Certainly! Let’s break down the steps to compute the test statistic for comparing the population proportions of males and females who think it is rude to interact with a smartphone in a restaurant.
### Step-by-Step Solution:
1. Identify the Sample Proportions:
- Number of males surveyed ([tex]\( n_m \)[/tex]): 108
- Number of males who responded "Yes" ([tex]\( x_m \)[/tex]): 62
- Number of females surveyed ([tex]\( n_f \)[/tex]): 92
- Number of females who responded "Yes" ([tex]\( x_f \)[/tex]): 57
Calculate the sample proportions:
[tex]\[
\hat{p}_m = \frac{x_m}{n_m} = \frac{62}{108} \approx 0.574
\][/tex]
[tex]\[
\hat{p}_f = \frac{x_f}{n_f} = \frac{57}{92} \approx 0.620
\][/tex]
2. Compute the Pooled Proportion:
The pooled proportion combines the successes and the total sample size from both groups:
[tex]\[
\hat{p} = \frac{x_m + x_f}{n_m + n_f} = \frac{62 + 57}{108 + 92} = \frac{119}{200} = 0.595
\][/tex]
3. Calculate the Standard Error (SE) of the Difference in Proportions:
The standard error is given by:
[tex]\[
SE = \sqrt{\hat{p}(1 - \hat{p})\left(\frac{1}{n_m} + \frac{1}{n_f}\right)}
\][/tex]
Substituting the values:
[tex]\[
SE = \sqrt{0.595 \cdot (1 - 0.595) \left( \frac{1}{108} + \frac{1}{92} \right)}
\][/tex]
[tex]\[
SE = \sqrt{0.595 \cdot 0.405 \left( \frac{1}{108} + \frac{1}{92} \right)}
\][/tex]
[tex]\[
SE = \sqrt{0.595 \cdot 0.405 \left( 0.009259 + 0.01087 \right)}
\][/tex]
[tex]\[
SE \approx \sqrt{0.595 \cdot 0.405 \cdot 0.020129} \approx \sqrt{0.004854} \approx 0.0697
\][/tex]
4. Compute the Test Statistic (Z-score):
The test statistic is computed as:
[tex]\[
Z = \frac{\hat{p}_m - \hat{p}_f}{SE} = \frac{0.574 - 0.620}{0.0697} \approx -0.662
\][/tex]
5. Round the Test Statistic:
Round the Z-score to two decimal places:
[tex]\[
Z \approx -0.65
\][/tex]
### Conclusion:
The test statistic for comparing the population proportions of males and females who think interacting with a smartphone in a restaurant is rude is approximately -0.65.
