Answer :
Let's approach the problem step-by-step:
### Problem Analysis:
1. We are comparing the proportions of college students in 2016 and 2017 who believe that freedom of religion is secure or very secure.
2. We'll use a significance level of 0.01 to determine if there is a statistically significant difference between the two proportions.
### Step 1: Define Hypotheses
The null and alternative hypotheses are:
- Null hypothesis ([tex]\(H_0\)[/tex]): [tex]\( p_1 = p_2 \)[/tex] (the proportion of college students who believe that freedom of religion is secure is the same in both years).
- Alternative hypothesis ([tex]\(H_a\)[/tex]): [tex]\( p_1 \neq p_2 \)[/tex] (the proportion of college students who believe that freedom of religion is secure is different in 2016 and 2017).
So, the correct choice is:
- [tex]\( H_0: p_1 = p_2 \)[/tex]
- [tex]\( H_a: p_1 \neq p_2 \)[/tex]
### Step 2: Calculate Sample Proportions
- For 2016: [tex]\( n_1 = 3133 \)[/tex], [tex]\( x_1 = 2078 \)[/tex]
- Sample proportion for 2016 ([tex]\( \hat{p}_1 \)[/tex]): [tex]\( \hat{p}_1 = \frac{2078}{3133} \approx 0.6633 \)[/tex]
- For 2017: [tex]\( n_2 = 2953 \)[/tex], [tex]\( x_2 = 1929 \)[/tex]
- Sample proportion for 2017 ([tex]\( \hat{p}_2 \)[/tex]): [tex]\( \hat{p}_2 = \frac{1929}{2953} \approx 0.6532 \)[/tex]
### Step 3: Calculate Pooled Proportion
The pooled sample proportion ([tex]\( \hat{p}_{pooled} \)[/tex]) is given by:
[tex]\[ \hat{p}_{pooled} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{2078 + 1929}{3133 + 2953} \approx 0.6584 \][/tex]
### Step 4: Calculate Standard Error
The standard error (SE) of the sampling distribution of the difference between proportions is:
[tex]\[ SE = \sqrt{\hat{p}_{pooled}(1 - \hat{p}_{pooled}) \left(\frac{1}{n_1} + \frac{1}{n_2}\right)} \][/tex]
[tex]\[ SE \approx \sqrt{0.6584 \times (1 - 0.6584) \left(\frac{1}{3133} + \frac{1}{2953}\right)} \approx 0.0122 \][/tex]
### Step 5: Compute the Test Statistic (z-score)
The z-score for the difference in proportions is given by:
[tex]\[ z = \frac{\hat{p}_1 - \hat{p}_2}{SE} \][/tex]
[tex]\[ z \approx \frac{0.6633 - 0.6532}{0.0122} \approx 0.82 \][/tex]
### Step 6: Determine the p-value
For a two-tailed test, the p-value is found using the standard normal distribution:
[tex]\[ p\text{-value} = 2 \times \left(1 - \Phi(|z|)\right) \][/tex]
Where [tex]\( \Phi \)[/tex] is the cumulative distribution function (CDF) of the standard normal distribution.
Given [tex]\( z = 0.82 \)[/tex], we can find that:
[tex]\[ p\text{-value} \approx 2 \times (1 - \Phi(0.82)) \approx 0.410 \][/tex]
### Conclusion
Given that our p-value (approximately 0.410) is much greater than the significance level of 0.01, we fail to reject the null hypothesis. This implies that there is not enough evidence to conclude that the proportion of college students who believe that freedom of religion is secure or very secure has changed from 2016 to 2017.
#### Summarized Answers:
1. Hypotheses:
- Null hypothesis ([tex]\(H_0\)[/tex]): [tex]\( p_1 = p_2 \)[/tex]
- Alternative hypothesis ([tex]\(H_a\)[/tex]): [tex]\( p_1 \neq p_2 \)[/tex]
2. Test statistic:
- [tex]\( z = 0.82 \)[/tex]
3. p-value:
- [tex]\( p\text{-value} = 0.410 \)[/tex]
Since the p-value (0.410) is greater than the significance level of 0.01, we do not reject the null hypothesis.
