CHAPTER 13SOLUTIONS AND MINI-PROJECT NOTES
CHAPTER 13
STATISTICAL SIGNIFICANCE FOR 2×2 TABLES
EXERCISE SOLUTIONS
13.1A large sample is much more likely to result in a statistically significant relationship. The more data available, the easier it is to rule out chance as an explanation for a relationship observed in the sample.
13.2They are equivalent; the probability is usually set at 0.05.
13.3a.Yes, there is definitely a relationship in the sample. If there weren’t, the chi-square statistic would be 0 and the p-value would be 1.0.
b.There isn’t necessarily a relationship in the population. Using the usual significance level of 5%, in about 5% of tests where there is no relationship in the population, the sample will produce a statistically significant relationship.
c.Yes, it is likely that there is a relationship in the population.
13.4Null and alternative hypotheses are never about samples, they are always about populations. We know whether there is a relationship in the sample because we have the data. We don’t have to make hypotheses about samples.
13.5They are expected if the null hypothesis is true.
13.6a.Yes.
b.No.
c.No.
d.Yes.
e.No.
f.Yes.
13.7a.No.
b.No, we do not talk about rejecting the alternative hypothesis in any case.
c.No, the correct terminology is that we fail to reject the null hypothesis.
d.No.
e.Yes.
f.No.
13.8a.No, 1.42 < 3.84.
b.Yes, 4.5 > 3.84.
c.Yes, 0.01 < 0.05.
d.No, 0.15 > 0.05.
13.9a.The population is all registered voters over age 25. The two variables and their categories are (1)Graduated from college (yes, no); (2)Voted in the last presidential election (yes, no).
b.The population is people who were married during the 1970s and 1980s. The two variables and their categories are (1)Smoker (yes, no) and (2)Got divorced (yes, no).
c.The population is midsize cities in the United States. The two variables and their categories are (1)The city’s median family income compared to median family income for the state (higher, lower); (2)One of the 30 busiest airports in the country is within 50 miles of the city (yes, no).
13.10a.Null hypothesis is that there is no relationship between having graduated from college and voting in the last presidential election for the population of registered voters over age 25. The alternative hypothesis is that there is a relationship between having graduated from college and voting in the last presidential election for voters over age 25.
b.The null hypothesis is that there is no relationship between smoking and getting divorced for people who were married during the 1970s and 1980s. The alternative hypothesis is that there is a relationship.
c.The null hypothesis is that median family income for mid-sized cities does not depend on whether there is a major airport within 50 miles. The alternative hypothesis is that median family income does depend on whether there is a major airport within 50 miles.
13.11a.Null hypothesis: There is no relationship between sex and preferred candidate for the population of voters. Alternative hypothesis: There is a relationship between sex and preferred candidate for the population of voters.
b.The expected counts are
(450)(500)/1000 = 225 / (450)(500)/1000 = 225(550)(500)/1000 = 275 / (550)(500)/1000 = 275
c.Half of the people asked said they would vote for Candidate A and half for Candidate B. Therefore, if there is no relationship between sex and preferred candidate, we would expect half of the males to vote for each candidate and half of the females to vote for each candidate. Thus, we would expect 225 males and 275 females to vote for each candidate.
d.Chi-square statistic = = 10.1.
e.Reject the null hypothesis because 10.1 > 3.84. There appears to be a relationship between sex and preferred candidate in the population.
13.12The hypotheses are given on page 246. The expected counts and chi-square statistic are shown in the box below. (These were computed using the statistical software Minitab, but could have been hand-calculated.)
Expected counts are printed below observed countsHeart attack?
Yes No Total
Aspirin 104 10933 11037
146.52 10890.48
Placebo 189 10845 11034
146.48 10887.52
Total 293 21778 22071
Chi-Sq = 12.339 + 0.166 +
12.343 + 0.166 = 25.014
The chi-square statistic is 25.014, much larger than 3.84, so the relationship is statistically significant. We can conclude that there is a relationship between taking aspirin and risk of a heart attack in the population.
13.13Because the chi-squared statistic of 11.967 exceeds the required 3.84, we can say that there is a statistically significant relationship between age group and reportedly seeing a ghost.
13.14Follow the steps in the box on page 249 to compute the chi-square statistic.
(1)Compute the expected counts:
Lung Cancer / No Lung Cancer / TotalBird / (239)(199)/668 = 71.2 / 199 − 71.2 = 127.8 / 199
No Bird / 239 − 71.2 = 167.8 / 469 − 167.8 = 301.2 / 469
Total / 239 / 429 / 668
(2) and (3)Compare observed and expected numbers and compute the chi-squared statistic:
(98 − 71.2)2/71.2 + (101 − 127.8)2/127.8 + (141 − 167.8)2/167.8 + (328 − 301.2)2/301.2 = 10.09 + 5.62 + 4.28 + 2.39 = 22.37
Because the chi-squared statistic is (much) larger than 3.84, the relationship is statistically significant. There is indeed a relationship between bird ownership and lung cancer that is not likely to be due to chance.
13.15a.0.28 or 28% for males, 0.23 or 23% for females, so the proportion of males who helped is higher.
b.The chi-squared statistic is 8.64 so the relationship is statistically significant. Researchers have found that males are more likely than females to help someone who has dropped a handful of pencils on an elevator, and that the results observed in the study are strong enough to generalize them to the population.
c.No, the result would not have been statistically significant because the chi-squared statistic would be about 0.86. When conducting research to determine the existence of a weak relationship it is important to take a large enough sample.
