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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]
\[
\begin{array}{|c|cccccccccc|}
\hline
\text{Actress (years)} & 26 & 28 & 34 & 30 & 38 & 29 & 23 & 44 & 31 & 31 \\
\hline
\text{Actor (years)} & 63 & 39 & 34 & 40 & 33 & 31 & 48 & 41 & 39 & 45 \\
\hline
\end{array}
\]
[/tex]

a. 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, [tex]\(\mu_d\)[/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]
\[
\begin{array}{l}
H_0: \mu_d = 0 \text{ year(s)} \\
H_1: \mu_d \ < \ 0 \text{ year(s)}
\end{array}
\]
[/tex]

(Type integers or decimals. Do not round.)

Identify the test statistic.

[tex]\(t = -2.40\)[/tex] (Round to two decimal places as needed)

Identify the P-value.

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

Answer :

To solve the problem, we need to analyze the provided ages and conduct a hypothesis test. Here's how to approach it step by step:

### Step 1: State the Hypotheses
We are given that each difference [tex]\(d\)[/tex] is calculated as the actress's age minus the actor's age. We want to test the claim that the mean difference [tex]\(\mu_d\)[/tex] is less than 0.

- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\mu_d = 0\)[/tex] (There is no difference in ages.)
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): [tex]\(\mu_d < 0\)[/tex] (Actresses are generally younger than actors.)

### Step 2: Calculate the Differences
Calculate the difference for each pair:
- Differences ([tex]\(d\)[/tex]): [tex]\(26-63, 28-39, 34-34, 30-40, 38-33, 29-31, 23-48, 44-41, 31-39, 31-45\)[/tex]
- Which results in: [tex]\([-37, -11, 0, -10, 5, -2, -25, 3, -8, -14]\)[/tex]

### Step 3: Analyze the Differences
Calculate the mean and standard deviation of these differences:
- Mean ([tex]\(\bar{d}\)[/tex]): [tex]\(-9.9\)[/tex] years
- Standard Deviation (s) of differences: [tex]\(13.05\)[/tex]

### Step 4: Calculate the Test Statistic
For a t-test:
- Use the formula for the t-statistic: [tex]\[ t = \frac{\bar{d} - 0}{s/\sqrt{n}} \][/tex]
- Where [tex]\(n\)[/tex] = number of pairs = 10
- Substitute the known values: [tex]\(-2.40\)[/tex]

### Step 5: Determine the P-Value
Using the calculated t-statistic of [tex]\(-2.40\)[/tex], find the p-value for a one-tailed test with [tex]\(n-1 = 9\)[/tex] degrees of freedom.

- P-value: [tex]\(0.020\)[/tex]

### Step 6: Conclusion
Compare the P-value to the significance level of [tex]\(0.01\)[/tex]:

- Decision: Since the P-value [tex]\(0.020 > 0.01\)[/tex], we do not reject the null hypothesis at the 0.01 significance level.

- Conclusion: There isn't sufficient evidence to support the claim that the mean age of Best Actresses is less than that of Best Actors at a significance level of 0.01. It is indicated by the P-value being greater than the set significance level.

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