Step 1 :Given the ages of actresses and actors when they won the award, we are to test the claim that actresses are generally younger than actors when they won the award. We define the individual difference, \(d\), as the actress's age minus the actor's age. The mean value of these differences for the population of all pairs of data is denoted as \(\mu_{d}\).
Step 2 :The null hypothesis, \(H_{0}\), and the alternative hypothesis, \(H_{1}\), for the hypothesis test are: \[ \begin{array}{ll} H_{0}: \mu_{d} = 0 \text{ year(s)} \ H_{1}: \mu_{d} < 0 \text{ year(s)} \end{array} \]
Step 3 :We first calculate the differences between the ages of actresses and actors. The differences are: \[-29, -11, 0, -11, 6, -8, -23, -3, -6, -14\]
Step 4 :We then calculate the mean and standard deviation of these differences. The mean difference is \(-9.9\) years and the standard deviation of the differences is approximately \(10.4\) years.
Step 5 :The test statistic is calculated as the mean difference divided by the standard error of the mean difference. The test statistic is \(-3.01\).
Step 6 :The P-value is calculated using the cumulative distribution function of the t-distribution. The P-value is \(0.007\).
Step 7 :Since the P-value is less than the significance level of \(0.01\), we reject the null hypothesis. There is sufficient evidence to support the claim that actresses are generally younger when they won the award than actors. Hence, the final answer is \(\boxed{\text{Reject } H_{0}}\).