Answer :
Given the details of the problem, let's address the parts step-by-step:
### Part A: Hypothesis Testing
Hypotheses:
We need to evaluate the following hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The proportion of students in 2016 who believed freedom of the press was secure or very secure ([tex]\(P_1\)[/tex]) is equal to the proportion of students in 2017 who believed the same ([tex]\(P_2\)[/tex]).
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): The proportions are not equal.
In symbols:
- [tex]\(H_0: P_1 = P_2\)[/tex]
- [tex]\(H_a: P_1 \neq P_2\)[/tex]
Test Statistic:
The test statistic for comparing two proportions is given by [tex]\(z\)[/tex].
Given the data:
- The number of students surveyed in 2016 ([tex]\(n_1\)[/tex]) is [tex]\(3075\)[/tex] and those who felt freedom of the press was secure ([tex]\(x_1\)[/tex]) is [tex]\(2490\)[/tex].
- The number of students surveyed in 2017 ([tex]\(n_2\)[/tex]) is [tex]\(2026\)[/tex] and those who felt the same ([tex]\(x_2\)[/tex]) is [tex]\(1810\)[/tex].
The proportions are:
- 2016 proportion ([tex]\(p_1\)[/tex]) = [tex]\( \frac{2490}{3075} = 0.8098\)[/tex]
- 2017 proportion ([tex]\(p_2\)[/tex]) = [tex]\( \frac{1810}{2026} = 0.8934\)[/tex]
The test statistic value calculated is:
[tex]\[ z = -8.03 \][/tex]
p-value:
The corresponding p-value is used to determine the significance of the results. The p-value is:
[tex]\[ p\text{-value} = 0.000 \][/tex] (rounded to three decimal places).
Conclusion based on p-value:
Since the p-value ([tex]\(0.000\)[/tex]) is less than the significance level ([tex]\(\alpha = 0.01\)[/tex]), we reject the null hypothesis. This indicates that there is significant evidence to suggest that the proportion of students in 2016 who believed freedom of the press was secure differs from the proportion in 2017.
### Part B: Confidence Interval
98% Confidence Interval:
We construct a 98% confidence interval for the difference in proportions between 2016 and 2017.
The confidence interval is given by:
[tex]\[ \text{CI} = \left( -0.107, -0.061 \right) \][/tex] (rounded to three decimal places).
Interpretation:
Since the 98% confidence interval for the difference in proportions ([tex]\(-0.107\)[/tex] to [tex]\(-0.061\)[/tex]) does not contain 0, it supports the conclusion from the hypothesis test, suggesting a significant difference between the proportions of students in 2016 and 2017 who believed freedom of the press was secure or very secure.
### Conclusion
Given the data:
- [tex]\(p_1 = 0.8098\)[/tex], [tex]\(p_2 = 0.8934\)[/tex]
- Test statistic ([tex]\(z\)[/tex]) = [tex]\(-8.03\)[/tex]
- p-value = [tex]\(0.0\)[/tex]
- 98% confidence interval = [tex]\((-0.107, -0.061)\)[/tex]
Since the p-value [tex]\(0.0\)[/tex] is less than the significance level [tex]\(0.01\)[/tex], we reject the null hypothesis. This concludes that there is significant evidence to support that the proportions regarding the belief in the security of press freedom are different between 2016 and 2017. Additionally, the confidence interval further supports this conclusion by not including zero, indicating a significant difference in proportions.
### Part A: Hypothesis Testing
Hypotheses:
We need to evaluate the following hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The proportion of students in 2016 who believed freedom of the press was secure or very secure ([tex]\(P_1\)[/tex]) is equal to the proportion of students in 2017 who believed the same ([tex]\(P_2\)[/tex]).
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): The proportions are not equal.
In symbols:
- [tex]\(H_0: P_1 = P_2\)[/tex]
- [tex]\(H_a: P_1 \neq P_2\)[/tex]
Test Statistic:
The test statistic for comparing two proportions is given by [tex]\(z\)[/tex].
Given the data:
- The number of students surveyed in 2016 ([tex]\(n_1\)[/tex]) is [tex]\(3075\)[/tex] and those who felt freedom of the press was secure ([tex]\(x_1\)[/tex]) is [tex]\(2490\)[/tex].
- The number of students surveyed in 2017 ([tex]\(n_2\)[/tex]) is [tex]\(2026\)[/tex] and those who felt the same ([tex]\(x_2\)[/tex]) is [tex]\(1810\)[/tex].
The proportions are:
- 2016 proportion ([tex]\(p_1\)[/tex]) = [tex]\( \frac{2490}{3075} = 0.8098\)[/tex]
- 2017 proportion ([tex]\(p_2\)[/tex]) = [tex]\( \frac{1810}{2026} = 0.8934\)[/tex]
The test statistic value calculated is:
[tex]\[ z = -8.03 \][/tex]
p-value:
The corresponding p-value is used to determine the significance of the results. The p-value is:
[tex]\[ p\text{-value} = 0.000 \][/tex] (rounded to three decimal places).
Conclusion based on p-value:
Since the p-value ([tex]\(0.000\)[/tex]) is less than the significance level ([tex]\(\alpha = 0.01\)[/tex]), we reject the null hypothesis. This indicates that there is significant evidence to suggest that the proportion of students in 2016 who believed freedom of the press was secure differs from the proportion in 2017.
### Part B: Confidence Interval
98% Confidence Interval:
We construct a 98% confidence interval for the difference in proportions between 2016 and 2017.
The confidence interval is given by:
[tex]\[ \text{CI} = \left( -0.107, -0.061 \right) \][/tex] (rounded to three decimal places).
Interpretation:
Since the 98% confidence interval for the difference in proportions ([tex]\(-0.107\)[/tex] to [tex]\(-0.061\)[/tex]) does not contain 0, it supports the conclusion from the hypothesis test, suggesting a significant difference between the proportions of students in 2016 and 2017 who believed freedom of the press was secure or very secure.
### Conclusion
Given the data:
- [tex]\(p_1 = 0.8098\)[/tex], [tex]\(p_2 = 0.8934\)[/tex]
- Test statistic ([tex]\(z\)[/tex]) = [tex]\(-8.03\)[/tex]
- p-value = [tex]\(0.0\)[/tex]
- 98% confidence interval = [tex]\((-0.107, -0.061)\)[/tex]
Since the p-value [tex]\(0.0\)[/tex] is less than the significance level [tex]\(0.01\)[/tex], we reject the null hypothesis. This concludes that there is significant evidence to support that the proportions regarding the belief in the security of press freedom are different between 2016 and 2017. Additionally, the confidence interval further supports this conclusion by not including zero, indicating a significant difference in proportions.
