The following data lists the ages of a random selection of actresses when they won an award in the category of Best Actress, along with the ages of actors when they won in the category of Best Actor. The ages are matched according to the year that the awards were presented. Complete parts (a) and (b) below.
\begin{tabular}{lllllllllll}
\hline Actress (years) & 27 & 25 & 34 & 29 & 33 & 26 & 29 & 39 & 32 & 32 \\
\hline Actor (years) & 56 & 36 & 34 & 40 & 27 & 34 & 52 & 42 & 38 & 46 \\
\hline
\end{tabular}
Use the sample data with a 0.01 significance level to test the claim that for the population of ages of Best Actresses and Best Actors, the differences have a mean less than 0 (indicating that the Best Actresses are generally younger than Best Actors).
In this example, $\mu_{d}$ is the mean value of the differences $d$ for the population of all pairs of data, where each individual difference $d$ is defined as the actress's age minus the actor's age. What are the null and alternative hypotheses for the hypothesis test?
\[
\begin{array}{ll}
\mathrm{H}_{0}: \mu_{\mathrm{d}} & \mathbf{\nabla} \square \text { year(s) } \\
\mathrm{H}_{1} \cdot \mu_{\mathrm{d}} & \square \square \text { year(s) }
\end{array}
\]
(Type integers or decimals. Do not round.)
Identify the test statistic.
$\mathrm{t}=\square$ (Round to two decimal places as needed.)
Identify the P-value.
P-value $=\square$ (Round to three decimal places as needed. $)$
What is the conclusion based on the hypothesis test?
Since the P-value is the significance level, the null hypothesis. There sufficient evidence to support the claim that actresses are generally younger when they won the award than actors.
Answer
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 \(\boxed{\text{Reject } H_{0}}\).
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 \(\boxed{\text{Reject } H_{0}}\).
