1 CHAPTER 1 Section 1.1 1. a. Houston Chronicle , Des Moines Register, Chicago Tribune , Washington Post b. Capital One, Campbell Soup, Merrill Lynch, Pulitzer c. Bill Jasper, Kay Reinke, Helen Ford, David Menendez d. 1.78, 2.44, 3.50, 3.04 2. a. 29.1 yd., 28.3 yd., 24.7 yd., 31.0 yd. b. 432, 196, 184, 321 c. 2.1, 4.0, 3.2, 6.3 d. 0.07 g, 1.58 g, 7.1 g, 27.2 g 3. a. In a sample of 100 phones, what are the chances that more than 20 need service while under warranty? What are the chances than none need service while still under warranty? b. What proportion of all phones of this brand and model will need service within the warranty period? 4. Concrete populations include : all living U.S. Citizens, all mutual funds marketed in the U.S., all books published in 2019 . For the population of all mutual funds, possible questions include • Probability : In a sample of 5 mutual funds, what is the chance that all 5 have rates of return which exceeded 10% last year? • Statistical inference : If previous year rates -of-return for 5 mutual funds were 9. 6, 14.5, 8.3, 9.9 and 10.2, can we conclude that the average rate for all funds was below 10%? Hypothetical populations include: all grade point averages for University of California undergraduates during the next academic year, page lengths for all boo ks published during the next calendar year, batting averages for all major league players during the next baseball season. For the page length population, possible questions include • Probability: In a sample of 10 books to be published next year, how likely is it that the average number of pages for the 10 is between 200 and 250? • Statistical inference: If the sample average number of pages for 10 books is 227, can we be highly confident that the average for all books is between 200 and 245? SOLUTION MANUAL for Modern Mathematical Statistics with Applications 3rd Edition by Jay L. Devore Chapter 1: Overview and Descriptive Statistics 2 5. a. Two variables (at least) were recorded: skin color and hourly wages. b. Skin color is categorical (with four categories), while hourly wages is quantitative (units: $/hr). 6. a. Two variables were recorded: brand and percent of polyunsaturated fat. b. Brand is categorical (with as many categories as there were brands in the sample), while polyunsaturated fat is quantitative (units: percent). c. Based on the sample data, can we conclude that the mean percent of polyunsaturated fat in all margarine containers exceeds 20%? Can we conclude that the population mean polyunsaturated fat percent differs by brand? What proportion of all margarine containers have a polyunsaturated fat percent between 15% and 25%? d. Since this only describes observations in the sample and does not draw a conclusion about all Parkay margarine containers, this is an example of descriptive statistics. 7. a. categorical b. quantitative c. categorical d. categorical e. categorical 8. a. categorical b. quantitative c. quantitative d. categorical e. quantitative f. quantitative 9. a. No, the relevant conceptual population is all scores of all students who participate in the SI in conjunction with this particular statistics course. b. The advantage to randomly assigning students to the two groups is that the two groups should then be fairly comparable before the study. If the two groups perform differently in the class, we can reasonably attribute this to the treatments (SI and control). If it were left to students to c hoose, stronger or more dedicated students might gravitate toward SI, confounding the results. c. If all students were put in the treatment group there would be no results with which to compare the treatments. 10. One could take a simple random sample of students from all students in the California State University system and ask each student in the sample to report the distance from their hometown to campus. Alternatively, the sample could be generated by taking a stratified random sample by taking a simple random sample from each of the 23 campuses and again asking each student in the sample to report the distance from their hometown to campus. Certain problems might arise with self -reporting of distances, such a s recording error or poor recall. This study is enumerative because there exists a finite, identifiable population of objects from which to sample. Chapter 1: Overview and Descriptive Statistics 3 11. One could generate a simple random sample of all single -family homes in the city or a stratified random sample by taking a simple random sample from each of the 10 district neighborhoods. From each of the homes in the sample the necessary variables would be collected. This would be an enumerative study because there exists a finite, identifiable population of objects from which to sample. 12. a. Number observations equal 2 x 2 x 2 = 8 b. This could be called a conceptual population because the data would be collected on an existing process. There is no sampling frame. 13. a. There could be several explanations for t he variability of the measurements. Among them could be measuring error, (due to mechanical or technical changes across measurements), recording error, differences in weather conditions at time of measurements, etc. b. This could be called a conceptual beca use there is no sampling frame. Section 1.2 14. a. Truncate each observation at the decimal point and use two rows for each tens place. So, the first row will represent values 30 -34, the next row 35 -39, then 40 -44, and so on. 3L 444 Stem: tens digit 3H 66 Leaf: ones digit 4L 123 4H 5777889 5L 223 5H 6 b. The distribution shows some slight asymmetry, extending further toward lower values (from a “center” of about 47) than toward higher values. We might describe this as a (somewhat) negatively -skewed distribution. c. No, none of the data values in the stem -and-leaf display appear to be outliers. d. Ten of the 19 times exceed 45 minutes, for a proportion of 10/19 ≈ .526. Chapter 1: Overview and Descriptive Statistics 4 15. 6L 034 6H 667899 7L 00122244 7H Stem=Tens 8L 001111122344 Leaf=Ones 8H 5557899 9L 03 9H 58 This display brings out the gap in the data: There are no scores in the high 70s. 16. The accompanying stem -and-leaf display divides each tens place into five groups: 0 -1, 2-3, 4-5, 6-7, 8-9. The graph suggests that noise levels followed a positively -skewed distribution, ranging from about 55 to 83 dBA. There are a few gaps in the distribution, and the three 83.0 dBA observations might be high outliers. 5 5555555 5 66666666777777777 Stem: Tens digit 5 888999 Leaf: Ones digit 6 6 22333333 6 4445555555 6 7777 6 88889 7 00111 7 3333 7 4444 7 7 9999 8 8 333 17. a. The stem -and-leaf display appears at the top of the next page. b. Arguably, a representative crack depth might be around 9- 10 μm. c. This is somewhat subjective, but the display appears quite spread out. d. No, the distribution is certainly not symmetric. Rather, crack depths appear to be strongly positively skewed. e. Yes: All of the values 66.5, 76.1, and 81.1 μm appear to be high outliers. (Using an outlier convention described later in the chapter, even the values in the 50s would be considered outliers!)
