Question

In a survey of 1,000 women ages 22 to 35 who work full-time, 540 indicated that they would be willing to give up some personal time in order to make more money (USA Today, March 4, 2010). The sample was selected to be representative of women in the targeted age group. a. Do the sample data provide convincing evidence that a majority of women ages 22 to 35 who work fulltime would be willing to give up some personal time for more money? Test the relevant hypotheses using \(\alpha=0.01\) b. Would it be reasonable to generalize the conclusion from Part (a) to all working women? Explain why or why not.

Short answer

a. Yes, the sample data does provide convincing evidence that a majority of women ages 22 to 35 who work fulltime would be willing to give up some personal time for more money, as the p-value is less than 0.01. b. It would not be reasonable to generalize this to all working women, as the sample specifically includes only full-time workers within a certain age group, who may have different preferences compared to part-time workers or those in different age groups.

Step-by-step solution

Formulating the Hypotheses

The null hypothesis is that the majority (more than 0.5) of women ages 22 to 35 who work full time are not willing to give up personal time for more money, which can be written as H0: \(p \leq 0.5\). The alternative hypothesis, which we are testing for, is that the majority will give up personal time for more money, or H1: \(p > 0.5\).

Calculating the Test Statistic

The test statistic is calculated using the formula \((\hat{p} - p_0) / \sqrt{p_0(1 - p_0) / n}\), where \(\hat{p}\) is the sample proportion, \(p_0\) is the assumed population proportion, and \(n\) is the sample size. In this case, \(\hat{p} = 540/1000 = 0.54\), \(p_0 = 0.5\), and \(n = 1000\). Plugging these values into the test statistic formula gives a test statistic roughly equal to 2.83.

The p-value

The p-value, which represents the chance of obtaining a test statistic as extreme or more extreme than what was actually observed in case the null hypothesis is true, is calculated using the normal distribution. Given that our test in this case is one-tailed (with the alternate hypothesis suggesting that \(p > 0.5\)), we calculate the p-value as the probability of getting a value as extreme or more than our test statistic in the tail of the distribution. Considering the test statistic 2.83, the p-value will be smaller than \(\alpha=0.01\), typically found in normal distribution tables or by using software.

Conclusion of the Hypothesis Test

As the p-value obtained is less than the level of significance \(\alpha = 0.01\), the null hypothesis is rejected, providing convincing evidence that a majority of women ages 22 to 35 who work full time are willing to give up some personal time for more money.

Generalizability

While it might be tempting to generalize this result to all working women, it might not be reasonable to do so. The reason is that the sample was selected specifically from women ages 22 to 35 who work full time. This group may have different preferences and pressures compared to other working women such as part-time workers or those of different age groups, thus generalizing may lead to biased or wrongly inferred conclusions.