Answer :
### Solution to Part (a)
#### Step 1: Formulate the Null and Alternative Hypotheses
The given problem asks us to determine whether the proportion of college students who believe freedom of religion is secure or very secure has changed from 2016 to 2017.
- The null hypothesis ([tex]\( H_0 \)[/tex]) states that there is no difference in the proportions from 2016 to 2017.
[tex]\[ H_0: p_1 = p_2 \][/tex]
- The alternative hypothesis ([tex]\( H_a \)[/tex]) states that there is a difference in the proportions from 2016 to 2017.
[tex]\[ H_a: p_1 \neq p_2 \][/tex]
These hypotheses can be matched with option E:
[tex]\[ H_0: p_1 = p_2 \quad \text{and} \quad H_a: p_1 \neq p_2 \][/tex]
#### Step 2: Identify the Test Statistic
We need to calculate the [tex]\( z \)[/tex]-statistic for the difference in proportions. This test statistic is computed from the proportions of successes in each sample, along with the pooled proportion and its standard error.
Given:
- [tex]\( p_1 = \frac{2078}{3133} = 0.663262 \)[/tex]
- [tex]\( p_2 = \frac{1929}{2953} = 0.653234 \)[/tex]
- [tex]\( p_{\text{combined}} = \frac{2078 + 1929}{3133 + 2953} = 0.658396 \)[/tex]
- Standard error = 0.012164
The [tex]\( z \)[/tex]-statistic is:
[tex]\[ z = \frac{p_1 - p_2}{\text{standard error}} = \frac{0.663262 - 0.653234}{0.012164} \approx 0.82 \][/tex]
#### Step 3: Identify the p-value
The p-value for the computed [tex]\( z \)[/tex]-statistic (0.82) is:
[tex]\[ \text{p-value} \approx 1.590 \][/tex]
Since this is a two-tailed test, the p-value must be doubled. Therefore, the correct p-value rounded to three decimal places is:
[tex]\[ \text{p-value} = 0.412 \][/tex]
#### Step 4: Compare the p-value with the Significance Level
Given the significance level [tex]\( \alpha = 0.01 \)[/tex]:
[tex]\[ \text{p-value} = 0.412 > 0.01 \][/tex]
Since the p-value is greater than the significance level, we fail to reject the null hypothesis. Thus, there is insufficient evidence to support the claim that the proportion of college students who believe freedom of religion is secure has changed from 2016 to 2017.
### Solution to Part (b)
#### Step 5: Construct the Confidence Interval
The 99% confidence interval for the difference in the proportions is calculated as follows.
- The critical z-value for a 99% confidence interval ([tex]\( \alpha = 0.01 \)[/tex]) is approximately 2.576.
- The margin of error (ME) is:
[tex]\[ \text{ME} = z_{\text{critical}} \times \text{standard error} = 2.576 \times 0.012164 = 0.031531 \][/tex]
The 99% confidence interval for [tex]\( p_1 - p_2 \)[/tex] is:
[tex]\[ \left( (p_1 - p_2) - \text{ME}, (p_1 - p_2) + \text{ME} \right) = \left( (0.663262 - 0.653234) - 0.031531, (0.663262 - 0.653234) + 0.031531 \right) \][/tex]
[tex]\[ \approx \left( -0.021303, 0.041359 \right) \][/tex]
#### Step 6: Interpretation of the Confidence Interval
The 99% confidence interval for the difference in proportions is:
[tex]\[ (-0.021, 0.041) \][/tex]
Since this interval contains 0, it supports the conclusion from the hypothesis test that there is no significant difference in the proportions of college students who believe freedom of religion is secure between 2016 and 2017 at the 1% significance level.
#### Step 1: Formulate the Null and Alternative Hypotheses
The given problem asks us to determine whether the proportion of college students who believe freedom of religion is secure or very secure has changed from 2016 to 2017.
- The null hypothesis ([tex]\( H_0 \)[/tex]) states that there is no difference in the proportions from 2016 to 2017.
[tex]\[ H_0: p_1 = p_2 \][/tex]
- The alternative hypothesis ([tex]\( H_a \)[/tex]) states that there is a difference in the proportions from 2016 to 2017.
[tex]\[ H_a: p_1 \neq p_2 \][/tex]
These hypotheses can be matched with option E:
[tex]\[ H_0: p_1 = p_2 \quad \text{and} \quad H_a: p_1 \neq p_2 \][/tex]
#### Step 2: Identify the Test Statistic
We need to calculate the [tex]\( z \)[/tex]-statistic for the difference in proportions. This test statistic is computed from the proportions of successes in each sample, along with the pooled proportion and its standard error.
Given:
- [tex]\( p_1 = \frac{2078}{3133} = 0.663262 \)[/tex]
- [tex]\( p_2 = \frac{1929}{2953} = 0.653234 \)[/tex]
- [tex]\( p_{\text{combined}} = \frac{2078 + 1929}{3133 + 2953} = 0.658396 \)[/tex]
- Standard error = 0.012164
The [tex]\( z \)[/tex]-statistic is:
[tex]\[ z = \frac{p_1 - p_2}{\text{standard error}} = \frac{0.663262 - 0.653234}{0.012164} \approx 0.82 \][/tex]
#### Step 3: Identify the p-value
The p-value for the computed [tex]\( z \)[/tex]-statistic (0.82) is:
[tex]\[ \text{p-value} \approx 1.590 \][/tex]
Since this is a two-tailed test, the p-value must be doubled. Therefore, the correct p-value rounded to three decimal places is:
[tex]\[ \text{p-value} = 0.412 \][/tex]
#### Step 4: Compare the p-value with the Significance Level
Given the significance level [tex]\( \alpha = 0.01 \)[/tex]:
[tex]\[ \text{p-value} = 0.412 > 0.01 \][/tex]
Since the p-value is greater than the significance level, we fail to reject the null hypothesis. Thus, there is insufficient evidence to support the claim that the proportion of college students who believe freedom of religion is secure has changed from 2016 to 2017.
### Solution to Part (b)
#### Step 5: Construct the Confidence Interval
The 99% confidence interval for the difference in the proportions is calculated as follows.
- The critical z-value for a 99% confidence interval ([tex]\( \alpha = 0.01 \)[/tex]) is approximately 2.576.
- The margin of error (ME) is:
[tex]\[ \text{ME} = z_{\text{critical}} \times \text{standard error} = 2.576 \times 0.012164 = 0.031531 \][/tex]
The 99% confidence interval for [tex]\( p_1 - p_2 \)[/tex] is:
[tex]\[ \left( (p_1 - p_2) - \text{ME}, (p_1 - p_2) + \text{ME} \right) = \left( (0.663262 - 0.653234) - 0.031531, (0.663262 - 0.653234) + 0.031531 \right) \][/tex]
[tex]\[ \approx \left( -0.021303, 0.041359 \right) \][/tex]
#### Step 6: Interpretation of the Confidence Interval
The 99% confidence interval for the difference in proportions is:
[tex]\[ (-0.021, 0.041) \][/tex]
Since this interval contains 0, it supports the conclusion from the hypothesis test that there is no significant difference in the proportions of college students who believe freedom of religion is secure between 2016 and 2017 at the 1% significance level.
