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Evaluate the following formula for [tex]\( p_1 - p_2 = 0 \)[/tex], [tex]\( x_1 = 73 \)[/tex], [tex]\( x_2 = 17 \)[/tex], [tex]\( n_1 = 299 \)[/tex], [tex]\( n_2 = 294 \)[/tex]:

[tex]\[
\hat{p}_1 = \frac{x_1}{n_1}, \quad \hat{p}_2 = \frac{x_2}{n_2}, \quad \bar{p} = \frac{x_1 + x_2}{n_1 + n_2}, \quad \text{and} \quad \bar{q} = 1 - \bar{p}
\][/tex]

[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]

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



Answer :

GinnyAnswer
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]

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