b1 (year) What is b0? (report to 3 decimal places; remember to include signs if applicable.)

-18.589 31. For the previous problem 29: What is b1? (report to 3 decimal places; remember to include signs if applicable.) 0.011 32. For question 29, find the 95% margin of error for the slope: (report to 3 decimal places) HINT: SEb1 can be found in the software output or can be found by hand. Find the margin of error b1 ± __0.003_____________

33. Kiplinger's "Best Values in Public Colleges" provided a ranking of U.S. public schools based on a combination of various measures of academics and affordability. Let's focus on the relationship between the average debt in dollars at graduation (AveDebt) and the in-state cost per year after need-based aid (InCostAid). Below is simple linear regression output for a sample of 24 colleges (Figure 10.13 from the book). Summary of Fit RSquare 0.298317 RSquare Adj 0.266422 Root Mean Square Error 3665.63 Mean of Response 24004.63 Observations (or Sum Wgts) 24 Analysis of Variance Source DF SS MS F P Model 1 125685610 125685610 9.3532 0.0058 Error 22 295629950 13437725 Total 23 421315560 Parameter Estimates Term Estimate Std Error t Ratio Prob > |t| Intercept 14179.871 3298.484 4.30 0.0003 InCostAid 0.7768753 0.254022 3.06 0.0058 Find the equation of the line: AveDebt = b0 + b1 (InCostAid). What is b0? (report to 3 decimal places) 14179.871 What is b1? (report to 3 decimal places) 0.777 34. Following the previous problem 33: Find the residual for the University of North Florida if the instate cost is $11,421 and the average debt is $17,617. (report as an integer)

-5436 35. For problem 33, find the 95% margin of error for the slope. (Report to 3 decimal places; HINT: SEb1 can be found in the output table.) b1 ± ____0.527____________

36. How are returns on common stocks in overseas markets related to returns in the US markets? Consider measuring US returns by the annual rate of return on the Standard & Poor's 500 stock index and overseas returns by the annual rate of return on the Morgan Stanley EAFE index. Both are recorded in percents. We will find the linear relationship between the EAFE returns and the S&P 500 returns for the years 1989 to 2014. Here is the Minitab output for this regression: The regression equation is: EAFE = -3.19 + 0.813 S&P Analysis of Variance Source DF SS (1 decimal MS (1 decimal F (2 decimal place) place) places) Regression 1 5552.9 5552.9 29.45 Error 24 4525.0 188.5 Total 25 10077.9 Fill in the missing information above. 37. For question 36, Is there evidence of a relationship between the EAFE returns and the S&P 500 returns? YES NO 38. For question 36, what is R2? (report to 2 decimal places) 0.55 39. Use the information and general form of the ANOVA table for multiple regression from class or from page 613 of the book to complete the table, find the F-statistic, and find R2. Source DF SS MS F Model 4 70 17.5 1.1178 Error 29 454 15.65 Total 33 524 F=1.1178 R2= 0.1336 40. For question 39, what percentage of the variation in the response variable is explained by the explanatory variables? (report as %) – decimals? 13.36%

41. What is the range for the p-value for question 39? a. >0.100 b. 0.050 to 0.100 c. 0.025 to 0.050 d. 0.010 to 0.025 e. 0.001 to 0.010 f....