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Answer :
Certainly! Let's analyze the problem step-by-step to understand the solution. We're testing a hypothesis about the ages of actresses and actors when they won Best Actress and Best Actor awards, respectively.
### Step 1: Understanding the Hypotheses
The problem asks us to test the claim that the ages of Best Actresses are generally younger than those of Best Actors.
- Null Hypothesis ([tex]\( H_0 \)[/tex]): The mean difference in ages ([tex]\( \mu_d \)[/tex]) between actresses and actors is zero or greater. Mathematically, [tex]\( \mu_d \geq 0 \)[/tex].
- Alternative Hypothesis ([tex]\( H_1 \)[/tex]): The mean difference in ages ([tex]\( \mu_d \)[/tex]) is less than zero. Mathematically, [tex]\( \mu_d < 0 \)[/tex].
### Step 2: Calculate the Differences
For each pair of ages from the actresses and actors, we calculate the difference:
[tex]\[
\text{Difference} = \text{Age of Actress} - \text{Age of Actor}
\][/tex]
Using the data provided:
- Actresses' ages: [25, 29, 31, 26, 34, 28, 29, 43, 32, 31]
- Actors' ages: [65, 36, 39, 34, 33, 34, 49, 36, 37, 43]
The list of differences computed would be:
[tex]\[
[-40, -7, -8, -8, 1, -6, -20, 7, -5, -12]
\][/tex]
### Step 3: Calculate the Mean of Differences
Next, we find the mean of these differences:
- Mean difference = [tex]\(-9.8\)[/tex]
### Step 4: Perform a Hypothesis Test
We conduct a one-sample t-test for the mean of these differences to determine if it's significantly less than zero.
The results of our t-test are:
- t-statistic: [tex]\(-2.42\)[/tex]
- p-value: [tex]\(0.019\)[/tex]
### Step 5: Make a Conclusion
Our significance level ([tex]\( \alpha \)[/tex]) is [tex]\(0.05\)[/tex]. We compare the p-value against the significance level:
- Since the [tex]\( \text{p-value} = 0.019 \)[/tex] is less than [tex]\( \alpha = 0.05 \)[/tex], we have enough evidence to reject the null hypothesis.
### Conclusion
We reject the null hypothesis, supporting the claim that Best Actresses are generally younger than Best Actors when they won the award. This analysis indicates a statistically significant difference in the mean ages, favoring that Best Actresses tend to be younger.
### Step 1: Understanding the Hypotheses
The problem asks us to test the claim that the ages of Best Actresses are generally younger than those of Best Actors.
- Null Hypothesis ([tex]\( H_0 \)[/tex]): The mean difference in ages ([tex]\( \mu_d \)[/tex]) between actresses and actors is zero or greater. Mathematically, [tex]\( \mu_d \geq 0 \)[/tex].
- Alternative Hypothesis ([tex]\( H_1 \)[/tex]): The mean difference in ages ([tex]\( \mu_d \)[/tex]) is less than zero. Mathematically, [tex]\( \mu_d < 0 \)[/tex].
### Step 2: Calculate the Differences
For each pair of ages from the actresses and actors, we calculate the difference:
[tex]\[
\text{Difference} = \text{Age of Actress} - \text{Age of Actor}
\][/tex]
Using the data provided:
- Actresses' ages: [25, 29, 31, 26, 34, 28, 29, 43, 32, 31]
- Actors' ages: [65, 36, 39, 34, 33, 34, 49, 36, 37, 43]
The list of differences computed would be:
[tex]\[
[-40, -7, -8, -8, 1, -6, -20, 7, -5, -12]
\][/tex]
### Step 3: Calculate the Mean of Differences
Next, we find the mean of these differences:
- Mean difference = [tex]\(-9.8\)[/tex]
### Step 4: Perform a Hypothesis Test
We conduct a one-sample t-test for the mean of these differences to determine if it's significantly less than zero.
The results of our t-test are:
- t-statistic: [tex]\(-2.42\)[/tex]
- p-value: [tex]\(0.019\)[/tex]
### Step 5: Make a Conclusion
Our significance level ([tex]\( \alpha \)[/tex]) is [tex]\(0.05\)[/tex]. We compare the p-value against the significance level:
- Since the [tex]\( \text{p-value} = 0.019 \)[/tex] is less than [tex]\( \alpha = 0.05 \)[/tex], we have enough evidence to reject the null hypothesis.
### Conclusion
We reject the null hypothesis, supporting the claim that Best Actresses are generally younger than Best Actors when they won the award. This analysis indicates a statistically significant difference in the mean ages, favoring that Best Actresses tend to be younger.
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