Answer :
Certainly! Let's go through the solution step-by-step to address both parts of your question.
### Part (a): Hypothesis Testing
Step 1: Define the differences
First, we need to calculate the differences [tex]\(d\)[/tex] for each couple, where [tex]\(d\)[/tex] is the number of words spoken by the male minus the number of words spoken by the female.
[tex]\[ \begin{aligned} d_1 &= 16252 - 24371 = -8129 \\ d_2 &= 25714 - 13027 = 12687 \\ d_3 &= 1425 - 18375 = -16950 \\ d_4 &= 7602 - 18188 = -10623 \\ d_5 &= 19519 - 13418 = 6101 \\ d_6 &= 14922 - 16441 = -1519 \\ d_7 &= 13808 - 16846 = -3038 \\ d_8 &= 25418 - 19076 = 6342 \\ \end{aligned} \][/tex]
The differences are: [tex]\([-8129, 12687, -16950, -10623, 6101, -1519, -3038, 6342]\)[/tex].
Step 2: State the hypotheses
We need to test the claim that among couples, males speak fewer words than females. This can be expressed as:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The mean difference [tex]\( \mu_d \)[/tex] is equal to 0.
[tex]\[ H_0: \mu_d = 0 \][/tex]
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): The mean difference [tex]\( \mu_d \)[/tex] is less than 0.
[tex]\[ H_1 \mu_d < 0 \][/tex]
Since we're testing whether males speak fewer words, the alternative hypothesis is one-tailed (to the left).
Step 3: Calculate the mean and standard deviation of the differences
From the provided results:
- Mean of the differences ([tex]\(\bar{d}\)[/tex]) = [tex]\(-1885.25\)[/tex]
- Standard deviation of the differences (SD) = [tex]\(9905.198\)[/tex]
Step 4: Calculate the t-statistic
The formula for the t-statistic in a one-sample t-test is:
[tex]\[ t = \frac{\bar{d}}{SE} \quad \text{where} \quad SE = \frac{SD}{\sqrt{n}} \][/tex]
Given that [tex]\( n = 8 \)[/tex] (number of differences), the standard error (SE) is:
[tex]\[ SE = \frac{9905.198}{\sqrt{8}} \approx 3501.02 \][/tex]
The t-statistic is then:
[tex]\[ t = \frac{-1885.25}{3501.02} \approx -0.538 \][/tex]
Step 5: Determine the critical value and p-value
Given that the significance level ([tex]\(\alpha\)[/tex]) is 0.05 and degrees of freedom ([tex]\(df\)[/tex]) is [tex]\(n - 1 = 7\)[/tex], the critical t-value for a one-tailed test can be found in t-tables or using appropriate software:
[tex]\[ t_{\text{critical}} \approx -1.895 \][/tex]
The provided p-value is [tex]\(0.304\)[/tex].
Step 6: Make a decision
Based on the p-value method:
- If [tex]\( p \leq \alpha \)[/tex], we reject the null hypothesis.
- If [tex]\( p > \alpha \)[/tex], we fail to reject the null hypothesis.
In this case, [tex]\( p = 0.304 \)[/tex] which is greater than [tex]\(0.05\)[/tex]. Thus, we fail to reject the null hypothesis.
Therefore, with a significance level of 0.05, there is not enough evidence to support the claim that among couples, males speak fewer words a day than females.
### Summary of hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\( \mu_d = 0 \)[/tex]
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): [tex]\( \mu_d < 0 \)[/tex]
The results indicate that we fail to reject the null hypothesis, meaning there is insufficient evidence to conclude that males speak fewer words than females in a day.
---
If there is any part you would like me to explain further or if you have additional questions, feel free to ask!
### Part (a): Hypothesis Testing
Step 1: Define the differences
First, we need to calculate the differences [tex]\(d\)[/tex] for each couple, where [tex]\(d\)[/tex] is the number of words spoken by the male minus the number of words spoken by the female.
[tex]\[ \begin{aligned} d_1 &= 16252 - 24371 = -8129 \\ d_2 &= 25714 - 13027 = 12687 \\ d_3 &= 1425 - 18375 = -16950 \\ d_4 &= 7602 - 18188 = -10623 \\ d_5 &= 19519 - 13418 = 6101 \\ d_6 &= 14922 - 16441 = -1519 \\ d_7 &= 13808 - 16846 = -3038 \\ d_8 &= 25418 - 19076 = 6342 \\ \end{aligned} \][/tex]
The differences are: [tex]\([-8129, 12687, -16950, -10623, 6101, -1519, -3038, 6342]\)[/tex].
Step 2: State the hypotheses
We need to test the claim that among couples, males speak fewer words than females. This can be expressed as:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The mean difference [tex]\( \mu_d \)[/tex] is equal to 0.
[tex]\[ H_0: \mu_d = 0 \][/tex]
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): The mean difference [tex]\( \mu_d \)[/tex] is less than 0.
[tex]\[ H_1 \mu_d < 0 \][/tex]
Since we're testing whether males speak fewer words, the alternative hypothesis is one-tailed (to the left).
Step 3: Calculate the mean and standard deviation of the differences
From the provided results:
- Mean of the differences ([tex]\(\bar{d}\)[/tex]) = [tex]\(-1885.25\)[/tex]
- Standard deviation of the differences (SD) = [tex]\(9905.198\)[/tex]
Step 4: Calculate the t-statistic
The formula for the t-statistic in a one-sample t-test is:
[tex]\[ t = \frac{\bar{d}}{SE} \quad \text{where} \quad SE = \frac{SD}{\sqrt{n}} \][/tex]
Given that [tex]\( n = 8 \)[/tex] (number of differences), the standard error (SE) is:
[tex]\[ SE = \frac{9905.198}{\sqrt{8}} \approx 3501.02 \][/tex]
The t-statistic is then:
[tex]\[ t = \frac{-1885.25}{3501.02} \approx -0.538 \][/tex]
Step 5: Determine the critical value and p-value
Given that the significance level ([tex]\(\alpha\)[/tex]) is 0.05 and degrees of freedom ([tex]\(df\)[/tex]) is [tex]\(n - 1 = 7\)[/tex], the critical t-value for a one-tailed test can be found in t-tables or using appropriate software:
[tex]\[ t_{\text{critical}} \approx -1.895 \][/tex]
The provided p-value is [tex]\(0.304\)[/tex].
Step 6: Make a decision
Based on the p-value method:
- If [tex]\( p \leq \alpha \)[/tex], we reject the null hypothesis.
- If [tex]\( p > \alpha \)[/tex], we fail to reject the null hypothesis.
In this case, [tex]\( p = 0.304 \)[/tex] which is greater than [tex]\(0.05\)[/tex]. Thus, we fail to reject the null hypothesis.
Therefore, with a significance level of 0.05, there is not enough evidence to support the claim that among couples, males speak fewer words a day than females.
### Summary of hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\( \mu_d = 0 \)[/tex]
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): [tex]\( \mu_d < 0 \)[/tex]
The results indicate that we fail to reject the null hypothesis, meaning there is insufficient evidence to conclude that males speak fewer words than females in a day.
---
If there is any part you would like me to explain further or if you have additional questions, feel free to ask!
