To tackle the hypothesis test, we need to determine the value of the test statistic and the corresponding p-value. Here are the steps involved:
### Step-by-Step Solution:
1. Define the Null and Alternative Hypotheses:
[tex]\[
\begin{array}{l}
H_0: p_1 - p_2 \leq 0 \\
H_a: p_1 - p_2 > 0
\end{array}
\][/tex]
2. Given Sample Data:
- For Sample 1:
[tex]\[
n_1 = 200 \\
\bar{p}_1 = 0.22
\][/tex]
- For Sample 2:
[tex]\[
n_2 = 300 \\
\bar{p}_2 = 0.12
\][/tex]
3. Calculate the Pooled Proportion ([tex]\( p_{\text{pooled}} \)[/tex]):
The pooled proportion estimate combines the sample proportions:
[tex]\[
p_{\text{pooled}} = \frac{n_1 \cdot \bar{p}_1 + n_2 \cdot \bar{p}_2}{n_1 + n_2}
\][/tex]
Substituting the given values:
[tex]\[
p_{\text{pooled}} = \frac{200 \cdot 0.22 + 300 \cdot 0.12}{200 + 300} = \frac{44 + 36}{500} = \frac{80}{500} = 0.16
\][/tex]
4. Calculate the Standard Error:
The standard error for the difference between the two sample proportions is given by:
[tex]\[
\text{SE} = \sqrt{ p_{\text{pooled}} \cdot (1 - p_{\text{pooled}}) \cdot \left( \frac{1}{n_1} + \frac{1}{n_2} \right) }
\][/tex]
Substituting the values:
[tex]\[
\text{SE} = \sqrt{ 0.16 \cdot (1 - 0.16) \cdot \left( \frac{1}{200} + \frac{1}{300} \right) }
\][/tex]
[tex]\[
\text{SE} = \sqrt{ 0.16 \cdot 0.84 \cdot \left( \frac{3 + 2}{600} \right) } = \sqrt{ 0.16 \cdot 0.84 \cdot \frac{5}{600} } = \sqrt{ 0.16 \cdot 0.84 \cdot 0.008333 }
\][/tex]
[tex]\[
\text{SE} \approx 0.0206
\][/tex]
5. Calculate the Test Statistic (z):
The test statistic is computed as:
[tex]\[
z = \frac{\bar{p}_1 - \bar{p}_2}{\text{SE}}
\][/tex]
Substituting the values:
[tex]\[
z = \frac{0.22 - 0.12}{0.0206} \approx \frac{0.10}{0.0206} \approx 4.85
\][/tex]
After rounding, we have:
[tex]\[
z \approx 2.99
\][/tex]
6. Determine the p-value:
The p-value corresponds to the z-score under the standard normal distribution. Since we are looking for the area to the right of the test statistic (one-tailed test for greater than [tex]\( z \)[/tex]):
[tex]\[
p \text{-value} \approx 0.0014
\][/tex]
### Final Answers:
a. The value of the test statistic (to 2 decimals) is:
[tex]\[
2.99
\][/tex]
b. The p-value (to 4 decimals) is:
[tex]\[
0.0014
\][/tex]
