Answer :
Let's address the problem step by step:
### Step 1: Verify if the Normal Sampling Distribution can be Used
To use the normal approximation for determining the difference between two population proportions, we check if the sample size conditions are met. The conditions are:
- [tex]\( n_1 \cdot p_1 \geq 10 \)[/tex]
- [tex]\( n_1 \cdot (1 - p_1) \geq 10 \)[/tex]
- [tex]\( n_2 \cdot p_2 \geq 10 \)[/tex]
- [tex]\( n_2 \cdot (1 - p_2) \geq 10 \)[/tex]
First, calculate the sample proportions:
[tex]\[ p_1 = \frac{x_1}{n_1} = \frac{78}{159} \approx 0.4906 \][/tex]
[tex]\[ p_2 = \frac{x_2}{n_2} = \frac{88}{199} \approx 0.4422 \][/tex]
Now, check the conditions:
[tex]\[ n_1 \cdot p_1 = 159 \cdot 0.4906 \approx 78.07 \][/tex]
[tex]\[ n_1 \cdot (1 - p_1) = 159 \cdot (1 - 0.4906) \approx 80.93 \][/tex]
[tex]\[ n_2 \cdot p_2 = 199 \cdot 0.4422 \approx 88.00 \][/tex]
[tex]\[ n_2 \cdot (1 - p_2) = 199 \cdot (1 - 0.4422) \approx 111.00 \][/tex]
All conditions are satisfied:
[tex]\[ 78.07 \geq 10, \quad 80.93 \geq 10, \quad 88.00 \geq 10, \quad 111.00 \geq 10 \][/tex]
Since all conditions are met, the normal sampling distribution can be used.
### Step 2: Establish Hypotheses and Significance Level
- Null Hypothesis ([tex]\( H_0 \)[/tex]): [tex]\( p_1 = p_2 \)[/tex]
- Alternative Hypothesis ([tex]\( H_1 \)[/tex]): [tex]\( p_1 \neq p_2 \)[/tex]
- Significance level ([tex]\( \alpha \)[/tex]): 0.01
### Step 3: Calculate the Test Statistic
Calculate the pooled proportion ([tex]\( \hat{p} \)[/tex]):
[tex]\[ \hat{p} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{78 + 88}{159 + 199} = \frac{166}{358} \approx 0.4637 \][/tex]
Calculate the standard error ([tex]\( SE \)[/tex]):
[tex]\[ SE = \sqrt{\hat{p} \cdot (1 - \hat{p}) \cdot \left(\frac{1}{n_1} + \frac{1}{n_2}\right)} \][/tex]
[tex]\[ SE = \sqrt{0.4637 \cdot (1 - 0.4637) \cdot \left(\frac{1}{159} + \frac{1}{199}\right)} \approx 0.0530 \][/tex]
Calculate the test statistic ([tex]\( z \)[/tex]):
[tex]\[ z = \frac{p_1 - p_2}{SE} = \frac{0.4906 - 0.4422}{0.0530} \approx 0.9116 \][/tex]
### Step 4: Determine Critical Value and Decision Rule
Find the critical z-value for a two-tailed test at [tex]\( \alpha = 0.01 \)[/tex]:
[tex]\[ z_{\text{critical}} = \pm 2.5758 \][/tex]
### Step 5: Compare the Test Statistic to the Critical Value
- If [tex]\( |z| > z_{\text{critical}} \)[/tex], reject the null hypothesis.
- If [tex]\( |z| \leq z_{\text{critical}} \)[/tex], do not reject the null hypothesis.
In this case:
[tex]\[ |0.9116| \leq 2.5758 \][/tex]
Since the calculated test statistic (0.9116) is less than the critical value (2.5758), we do not reject the null hypothesis.
### Conclusion
There is not enough evidence at the [tex]\( \alpha = 0.01 \)[/tex] significance level to reject the claim that [tex]\( p_1 = p_2 \)[/tex].
