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### Proportions Question

A survey asked randomly selected people whether they thought it was rude for people to interact with their smartphones while in a restaurant. Among the 108 males surveyed, 62 responded "Yes." Of the 92 females surveyed, 57 responded "Yes."

Compute the test statistic for a hypothesis test to compare the population proportions of males and females that think interacting with a smartphone in a restaurant is rude. Round your answer to two decimal places.



Answer :

GinnyAnswer
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.

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