Answer :
Let's go through each part of the question step by step:
### Part (a): Hypothesis Testing
First, let's identify the null and alternative hypotheses for this hypothesis test.
The null hypothesis ([tex]\(H_0\)[/tex]) and the alternative hypothesis ([tex]\(H_1\)[/tex]) are:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The proportion of students in 2016 who believe that freedom of the press is secure or very secure is equal to the proportion of students in 2017 who feel the same way.
[tex]\[ H_0: P_1 = P_2 \][/tex]
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): The proportion of students in 2016 who believe that freedom of the press is secure or very secure is not equal to the proportion of students in 2017 who feel the same way.
[tex]\[ H_1: P_1 \neq P_2 \][/tex]
Among the provided options, the correct choice is:
- B. [tex]\(H_0: P1 = P2\)[/tex].
Identifying the test statistic ([tex]\(z\)[/tex]):
The calculated test statistic is:
[tex]\[ z = -8.03 \][/tex]
Identify the p-value:
The associated p-value is given as:
[tex]\[ p\text{-value} = 0.000 \][/tex]
Making a decision based on the p-value:
Since the p-value (0.000) is less than the significance level ([tex]\(\alpha = 0.01\)[/tex]), we reject the null hypothesis.
Conclusion:
- We reject the null hypothesis.
- There is significant evidence to support the claim that the 2016 proportion is different from the 2017 proportion.
So, the answers are:
1. Null Hypothesis: [tex]\(H_0: P_1 = P_2\)[/tex]
2. Test Statistic: [tex]\(z = -8.03\)[/tex]
3. p-value: [tex]\(0.000\)[/tex]
4. Decision: Since the p-value is [tex]\(0.000\)[/tex], which is less than the significance level of [tex]\(0.01\)[/tex], we reject the null hypothesis.
5. Conclusion: There is significant evidence to support the claim that the 2016 proportion is different from the 2017 proportion.
### Part (b): Confidence Interval
Use the sample data to construct a 95% confidence interval for the difference in the proportions of college students in 2016 and 2017 who felt that freedom of the press was secure or very secure.
The 95% confidence interval for the difference in proportions is given as:
[tex]\[ (-0.104, -0.063) \][/tex]
How does this confidence interval support your hypothesis test conclusion?
Because the confidence interval [tex]\((-0.104, -0.063)\)[/tex] does not contain 0, it reinforces the conclusion from the hypothesis test that there is a significant difference between the proportions of students in 2016 and 2017 who felt that freedom of the press was secure or very secure.
To summarize:
1. 95% Confidence Interval: [tex]\((-0.104, -0.063)\)[/tex]
2. Interpretation: Since the confidence interval does not contain 0, it appears that the two proportions are significantly different.
3. Conclusion: This conclusion supports the hypothesis test conclusion.
So, the confidence interval further supports our decision to reject the null hypothesis, indicating a significant difference between the proportions in 2016 and 2017.
### Part (a): Hypothesis Testing
First, let's identify the null and alternative hypotheses for this hypothesis test.
The null hypothesis ([tex]\(H_0\)[/tex]) and the alternative hypothesis ([tex]\(H_1\)[/tex]) are:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The proportion of students in 2016 who believe that freedom of the press is secure or very secure is equal to the proportion of students in 2017 who feel the same way.
[tex]\[ H_0: P_1 = P_2 \][/tex]
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): The proportion of students in 2016 who believe that freedom of the press is secure or very secure is not equal to the proportion of students in 2017 who feel the same way.
[tex]\[ H_1: P_1 \neq P_2 \][/tex]
Among the provided options, the correct choice is:
- B. [tex]\(H_0: P1 = P2\)[/tex].
Identifying the test statistic ([tex]\(z\)[/tex]):
The calculated test statistic is:
[tex]\[ z = -8.03 \][/tex]
Identify the p-value:
The associated p-value is given as:
[tex]\[ p\text{-value} = 0.000 \][/tex]
Making a decision based on the p-value:
Since the p-value (0.000) is less than the significance level ([tex]\(\alpha = 0.01\)[/tex]), we reject the null hypothesis.
Conclusion:
- We reject the null hypothesis.
- There is significant evidence to support the claim that the 2016 proportion is different from the 2017 proportion.
So, the answers are:
1. Null Hypothesis: [tex]\(H_0: P_1 = P_2\)[/tex]
2. Test Statistic: [tex]\(z = -8.03\)[/tex]
3. p-value: [tex]\(0.000\)[/tex]
4. Decision: Since the p-value is [tex]\(0.000\)[/tex], which is less than the significance level of [tex]\(0.01\)[/tex], we reject the null hypothesis.
5. Conclusion: There is significant evidence to support the claim that the 2016 proportion is different from the 2017 proportion.
### Part (b): Confidence Interval
Use the sample data to construct a 95% confidence interval for the difference in the proportions of college students in 2016 and 2017 who felt that freedom of the press was secure or very secure.
The 95% confidence interval for the difference in proportions is given as:
[tex]\[ (-0.104, -0.063) \][/tex]
How does this confidence interval support your hypothesis test conclusion?
Because the confidence interval [tex]\((-0.104, -0.063)\)[/tex] does not contain 0, it reinforces the conclusion from the hypothesis test that there is a significant difference between the proportions of students in 2016 and 2017 who felt that freedom of the press was secure or very secure.
To summarize:
1. 95% Confidence Interval: [tex]\((-0.104, -0.063)\)[/tex]
2. Interpretation: Since the confidence interval does not contain 0, it appears that the two proportions are significantly different.
3. Conclusion: This conclusion supports the hypothesis test conclusion.
So, the confidence interval further supports our decision to reject the null hypothesis, indicating a significant difference between the proportions in 2016 and 2017.
