To solve the given problem, we'll go through each step of evaluating the formula for the provided values:
### Given Values
- [tex]\(\hat{p}_1 = 0.9\)[/tex]
- [tex]\(\hat{p}_2 = 0.3\)[/tex]
- [tex]\(p_1 - p_2 = 0\)[/tex]
- [tex]\(\bar{p} = 0.798721\)[/tex]
- [tex]\(\bar{q} = 0.201279\)[/tex]
- [tex]\(n_1 = 35\)[/tex]
- [tex]\(n_2 = 32\)[/tex]
### Formula to Evaluate
The formula to evaluate is:
[tex]\[ z = \frac{(\hat{p}_1 - \hat{p}_2) - (p_1 - p_2)}{\sqrt{\frac{\bar{p} \cdot \bar{q}}{n_1} + \frac{\bar{p} \cdot \bar{q}}{n_2}}} \][/tex]
### Step-by-Step Solution
1. Calculate the numerator:
[tex]\[ (\hat{p}_1 - \hat{p}_2) - (p_1 - p_2) = (0.9 - 0.3) - 0 = 0.6 \][/tex]
2. Calculate the first denominator term:
[tex]\[ \frac{\bar{p} \cdot \bar{q}}{n_1} = \frac{0.798721 \cdot 0.201279}{35} \][/tex]
3. Calculate the second denominator term:
[tex]\[ \frac{\bar{p} \cdot \bar{q}}{n_2} = \frac{0.798721 \cdot 0.201279}{32} \][/tex]
4. Sum the two denominator terms:
[tex]\[ \frac{\bar{p} \cdot \bar{q}}{n_1} + \frac{\bar{p} \cdot \bar{q}}{n_2} \approx \frac{0.798721 0.201279}{35} + \frac{0.798721 0.201279}{32} \approx 0.045920630 \][/tex]
5. Take the square root of the summed denominator terms:
[tex]\[ \sqrt{\frac{\bar{p} \cdot \bar{q}}{n_1} + \frac{\bar{p} \cdot \bar{q}}{n_2}} \approx \sqrt{0.045920630} \approx 0.098067516 \][/tex]
6. Divide the numerator by the denominator to find z:
[tex]\[ z = \frac{0.6}{0.098067516} \approx 6.12 \][/tex]
### Final Rounded Answer
[tex]\[ z \approx 6.12 \][/tex]
By following this detailed step-by-step solution, we found that the value of [tex]\( z \)[/tex] rounded to two decimal places is:
[tex]\[ z = 6.12 \][/tex]