### Step 1: Verify if the Normal Sampling Distribution can be Used
To use the normal approximation for determining the difference between two population proportions, we check if the sample size conditions are met. The conditions are:
- [tex]\( n_1 \cdot p_1 \geq 10 \)[/tex]
- [tex]\( n_1 \cdot (1 - p_1) \geq 10 \)[/tex]
- [tex]\( n_2 \cdot p_2 \geq 10 \)[/tex]
- [tex]\( n_2 \cdot (1 - p_2) \geq 10 \)[/tex]
First, calculate the sample proportions:
[tex]\[ p_1 = \frac{x_1}{n_1} = \frac{78}{159} \approx 0.4906 \][/tex]
[tex]\[ p_2 = \frac{x_2}{n_2} = \frac{88}{199} \approx 0.4422 \][/tex]
Now, check the conditions:
[tex]\[ n_1 \cdot p_1 = 159 \cdot 0.4906 \approx 78.07 \][/tex]
[tex]\[ n_1 \cdot (1 - p_1) = 159 \cdot (1 - 0.4906) \approx 80.93 \][/tex]
[tex]\[ n_2 \cdot p_2 = 199 \cdot 0.4422 \approx 88.00 \][/tex]
[tex]\[ n_2 \cdot (1 - p_2) = 199 \cdot (1 - 0.4422) \approx 111.00 \][/tex]
All conditions are satisfied:
[tex]\[ 78.07 \geq 10, \quad 80.93 \geq 10, \quad 88.00 \geq 10, \quad 111.00 \geq 10 \][/tex]
Since all conditions are met, the normal sampling distribution can be used.
### Step 2: Establish Hypotheses and Significance Level
- Null Hypothesis ([tex]\( H_0 \)[/tex]): [tex]\( p_1 = p_2 \)[/tex]
- Alternative Hypothesis ([tex]\( H_1 \)[/tex]): [tex]\( p_1 \neq p_2 \)[/tex]
- Significance level ([tex]\( \alpha \)[/tex]): 0.01
### Step 3: Calculate the Test Statistic
Calculate the pooled proportion ([tex]\( \hat{p} \)[/tex]):
[tex]\[ \hat{p} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{78 + 88}{159 + 199} = \frac{166}{358} \approx 0.4637 \][/tex]
Calculate the standard error ([tex]\( SE \)[/tex]):
[tex]\[ SE = \sqrt{\hat{p} \cdot (1 - \hat{p}) \cdot \left(\frac{1}{n_1} + \frac{1}{n_2}\right)} \][/tex]
[tex]\[ SE = \sqrt{0.4637 \cdot (1 - 0.4637) \cdot \left(\frac{1}{159} + \frac{1}{199}\right)} \approx 0.0530 \][/tex]
Calculate the test statistic ([tex]\( z \)[/tex]):
[tex]\[ z = \frac{p_1 - p_2}{SE} = \frac{0.4906 - 0.4422}{0.0530} \approx 0.9116 \][/tex]
### Step 4: Determine Critical Value and Decision Rule
Find the critical z-value for a two-tailed test at [tex]\( \alpha = 0.01 \)[/tex]:
[tex]\[ z_{\text{critical}} = \pm 2.5758 \][/tex]
### Step 5: Compare the Test Statistic to the Critical Value
- If [tex]\( |z| > z_{\text{critical}} \)[/tex], reject the null hypothesis.
- If [tex]\( |z| \leq z_{\text{critical}} \)[/tex], do not reject the null hypothesis.
In this case:
[tex]\[ |0.9116| \leq 2.5758 \][/tex]
Since the calculated test statistic (0.9116) is less than the critical value (2.5758), we do not reject the null hypothesis.
### Conclusion
There is not enough evidence at the [tex]\( \alpha = 0.01 \)[/tex] significance level to reject the claim that [tex]\( p_1 = p_2 \)[/tex].
