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Evaluate the following formula for the given values: [tex]\( p_1 - p_2 = 0 \)[/tex], [tex]\( \bar{p} = 0.144975 \)[/tex], [tex]\( \bar{q} = 1 - \bar{p} \)[/tex], [tex]\( x_1 = 77 \)[/tex], [tex]\( x_2 = 11 \)[/tex], [tex]\( n_1 = 293 \)[/tex], [tex]\( n_2 = 314 \)[/tex], [tex]\( \hat{p}_1 = \frac{x_1}{n_1} \)[/tex], and [tex]\( \hat{p}_2 = \frac{x_2}{n_2} \)[/tex].

[tex]\[ z = \frac{\left(\hat{p}_1 - \hat{p}_2\right) - \left(p_1 - p_2\right)}{\sqrt{\frac{\bar{p} \cdot \bar{q}}{n_1} + \frac{\bar{p} \cdot \bar{q}}{n_2}}} \][/tex]

[tex]\[ z = \square \][/tex] (Round to two decimal places as needed.)



Answer :

GinnyAnswer
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].

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