13.16The data provide overwhelming evidence that there is a relationship between ethnic group and whether or not someone is laid off. The results from this sample are so strong that we can clearly conclude that there is a relationship in the population of such workers.
13.17a.Here is the table:
Application approved / Application not approved / TotalBlack / 30 / 10 / 40
White / 720 / 130 / 850
Total / 750 / 140 / 890
- The chi-squared statistic is found as follows. Expected numbers are given below the observed numbers.
Approved / Denied / Total
Black / 30
33.71 / 10
6.29 / 40
White / 720
716.29 / 130
133.71 / 850
Total / 750 / 140 / 890
Chi-squared statistic = 0.408 + 2.185 + 0.019 + 0.103 = 2.715
c.We would now conclude that there is not a statistically significant relationship between ethnicity and mortgage approval. Yet the relationship for the full sample was very strong. This example illustrates the problem with trying to demonstrate a relationship based on a small sample.
13.18The participants were students who drink alcohol and were at a university in the Midwestern United States. They were enrolled in an Introductory Psychology course. Therefore, the population represented by these students is probably the population of students in similar courses at similar universities.
13.19a.The first null hypothesis: For the population of college-age students who drink there is no relationship between gender and experiencing vomiting as a hangover symptom. For the second null hypothesis, replace “vomiting” with “sweating more than usual.” The alternative hypothesis in each case is that there is a relationship.
b.In both cases the p-value given is less than 0.05, so the null hypothesis would be rejected. We can conclude that there is a relationship between gender and experiencing vomiting as a hangover symptom in the population of college students similar to the ones in the study. The same conclusion can be made about sweating more than usual as a hangover symptom.
13.20a.Null hypothesis: In the population of college students similar to the ones in this study, men and women are equally likely to experience at least one hangover symptom in a year. Alternative hypothesis: In the population of college students similar to the ones in this study, men and women are not equally likely to experience at least one hangover symptom in a year.
b.No, it is not exact, as shown by the reported 89% for men and 87% for women. The conclusion should have been written as “There was no significant difference in the proportions of men and women in the study who experienced at least one of the hangover symptoms in the past year.”
13.21a.Null hypothesis: In the population of college students similar to the ones in this study, men and women are equally likely to experience hangover symptoms at least once a month. Alternative hypothesis: In the population of college students similar to the ones in this study, men and women are not equally likely to experience hangover symptoms at least once a month.
b.Expected count = (466)(895)/1215 = 343.27.
c.Chi-square statistic = 5.35, p-value = 0.021. Reject the null hypothesis. We can conclude that men and women similar to those in the study are not equally likely to experience hangover symptoms at least once a month.
13.22a.Null hypothesis: In the population of drivers similar to these, there is no relationship between sex and being distracted by grooming. Alternative hypothesis: There is a relationship.
b.The footnote for grooming means that there was a statistically significant relationship between sex and being distracted by grooming, and the associated p-value was less than 0.01.
13.23a.The expected count for “Male, Yes” is (35)(60)/70 = 30, for “Male, No” it’s 5, for “Female, Yes” it’s 30 and for “Female, No” it’s 5.
b.The null hypothesis is that there is no relationship between sex and external distractions for the population of drivers similar to those in this study. The alternative hypothesis is that there is a relationship.
c.Chi-square statistic = 4.20, p-value is 0.04. Reject the null hypothesis. Conclude that there is a relationship between sex and external distractions while driving, for the population of drivers similar to those in the study.
d.The footnote says that the p-value is less than 0.05. The actual p-value is given as 0.04, which is indeed less than 0.05.
13.24a.The null hypothesis is that there is no relationship between sex and being observed conversing for the population of drivers similar to those in this study. The alternative hypothesis is that there is a relationship.
b.Expected count for “Male, Yes” is 27. Therefore, for “Male, No” it’s 8, for “Female, Yes” it’s 27 and for “Female, No” it’s 8.
c.The chi-square statistic is 0.324.
d.The chi-square statistic is much less than 3.84, therefore, the null hypothesis cannot be rejected. There is not sufficient evidence to conclude that there is a relationship between sex and being observed conversing while driving, for the population of drivers similar to those in the study.
NOTES ABOUT THE MINI-PROJECTS FOR CHAPTER 13
Mini-Project 13.1
This project is relatively straightforward in that it involves the same kinds of computations as the exercises. Be careful about collecting the data, to be sure the individuals are independent of each other and the same variables are measured on each one. Some guidelines to check are as follows. For part a, make sure the convention of putting the explanatory variable in the rows and the response in the columns is followed, if that distinction can be made. For part b make sure all parts are answered and that relative risk is computed rather than increased risk. For part c remember that the chi-squared statistic must exceed 3.84 to achieve statistical significance. For part d it is likely that the sample for the project will have been too small to achieve statistical significance, and this fact should be discussed. If the sample did achieve significance and the sample was small, it should be noted that the relationship is probably very strong.
Mini-Project 13.2
This project is relatively straightforward. Make sure the hypotheses are stated in terms of populations at the times the polls are taken.
Mini-Project 13.3
The purpose of this project is to get experience with reading journal articles and recognizing familiar procedures. Make sure the hypotheses are written about the population and not the sample, and that the population is specified. The paragraph written for part d should also be clear about the distinction between the sample results and the conclusions that can be made about the population.
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