Let's break down the problem step-by-step to evaluate the given formula and find [tex]\(z\)[/tex].
1. Determine Sample Proportions:
First, we calculate [tex]\(\hat{p}_1\)[/tex] and [tex]\(\hat{p}_2\)[/tex]:
[tex]\[
\hat{p}_1 = \frac{x_1}{n_1} = \frac{73}{299} = 0.24414715719063546
\][/tex]
[tex]\[
\hat{p}_2 = \frac{x_2}{n_2} = \frac{17}{294} = 0.05782312925170068
\][/tex]
2. Calculate Pooled Proportion ([tex]\(\bar{p}\)[/tex]) and its Complement ([tex]\(\bar{q}\)[/tex]):
The pooled proportion [tex]\(\bar{p}\)[/tex] is calculated as follows:
[tex]\[
\bar{p} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{73 + 17}{299 + 294} = \frac{90}{593} = 0.15177065767284992
\][/tex]
Next, we find [tex]\(\bar{q}\)[/tex], which is:
[tex]\[
\bar{q} = 1 - \bar{p} = 1 - 0.15177065767284992 = 0.8482293423271501
\][/tex]
3. Calculate the Standard Error for the Difference in Proportions:
The standard error is given by:
[tex]\[
\text{Standard Error} = \sqrt{\frac{\bar{p} \cdot \bar{q}}{n_1} + \frac{\bar{p} \cdot \bar{q}}{n_2}}
\][/tex]
Substitute the values:
[tex]\[
\text{Standard Error} = \sqrt{\frac{0.15177065767284992 \cdot 0.8482293423271501}{299} + \frac{0.15177065767284992 \cdot 0.8482293423271501}{294}}
\][/tex]
[tex]\[
\text{Standard Error} = 0.02946922001557229
\][/tex]
4. Compute the Z-Score:
Given [tex]\( p_1 - p_2 = 0 \)[/tex], the z-score formula becomes:
[tex]\[
z = \frac{\left(\hat{p}_1 - \hat{p}_2\right) - 0}{\text{Standard Error}}
\][/tex]
Substitute the values into the formula:
[tex]\[
z = \frac{0.24414715719063546 - 0.05782312925170068}{0.02946922001557229} = 6.322665745495686
\][/tex]
5. Round the Z-Score to Two Decimal Places:
Finally, rounding the z-score to two decimal places gives us:
[tex]\[
z \approx 6.32
\][/tex]
Thus, the evaluated z-score is:
[tex]\[
z = 6.32
\][/tex]
