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Answer :
To solve this problem, we need to test the claim that the mean difference in ages, when actresses and actors win their respective Best Actress and Best Actor awards, is less than 0. This would imply that, on average, Best Actresses are younger than Best Actors. Let's walk through the steps to test this hypothesis using the provided data.
### Step 1: Define the Hypotheses
First, let's set up the null and alternative hypotheses. We define the individual difference [tex]\( d \)[/tex] as the actress's age minus the actor's age. We are testing whether the mean of these differences is less than 0.
- Null Hypothesis ([tex]\( H_0 \)[/tex]): The mean difference in ages, [tex]\( \mu_d \)[/tex], is equal to 0. This would suggest that there is no age difference between Best Actresses and Best Actors.
[tex]\[
H_0: \mu_d = 0
\][/tex]
- Alternative Hypothesis ([tex]\( H_1 \)[/tex]): The mean difference in ages, [tex]\( \mu_d \)[/tex], is less than 0. This suggests that Best Actresses are, on average, younger than Best Actors.
[tex]\[
H_1: \mu_d < 0
\][/tex]
### Step 2: Calculate the Differences and Mean Difference
Now, calculate the differences for each pair of ages and then find the mean of these differences.
Given data:
- Actress ages: 31, 30, 31, 26, 38, 24, 28, 41, 33, 37
- Actor ages: 66, 41, 32, 36, 26, 33, 48, 43, 39, 45
Calculating the differences [tex]\( d \)[/tex] (actress age - actor age):
[tex]\[ d = \{31-66, 30-41, 31-32, 26-36, 38-26, 24-33, 28-48, 41-43, 33-39, 37-45\} \][/tex]
The differences are:
[tex]\[ -35, -11, -1, -10, 12, -9, -20, -2, -6, -8 \][/tex]
Calculate the mean of the differences:
[tex]\[ \text{Mean difference} = -9.0 \][/tex]
### Step 3: Perform the Hypothesis Test
Now, we proceed to conduct a one-tailed t-test to see if the mean difference is statistically less than 0.
Given the results:
- Mean difference ([tex]\( \bar{d} \)[/tex]): -9.0
- t-score: -2.31
- Critical t-value for a one-tailed test at [tex]\( \alpha = 0.05 \)[/tex]: -1.83
- Degrees of freedom: 9
### Step 4: Decision
Compare the calculated t-score with the critical t-value:
- If the t-score is less than the critical value, we reject the null hypothesis in favor of the alternative.
Here, the t-score of -2.31 is indeed less than the critical value of -1.83. Therefore, we reject the null hypothesis.
### Conclusion
Based on the t-test, we can conclude that there is enough statistical evidence at the 0.05 significance level to support the claim that Best Actresses are generally younger than Best Actors when they win their respective awards.
### Step 1: Define the Hypotheses
First, let's set up the null and alternative hypotheses. We define the individual difference [tex]\( d \)[/tex] as the actress's age minus the actor's age. We are testing whether the mean of these differences is less than 0.
- Null Hypothesis ([tex]\( H_0 \)[/tex]): The mean difference in ages, [tex]\( \mu_d \)[/tex], is equal to 0. This would suggest that there is no age difference between Best Actresses and Best Actors.
[tex]\[
H_0: \mu_d = 0
\][/tex]
- Alternative Hypothesis ([tex]\( H_1 \)[/tex]): The mean difference in ages, [tex]\( \mu_d \)[/tex], is less than 0. This suggests that Best Actresses are, on average, younger than Best Actors.
[tex]\[
H_1: \mu_d < 0
\][/tex]
### Step 2: Calculate the Differences and Mean Difference
Now, calculate the differences for each pair of ages and then find the mean of these differences.
Given data:
- Actress ages: 31, 30, 31, 26, 38, 24, 28, 41, 33, 37
- Actor ages: 66, 41, 32, 36, 26, 33, 48, 43, 39, 45
Calculating the differences [tex]\( d \)[/tex] (actress age - actor age):
[tex]\[ d = \{31-66, 30-41, 31-32, 26-36, 38-26, 24-33, 28-48, 41-43, 33-39, 37-45\} \][/tex]
The differences are:
[tex]\[ -35, -11, -1, -10, 12, -9, -20, -2, -6, -8 \][/tex]
Calculate the mean of the differences:
[tex]\[ \text{Mean difference} = -9.0 \][/tex]
### Step 3: Perform the Hypothesis Test
Now, we proceed to conduct a one-tailed t-test to see if the mean difference is statistically less than 0.
Given the results:
- Mean difference ([tex]\( \bar{d} \)[/tex]): -9.0
- t-score: -2.31
- Critical t-value for a one-tailed test at [tex]\( \alpha = 0.05 \)[/tex]: -1.83
- Degrees of freedom: 9
### Step 4: Decision
Compare the calculated t-score with the critical t-value:
- If the t-score is less than the critical value, we reject the null hypothesis in favor of the alternative.
Here, the t-score of -2.31 is indeed less than the critical value of -1.83. Therefore, we reject the null hypothesis.
### Conclusion
Based on the t-test, we can conclude that there is enough statistical evidence at the 0.05 significance level to support the claim that Best Actresses are generally younger than Best Actors when they win their respective awards.
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