Answer :
### Step-by-Step Solution
#### Part (a): Hypothesis Testing
To determine whether the proportion of college students who believe that freedom of religion is secure or very secure has changed from 2016 to 2017, we need to set up and evaluate a hypothesis test.
##### Step 1: State the Null and Alternative Hypotheses
The question is about checking if there is a change in the proportions between 2016 and 2017. Therefore, we will use the following hypotheses:
- Null Hypothesis ([tex]\( H_0 \)[/tex]): The proportion of college students who believe that freedom of religion is secure or very secure is the same in 2016 and 2017:
[tex]\[ H_0: p_1 = p_2 \][/tex]
- Alternative Hypothesis ([tex]\( H_a \)[/tex]): The proportion of college students who believe that freedom of religion is secure or very secure has changed:
[tex]\[ H_a: p_1 \neq p_2 \][/tex]
From the provided options, the correct hypotheses are:
C.
[tex]\[ \begin{array}{l} H_0: p_1 = p_2 \\ H_a: p_1 \neq p_2 \end{array} \][/tex]
##### Step 2: Calculate the Sample Proportions
1. 2016 Survey:
- Total respondents ([tex]\( n_1 \)[/tex]): 3133
- Number who said "secure" or "very secure" ([tex]\( x_1 \)[/tex]): 2078
- Sample proportion ([tex]\( p_1 \)[/tex]):
[tex]\[ p_1 = \frac{x_1}{n_1} = \frac{2078}{3133} \approx 0.6633 \][/tex]
2. 2017 Survey:
- Total respondents ([tex]\( n_2 \)[/tex]): 2953
- Number who said "secure" or "very secure" ([tex]\( x_2 \)[/tex]): 1929
- Sample proportion ([tex]\( p_2 \)[/tex]):
[tex]\[ p_2 = \frac{x_2}{n_2} = \frac{1929}{2953} \approx 0.6532 \][/tex]
##### Step 3: Calculate the Pooled Sample Proportion
The pooled sample proportion ([tex]\( p_{pool} \)[/tex]) is calculated as:
[tex]\[ p_{pool} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{2078 + 1929}{3133 + 2953} \approx 0.6584 \][/tex]
##### Step 4: Calculate the Standard Error
The standard error of the sampling distribution is given by:
[tex]\[ SE = \sqrt{p_{pool} \cdot (1 - p_{pool}) \cdot \left( \frac{1}{n_1} + \frac{1}{n_2} \right)} \][/tex]
[tex]\[ SE = \sqrt{0.6584 \cdot (1 - 0.6584) \cdot \left( \frac{1}{3133} + \frac{1}{2953} \right)} \approx 0.0122 \][/tex]
##### Step 5: Calculate the Test Statistic
Using the test statistic formula,
[tex]\[ z = \frac{p_1 - p_2}{SE} \][/tex]
[tex]\[ z = \frac{0.6633 - 0.6532}{0.0122} \approx 0.82 \][/tex]
Thus, the test statistic [tex]\( z \)[/tex] is:
[tex]\[ z = 0.82 \][/tex]
So, the final answer for identifying the test statistic is:
[tex]\[ z = 0.82 \][/tex]
Given the significance level of 0.01 ([tex]\( \alpha = 0.01 \)[/tex]), we would compare this z-value with the critical values for a two-tailed test. However, since the question only requests the test statistic, our calculation above suffices for this part.
By using these steps, we have determined the hypotheses and calculated the test statistic necessary for the hypothesis test.
#### Part (a): Hypothesis Testing
To determine whether the proportion of college students who believe that freedom of religion is secure or very secure has changed from 2016 to 2017, we need to set up and evaluate a hypothesis test.
##### Step 1: State the Null and Alternative Hypotheses
The question is about checking if there is a change in the proportions between 2016 and 2017. Therefore, we will use the following hypotheses:
- Null Hypothesis ([tex]\( H_0 \)[/tex]): The proportion of college students who believe that freedom of religion is secure or very secure is the same in 2016 and 2017:
[tex]\[ H_0: p_1 = p_2 \][/tex]
- Alternative Hypothesis ([tex]\( H_a \)[/tex]): The proportion of college students who believe that freedom of religion is secure or very secure has changed:
[tex]\[ H_a: p_1 \neq p_2 \][/tex]
From the provided options, the correct hypotheses are:
C.
[tex]\[ \begin{array}{l} H_0: p_1 = p_2 \\ H_a: p_1 \neq p_2 \end{array} \][/tex]
##### Step 2: Calculate the Sample Proportions
1. 2016 Survey:
- Total respondents ([tex]\( n_1 \)[/tex]): 3133
- Number who said "secure" or "very secure" ([tex]\( x_1 \)[/tex]): 2078
- Sample proportion ([tex]\( p_1 \)[/tex]):
[tex]\[ p_1 = \frac{x_1}{n_1} = \frac{2078}{3133} \approx 0.6633 \][/tex]
2. 2017 Survey:
- Total respondents ([tex]\( n_2 \)[/tex]): 2953
- Number who said "secure" or "very secure" ([tex]\( x_2 \)[/tex]): 1929
- Sample proportion ([tex]\( p_2 \)[/tex]):
[tex]\[ p_2 = \frac{x_2}{n_2} = \frac{1929}{2953} \approx 0.6532 \][/tex]
##### Step 3: Calculate the Pooled Sample Proportion
The pooled sample proportion ([tex]\( p_{pool} \)[/tex]) is calculated as:
[tex]\[ p_{pool} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{2078 + 1929}{3133 + 2953} \approx 0.6584 \][/tex]
##### Step 4: Calculate the Standard Error
The standard error of the sampling distribution is given by:
[tex]\[ SE = \sqrt{p_{pool} \cdot (1 - p_{pool}) \cdot \left( \frac{1}{n_1} + \frac{1}{n_2} \right)} \][/tex]
[tex]\[ SE = \sqrt{0.6584 \cdot (1 - 0.6584) \cdot \left( \frac{1}{3133} + \frac{1}{2953} \right)} \approx 0.0122 \][/tex]
##### Step 5: Calculate the Test Statistic
Using the test statistic formula,
[tex]\[ z = \frac{p_1 - p_2}{SE} \][/tex]
[tex]\[ z = \frac{0.6633 - 0.6532}{0.0122} \approx 0.82 \][/tex]
Thus, the test statistic [tex]\( z \)[/tex] is:
[tex]\[ z = 0.82 \][/tex]
So, the final answer for identifying the test statistic is:
[tex]\[ z = 0.82 \][/tex]
Given the significance level of 0.01 ([tex]\( \alpha = 0.01 \)[/tex]), we would compare this z-value with the critical values for a two-tailed test. However, since the question only requests the test statistic, our calculation above suffices for this part.
By using these steps, we have determined the hypotheses and calculated the test statistic necessary for the hypothesis test.
