Chapter 13, Problem 32SE

Happiness Are women happier than men? A StatCrunch survey asked respondents to select a number from 1 (lowest) to 100 (highest) to measure their level of happiness. The sample mean for the 297 females was 71.15, and the sample mean for the 380 males was 67.08. To determine whether the population mean for women was higher than the population mean for men, we used a randomization test.

a. The histogram shows the results of 1000 randomizations of the data. In each randomization, 297 observations from the combined data were randomly marked “female,” and the rest were randomly marked “male.” We calculated the mean difference in happiness between these two randomly determined groups. Note that the distribution is centered at about 0, just as it should be, since we carried out the randomization in such a way that the null hypothesis is true. The red line shows the observed sample mean difference in happiness for the women minus the happiness for the men. From the graph, does it look like the observed mean difference is unusual for this data set? Explain.

b. The software output estimates the probability of having an observed difference of 4.07 or more. (See the output column labeled “Proportion $=>$ Observed”). Where does the value of 4.07 (or 4.0718) come from?

c. State the $p$ -value.

d. Using a significance level of 0.05, can we reject the null hypothesis that the means are equal and so conclude that women StatCrunch users tend to be happier than men StatCrunch users? (Assume the sample was randomly selected from the population of all StatCrunch users.)

(Source: StatCrunch: Responses to Happiness survey. Owner: Webster West)