Problem
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) & 26 & 27 & 31 & 26 & 34 & 28 & 29 & 45 & 32 & 32 \\ \hline Actor (years) & 61 & 33 & 35 & 42 & 27 & 33 & 46 & 38 & 36 & 43 \\ \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? \[ \begin{array}{l} \mathrm{H}_{0}: \mu_{\mathrm{d}} \nabla \square \text { year(s) } \\ \mathrm{H}_{1} \mu_{\mathrm{d}} \square \square \text { year(s) } \end{array} \] (Type integers or decimals. Do not round.)
Solution
Step 1 ：The null hypothesis (H0) is usually a statement of no effect or no difference. In this case, it would be that the mean difference in ages is equal to 0.
Step 2 ：The alternative hypothesis (H1) is what we are testing against the null hypothesis. In this case, it would be that the mean difference in ages is less than 0. This would indicate that the Best Actresses are generally younger than Best Actors.
Step 3 ：Final Answer: \n\[\begin{array}{l}\mathrm{H}_{0}: \mu_{\mathrm{d}} = 0 \text { year(s) } \\mathrm{H}_{1}: \mu_{\mathrm{d}} < 0 \text { year(s) }\end{array}\]