### Problem Analysis:
1. We are comparing the proportions of college students in 2016 and 2017 who believe that freedom of religion is secure or very secure.
2. We'll use a significance level of 0.01 to determine if there is a statistically significant difference between the two proportions.
### Step 1: Define Hypotheses
The null and alternative hypotheses are:
- Null hypothesis ([tex]\(H_0\)[/tex]): [tex]\( p_1 = p_2 \)[/tex] (the proportion of college students who believe that freedom of religion is secure is the same in both years).
- Alternative hypothesis ([tex]\(H_a\)[/tex]): [tex]\( p_1 \neq p_2 \)[/tex] (the proportion of college students who believe that freedom of religion is secure is different in 2016 and 2017).
So, the correct choice is:
- [tex]\( H_0: p_1 = p_2 \)[/tex]
- [tex]\( H_a: p_1 \neq p_2 \)[/tex]
### Step 2: Calculate Sample Proportions
- For 2016: [tex]\( n_1 = 3133 \)[/tex], [tex]\( x_1 = 2078 \)[/tex]
- Sample proportion for 2016 ([tex]\( \hat{p}_1 \)[/tex]): [tex]\( \hat{p}_1 = \frac{2078}{3133} \approx 0.6633 \)[/tex]
- For 2017: [tex]\( n_2 = 2953 \)[/tex], [tex]\( x_2 = 1929 \)[/tex]
- Sample proportion for 2017 ([tex]\( \hat{p}_2 \)[/tex]): [tex]\( \hat{p}_2 = \frac{1929}{2953} \approx 0.6532 \)[/tex]
### Step 3: Calculate Pooled Proportion
The pooled sample proportion ([tex]\( \hat{p}_{pooled} \)[/tex]) is given by:
[tex]\[ \hat{p}_{pooled} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{2078 + 1929}{3133 + 2953} \approx 0.6584 \][/tex]
### Step 4: Calculate Standard Error
The standard error (SE) of the sampling distribution of the difference between proportions is:
[tex]\[ SE = \sqrt{\hat{p}_{pooled}(1 - \hat{p}_{pooled}) \left(\frac{1}{n_1} + \frac{1}{n_2}\right)} \][/tex]
[tex]\[ SE \approx \sqrt{0.6584 \times (1 - 0.6584) \left(\frac{1}{3133} + \frac{1}{2953}\right)} \approx 0.0122 \][/tex]
### Step 5: Compute the Test Statistic (z-score)
The z-score for the difference in proportions is given by:
[tex]\[ z = \frac{\hat{p}_1 - \hat{p}_2}{SE} \][/tex]
[tex]\[ z \approx \frac{0.6633 - 0.6532}{0.0122} \approx 0.82 \][/tex]
### Step 6: Determine the p-value
For a two-tailed test, the p-value is found using the standard normal distribution:
[tex]\[ p\text{-value} = 2 \times \left(1 - \Phi(|z|)\right) \][/tex]
Where [tex]\( \Phi \)[/tex] is the cumulative distribution function (CDF) of the standard normal distribution.
Given [tex]\( z = 0.82 \)[/tex], we can find that:
[tex]\[ p\text{-value} \approx 2 \times (1 - \Phi(0.82)) \approx 0.410 \][/tex]
### Conclusion
Given that our p-value (approximately 0.410) is much greater than the significance level of 0.01, we fail to reject the null hypothesis. This implies that there is not enough evidence to conclude that the proportion of college students who believe that freedom of religion is secure or very secure has changed from 2016 to 2017.
#### Summarized Answers:
1. Hypotheses:
- Null hypothesis ([tex]\(H_0\)[/tex]): [tex]\( p_1 = p_2 \)[/tex]
- Alternative hypothesis ([tex]\(H_a\)[/tex]): [tex]\( p_1 \neq p_2 \)[/tex]
2. Test statistic:
- [tex]\( z = 0.82 \)[/tex]
3. p-value:
- [tex]\( p\text{-value} = 0.410 \)[/tex]
Since the p-value (0.410) is greater than the significance level of 0.01, we do not reject the null hypothesis.
