With [tex]\hat{p}_1 - \hat{p}_2 = 0.8 - 0.7 = 0.10[/tex] for different sample sizes, in parts (a) and (b), use StatKey or other technology to find the [tex]p[/tex]-value.

(a) [tex]\hat{p}_1 = \frac{24}{30} = 0.8[/tex] and [tex]\hat{p}_2 = \frac{14}{20} = 0.7[/tex]

Round your answer to two decimal places.

To find the p-value for the difference in sample proportions with the given data, we'll follow these steps:

1. Identify the Sample Proportions:
[tex]\[
\hat{p}_1 = \frac{24}{30} = 0.8
\][/tex]
[tex]\[
\hat{p}_2 = \frac{14}{20} = 0.7
\][/tex]

2. Calculate the Difference in Sample Proportions:
[tex]\[
\hat{p}_1 - \hat{p}_2 = 0.8 - 0.7 = 0.10
\][/tex]

3. Determine the Sample Sizes:
[tex]\[
n_1 = 30, \quad n_2 = 20
\][/tex]

4. Calculate the Standard Error (SE) for the Difference in Proportions:
The formula for the standard error of the difference between two proportions is:
[tex]\[
\text{SE} = \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}
\][/tex]
Substituting the values:
[tex]\[
\text{SE} = \sqrt{\frac{0.8 \times 0.2}{30} + \frac{0.7 \times 0.3}{20}} = 0.13
\][/tex]

5. Calculate the Z-Score:
The z-score for the difference in proportions is calculated as:
[tex]\[
z = \frac{\hat{p}_1 - \hat{p}_2}{\text{SE}} = \frac{0.10}{0.13} \approx 0.79
\][/tex]

6. Find the P-Value:
The p-value corresponds to the probability of obtaining a z-score as extreme as the observed value under the null hypothesis. Based on the z-score given, the p-value is approximately 0.16.

Therefore, the p-value for the difference between the two sample proportions is 0.16 when rounded to two decimal places.