Westonci.ca is the Q&A platform that connects you with experts who provide accurate and detailed answers. Explore our Q&A platform to find in-depth answers from a wide range of experts in different fields. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
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!
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.
