Certainly! Let's go through the solution step-by-step:
1. Given Values:
- [tex]\( p_1 - p_2 = 0 \)[/tex]
- [tex]\(\bar{p} = 0.144975\)[/tex]
- [tex]\(\bar{q} = 1 - \bar{p} = 1 - 0.144975 = 0.855025\)[/tex]
- [tex]\( x_1 = 77 \)[/tex]
- [tex]\( x_2 = 11 \)[/tex]
- [tex]\( n_1 = 293 \)[/tex]
- [tex]\( n_2 = 314 \)[/tex]
- [tex]\(\hat{p}_1 = \frac{x_1}{n_1} = \frac{77}{293}\)[/tex]
- [tex]\(\hat{p}_2 = \frac{x_2}{n_2} = \frac{11}{314}\)[/tex]
2. Calculate [tex]\(\hat{p}_1\)[/tex] and [tex]\(\hat{p}_2\)[/tex]:
[tex]\[
\hat{p}_1 = \frac{77}{293} \approx 0.262112
\][/tex]
[tex]\[
\hat{p}_2 = \frac{11}{314} \approx 0.035030
\][/tex]
3. Calculate the numerator of the z formula:
[tex]\[
\text{numerator} = (\hat{p}_1 - \hat{p}_2) - (p_1 - p_2) = (0.262112 - 0.035030) - 0 \approx 0.227082
\][/tex]
4. Calculate the denominator of the z formula:
[tex]\[
\text{denominator} = \sqrt{\frac{\bar{p} \cdot \bar{q}}{n_1} + \frac{\bar{p} \cdot \bar{q}}{n_2}}
\][/tex]
Substituting the values,
[tex]\[
\text{denominator} = \sqrt{\frac{0.144975 \cdot 0.855025}{293} + \frac{0.144975 \cdot 0.855025}{314}}
\][/tex]
[tex]\[
\text{denominator} \approx \sqrt{\frac{0.1239303125}{293} + \frac{0.1239303125}{314}}
\][/tex]
[tex]\[
\text{denominator} \approx \sqrt{0.000423 + 0.000395} \approx \sqrt{0.000818} \approx 0.028597
\][/tex]
5. Calculate the z value:
[tex]\[
z = \frac{\text{numerator}}{\text{denominator}} = \frac{0.227082}{0.028597} \approx 7.96
\][/tex]
6. Rounding to two decimal places as needed:
[tex]\[
z \approx 7.96
\][/tex]
Therefore, the value of [tex]\(z\)[/tex] is approximately [tex]\(7.96\)[/tex].
