5.1 The risks of sampling are that we might not have a representative sample, and sometimes this is difficult to know. If we didn’t realize that the sample was not representive, then we might draw conclusions about the population that are inaccurate.
5.2 The numbers in the fourth row, reading across, are 59808 08391 45427 26842 83609 49700 46058. Each person is assigned a number from 01 to 80. We then read the numbers from the table as two-digit numbers: 59, 80, 80, 83, 91, 45, 42, 72, 68, and so on. We ignore repeat numbers (e.g., 80) and numbers that exceed the sample of 80. So, the six people chosen would have the assigned numbers: 59, 80, 45, 42, 72, and 68.
5.3 Reading down from the first column, then the second, and so on, noting only the appearance of 0’s and 1’s, we see the numbers 0, 0, 0, 0, 1, and 0 (ending in the sixth column). Using these numbers, we could assign the first through the fourth people and the sixth person to the group designated as 0, and the fifth person to the group designated as 1. If we want an equal number of people in each of the two groups, we would assign the first three people to the 0 group and the last three to the 1 group, because we pulled three 0’s first.
5.4
a. The likely population is all patients who will undergo surgery; the researcher would not be able to access this population, and therefore random selection could not be used. Random assignment, however, could be used. The psychologist could randomly assign half of the patients to counseling and half to a control group.
b. The population is all children in this school system; the psychologist could identify all of these children and thus could use random selection. The psychologist could also use random assignment. She could randomly assign half the children to the interactive online textbook and half to the printed textbook.
c. The population is patients in therapy; because the whole population could not be identified, random selection could not be used. Moreover, random assignment could not be used. It is not possible to assign people to either have or not have a diagnosed personality disorder.
5.5 We regularly make personal assessments about how probable we think an event is, but we base these evaluations on the opinions about things rather than on systematically collected data. Statisticians are interested in objective probabilities that are based on unbiased research.
5.6
a. probability = successes/trials = 5/100 = 0.05
b. 8/50 = 0.16
c. 130/1044 = 0.12
5.7
a. In the short run, we might see a wide range of numbers of successes. It would not be surprising to have several in a row or none in a row. In the short run, the observations seem almost like chaos.
b. Given the assumptions listed for this problem, in the long run, we’d expect 0.50, or 50%, to be women, although there would likely be strings of men and of women along the way.
5.8 When we reject the null hypothesis, we are saying we reject the idea that there is no mean difference in the dependent variable across the levels of our independent variable. Rejecting the null hypothesis means we can support the research hypothesis that there is a mean difference.
5.9 The null hypothesis assumes that no mean difference would be observed, so the mean difference in grades would be zero.
5.10
a. The null hypothesis is that a decrease in temperature does not affect mean academic performance (or does not decrease mean academic performance).
b. The research hypothesis is that a decrease in temperature does affect mean academic performance (or decreases mean academic performance).
c. The researchers would reject the null hypothesis.
d. The researchers would fail to reject the null hypothesis.
5.11 We make a Type I error when we reject the null hypothesis but the null hypothesis is correct. We make a Type II error when we fail to reject the null hypothesis but the null hypothesis is false.
5.12 In this scenario, a Type I error would be imprisoning a person who is really innocent, and 7 convictions out of 280 innocent people calculates to be 0.025, or 2.5%.
5.13 In this scenario, a Type II error would be failing to convict a guilty person, and 11 acquittals for every 35 guilty people calculates to be 0.314, or 31.4%.
5.14
a. If the virtual-reality glasses really don’t have any effect, this is a Type I error, which is made when the null hypothesis is rejected but is really true.
b. If the virtual-reality glasses really do have an effect, this is a Type II error, which is made when the researchers fail to reject the null hypothesis but the null hypothesis is not true.