Answer :
### Step-by-Step Solution
#### (a) Hypothesis Testing
We are given the ages of actresses and actors when they won awards and need to test if there is a significant difference in their ages, particularly if actresses are generally younger than actors. Here we'll conduct a hypothesis test with a significance level of 0.05.
1. State the Hypotheses:
- Null Hypothesis ([tex]\( H_0 \)[/tex]): The mean difference in ages ([tex]\( \mu_d \)[/tex]) between actresses and actors is greater than or equal to 0. This implies actresses are not significantly younger than actors.
[tex]\[ H_0: \mu_d \geq 0 \][/tex]
- Alternative Hypothesis ([tex]\( H_1 \)[/tex]): The mean difference in ages ([tex]\( \mu_d \)[/tex]) is less than 0. This implies actresses are significantly younger than actors.
[tex]\[ H_1: \mu_d < 0 \][/tex]
- Here, [tex]\( \mu_d \)[/tex] represents the mean difference where each difference [tex]\( d \)[/tex] is calculated by [tex]\( \text{actress's age} - \text{actor's age} \)[/tex].
2. Sample Data:
The ages of actresses and actors are given as follows:
- Ages of Actresses: [tex]\( [30, 30, 30, 29, 35, 27, 26, 40, 30, 33] \)[/tex]
- Ages of Actors: [tex]\( [64, 40, 36, 36, 29, 35, 52, 37, 34, 38] \)[/tex]
3. Calculate the Differences:
Differences [tex]\( d \)[/tex] are calculated for matched pairs:
[tex]\[ [30-64, 30-40, 30-36, 29-36, 35-29, 27-35, 26-52, 40-37, 30-34, 33-38] \][/tex]
4. Sample Mean of Differences and Test Statistics:
From the result provided, the numerical values are:
- Mean of Differences ([tex]\( \bar{d} \)[/tex]): [tex]\( -9.1 \)[/tex]
- Test Statistic ([tex]\( t \)[/tex]-statistic): [tex]\( -2.359 \)[/tex]
- P-value: [tex]\( 0.0213 \)[/tex]
5. Comparison with Critical Value:
Since we are conducting a one-sample t-test with a one-tailed hypothesis (left-tailed), we compare the p-value to our significance level ([tex]\( \alpha = 0.05 \)[/tex]).
6. Make Decision:
- If the p-value < 0.05, we reject the null hypothesis.
- Here, [tex]\( 0.0213 < 0.05 \)[/tex], so we reject the null hypothesis.
7. Conclusion:
At a 0.05 significance level, there is sufficient evidence to support the claim that Best Actresses are generally younger than Best Actors.
### Answers for the Hypotheses
[tex]\[ \begin{array}{l} H_0: \mu_d \geq 0 \\ H_1: \mu_d < 0 \end{array} \][/tex]
#### (a) Hypothesis Testing
We are given the ages of actresses and actors when they won awards and need to test if there is a significant difference in their ages, particularly if actresses are generally younger than actors. Here we'll conduct a hypothesis test with a significance level of 0.05.
1. State the Hypotheses:
- Null Hypothesis ([tex]\( H_0 \)[/tex]): The mean difference in ages ([tex]\( \mu_d \)[/tex]) between actresses and actors is greater than or equal to 0. This implies actresses are not significantly younger than actors.
[tex]\[ H_0: \mu_d \geq 0 \][/tex]
- Alternative Hypothesis ([tex]\( H_1 \)[/tex]): The mean difference in ages ([tex]\( \mu_d \)[/tex]) is less than 0. This implies actresses are significantly younger than actors.
[tex]\[ H_1: \mu_d < 0 \][/tex]
- Here, [tex]\( \mu_d \)[/tex] represents the mean difference where each difference [tex]\( d \)[/tex] is calculated by [tex]\( \text{actress's age} - \text{actor's age} \)[/tex].
2. Sample Data:
The ages of actresses and actors are given as follows:
- Ages of Actresses: [tex]\( [30, 30, 30, 29, 35, 27, 26, 40, 30, 33] \)[/tex]
- Ages of Actors: [tex]\( [64, 40, 36, 36, 29, 35, 52, 37, 34, 38] \)[/tex]
3. Calculate the Differences:
Differences [tex]\( d \)[/tex] are calculated for matched pairs:
[tex]\[ [30-64, 30-40, 30-36, 29-36, 35-29, 27-35, 26-52, 40-37, 30-34, 33-38] \][/tex]
4. Sample Mean of Differences and Test Statistics:
From the result provided, the numerical values are:
- Mean of Differences ([tex]\( \bar{d} \)[/tex]): [tex]\( -9.1 \)[/tex]
- Test Statistic ([tex]\( t \)[/tex]-statistic): [tex]\( -2.359 \)[/tex]
- P-value: [tex]\( 0.0213 \)[/tex]
5. Comparison with Critical Value:
Since we are conducting a one-sample t-test with a one-tailed hypothesis (left-tailed), we compare the p-value to our significance level ([tex]\( \alpha = 0.05 \)[/tex]).
6. Make Decision:
- If the p-value < 0.05, we reject the null hypothesis.
- Here, [tex]\( 0.0213 < 0.05 \)[/tex], so we reject the null hypothesis.
7. Conclusion:
At a 0.05 significance level, there is sufficient evidence to support the claim that Best Actresses are generally younger than Best Actors.
### Answers for the Hypotheses
[tex]\[ \begin{array}{l} H_0: \mu_d \geq 0 \\ H_1: \mu_d < 0 \end{array} \][/tex]
