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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.

[tex]
\[
\begin{tabular}{lllllllllll}
\hline
Actress (years) & 31 & 25 & 34 & 26 & 39 & 25 & 28 & 43 & 28 & 32 \\
\hline
Actor (years) & 65 & 41 & 32 & 37 & 35 & 34 & 46 & 39 & 34 & 42 \\
\hline
\end{tabular}
\]
[/tex]

In this example, [tex]$H_1$[/tex] is the mean value of the differences [tex]$d$[/tex] for the population of all pairs of data, where each individual difference [tex]$d$[/tex] is defined as the actress's age minus the actor's age.

What are the null and alternative hypotheses for the hypothesis test?

[tex]${}^{16}0.1$[/tex] [tex]\[\square\][/tex] (Your answer)

[tex]$H_0$[/tex] and [tex]$H_a$[/tex] [tex]\[\square\][/tex] year(s)

(Type integers or decimals. Do not round.)

Identify the test statistic:

[tex]$t = $[/tex] [tex]\[\square\][/tex] (Round to two decimal places as needed.)

Identify the P-value:

[tex]$P = $[/tex] [tex]\[\square\][/tex] (Round to three decimal places as needed.)

What is the conclusion based on the hypothesis test?

Since the P-value is [tex]\[\square\][/tex] the significance level, [tex]\[\square\][/tex] the null hypothesis. There is [tex]\[\square\][/tex] sufficient evidence to support the claim that actresses are generally younger when they win.

Answer :

To solve this problem, we need to conduct a hypothesis test comparing the ages of actresses and actors when they won an award. Here's how we can do this step-by-step:

### (a) State the Null and Alternative Hypotheses
We define the differences [tex]\( d \)[/tex] as the actress's age minus the actor's age for each pair. The hypotheses are:

- Null Hypothesis ([tex]\( H_0 \)[/tex]): The mean of the differences [tex]\( \mu_d = 0 \)[/tex]. This means there is no age difference on average between actresses and actors.
- Alternative Hypothesis ([tex]\( H_1 \)[/tex]): The mean of the differences [tex]\( \mu_d \neq 0 \)[/tex]. This implies there is an age difference on average between actresses and actors.

### (b) Perform the Hypothesis Test
1. Calculate the Differences:
We have a list of ages for actresses and actors. For each pair of ages (actress, actor), we calculate the difference:

[tex]\[ d = \text{actress's age} - \text{actor's age} \][/tex]

After calculating, the differences are:

[tex]\[-34, -16, 2, -11, 4, -9, -18, 4, -6, -10 \][/tex]

2. Calculate the Mean of the Differences:
The mean of these differences is given as [tex]\(-9.4\)[/tex].

3. Calculate the Standard Deviation:
The standard deviation of these differences is [tex]\(11.67\)[/tex].

4. Determine the Sample Size:
There are 10 pairs, so [tex]\( n = 10 \)[/tex].

5. Calculate the t-Statistic:
The t-statistic is calculated using the formula:

[tex]\[
t = \frac{\text{mean difference}}{\frac{\text{standard deviation of differences}}{\sqrt{n}}}
\][/tex]

The calculated t-statistic is [tex]\(-2.55\)[/tex].

6. Determine the P-Value:
The given P-value for the test is [tex]\(0.0314\)[/tex].

### (c) Conclusion
- Compare the P-Value with the Significance Level:
Suppose we use a common significance level of [tex]\( \alpha = 0.05\)[/tex].

- Conclusion:
Since the P-value [tex]\(0.0314\)[/tex] is less than [tex]\(0.05\)[/tex], we reject the null hypothesis at the 5% significance level.

There is sufficient evidence to support the claim that there is a difference in ages when actresses and actors win the awards, with actresses generally being younger.

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