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) & 30 & 31 & 33 & 32 & 35 & 27 & 27 & 38 & 33 & 37 \\ \hline Actor (years) & 56 & 35 & 35 & 42 & 29 & 31 & 46 & 41 & 41 & 44 \\ \hline \end{tabular} a. Use the sample data with a 0.05 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? \[ \begin{array}{l} H_{0}: \mu_{d}=0 \text { year(s) } \\ H_{1}: \mu_{d}<0 \text { year(s) } \end{array} \] (Type integers or decimals. Do not round.) Identify the test statistic. \[ \mathrm{t}= \] (Round to two decimal places as needed.)

Step 1 ：We are given the ages of actresses and actors when they won an award in the category of Best Actress and Best Actor respectively. The ages are matched according to the year that the awards were presented.

Step 2 ：We are asked to use the sample data with a 0.05 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. This would indicate that the Best Actresses are generally younger than Best Actors.

Step 3 ：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.

Step 4 ：The null and alternative hypotheses for the hypothesis test are: \[\begin{array}{l} H_{0}: \mu_{d}=0 \text { year(s) } \\ H_{1}: \mu_{d}<0 \text { year(s) } \end{array}\]

Step 5 ：We calculate the differences between the ages of actresses and actors, and then calculate the mean and standard deviation of these differences.

Step 6 ：The test statistic is calculated as the mean difference divided by the standard deviation of the differences divided by the square root of the number of differences.

Step 7 ：The calculated test statistic is \(-2.69\).

Step 8 ：Final Answer: The test statistic is \(\boxed{-2.69}\).