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Chapter 5: sample problems for homework, class or exams
CHAPTER 5: SAMPLE PROBLEMS FOR HOMEWORK, CLASS OR EXAMS These problems are designed to be done without access to a computer, but they may require a calculator.
1. For each scenario below, choose the most likely method of analysis and write the corresponding letter in the blank. #1 two-sample t test
#5 McNemar’s test #6 two-sample test for medians
A. ___ Household incomes are often extremely right-skewed. Based on sample data from two neighborhoods, do typical household incomes differ for the two neighborhoods? B. ___ An investment counselor gives both the husband and the wife in each couple a questionnaire on risk tolerance. Are husbands typically more risk tolerant than the wives? C. ___ Grade point averages are collected for random samples of engineering majors and business majors. Do typical GPAs differ for the two groups? D. ___ Pollution levels, categorized as either Low or Elevated, are measured at a sample of lakes in the Southeast, and also at a sample of lakes in the Northeast. Do the two regions differ in the probability a lake will have an elevated pollution level? E. ___ A sample of lakes in the Southeast have their pollution level measured twice, once in the Spring and once in the Summer. Pollution level is categorized as either Low or Elevated. Does the probability a lake will have an elevated pollution level vary by season? F. ___ Diabetics about to undergo surgery are randomly assigned to one of two insulin protocols. Is one of the protocols more effective at reducing variability in the post-operative blood glucose levels? 2. Eight volunteers participate in an experiment where their reaction time (in milliseconds) is recorded in response to a flashed light. Each volunteer has one reaction time measured using a red light, and again using a blue light. The data is shown below. Give a 95% confidence interval for the difference in mean reaction times for the two light colors. Subject
Red 168 204 125 193 231 221 185 200 Blue 192 206 144 191 252 239 198 214 3. A researcher interviews a sample of a city’s residents to determine their attitudes towards the public school system. Among the 87 respondents who had school-age children, 39 expressed confidence in the school system. Among the 45 who did not have school-age children, 23 expressed confidence. At α = 5%, is there evidence that those who do and those who do not have children differ in their level of confidence in the school system?
Problems 4 through 6 are based on the following scenario.
4. At the end of their 4th grade year, 50 schoolchildren are tested for their reading ability. The children are randomly divided into two groups. One group is given a focused reading list for the summer, the other a broader list. At the beginning of the next school year, the children are tested again. The data for this study is summarized below. Loss is taken as the score at the end of the 4th grade – beginning of the 5th grade. The goal is to see whether one of the lists reduces the mean loss of reading ability that occurs over summer break. Group
a. At the end of the 4th grade, did the groups differ significantly with respect to the mean reading ability? Use α = 5%. b. At the end of the 4th grade, did the groups differ significantly with respect to the variability in their reading ability? Use α = 5%. 5. Refer to the data summarized in Problem 4. a. Did the children in the focused reading list group show a significant change in their mean reading scores during the summer? Use α = 5%. b. Did the children in the broader reading list group show a significant change in their mean reading scores during the summer? Use α = 5%. 6. Refer to the data summarized in Problem 4. a. Is there evidence, at α = 5%, that the groups differ with respect to the mean loss in reading ability over the summer? b. Is there evidence, at α = 5%, that the groups differ with respect to the variability in their loss of reading ability? 7. A company is testing two different monitors to see if they differ with respect to their ability to detect elevated carbon monoxide (CO) levels. The two monitors are mounted side-by-side, and then the pair are exposed to 50 different situations with elevated CO. The table below summarizes the results. Test the null hypothesis that the monitors have equal probabilities of detecting CO, using α = 5%.
8. Volunteers are randomly divided into two groups and given a cognition test. Participants in group A take the exam after a normal nights’ sleep. Volunteers in group B take the exam after going 24 hours without sleep. The scores on the exam seem to be right skewed, so the researchers decide to compare the medians in the groups. The overall median score was 42. In group A, 19 of the 25 participants scored above 42. In group B, 6 of the 25 participants scored above 42. At α = 1%, is there evidence that the medians differ? If so, which group seems to have the higher median? SOLUTIONS 1 A. #6
2. Paired samples. Taking the differences within samples as Blue – Red, the sample mean is 13.625 and the sample standard deviation is 9.180. With confidence 95%, the difference in mean reaction times is between 5.951 and 21.30, where the positive values indicate that Blue has a longer mean reaction time than Red. 3. two-sample test for proportions. Ho: p1 = p2 versus H1: p1 ≠ p2. Z = -0.686, p value = 0.4929 There is no significant evidence that the proportions feeling confident in the school system differ for those who do and do not have children. 4. a. two sample t-test. Using the pooled t-test, t = -0.56 with 48 df and p value 0.5778 There is no significant difference in the mean reading ability of the two groups initially. b. two sample F test. F = 1.381 with 24 and 24 df. P value = 0.4349 There is no significant difference in the variances of the reading ability in the two groups initially. 5. Paired t tests Ho: μD = 0 versus H1: μD ≠ 0 a. within the focused group: t = 13.38, p value = 0, the mean reading loss in the focused group differed significantly from 0. b. within the broader reading group: t = 9.09, p value = 0, the mean reading loss in the broader reading group differed significantly from 0. 6. a. applying the two sample t test to the losses, t = 0.137, p value = 0.8913 There is no significant evidence that the mean loss differs for the two types of reading lists. b. F = 2.09, there is no significant evidence that the variances differ in the two groups. 7 McNemars test. Out of 16 discordant pairs, only 4 were for B does not work/A works. Z = -2.0, p value =0.0455. There is significant evidence that the monitors differ in their effectiveness. Monitor B seems to be more reliable.
8. Compare the proportion of observations exceeding 42 for the two groups. Z = 3.68, p value = 0.0002, there is significant evidence that the medians in the two groups are different. There is apparently a higher median cognition score among people who have a normal night’s sleep.
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