To solve this problem, we need to compute the test statistic to compare the population proportions of males and females who responded "Yes" to paying higher taxes for gasoline when the revenue goes to improving roadways.
### Step-by-Step Solution:
Step 1: Define the sample proportions
- The number of males surveyed ([tex]\( n_1 \)[/tex]) is 41.
- The number of males who responded "Yes" ([tex]\( x_1 \)[/tex]) is 12.
- The sample proportion of males ([tex]\( p_1 \)[/tex]) is calculated as:
[tex]\[
p_1 = \frac{x_1}{n_1} = \frac{12}{41} \approx 0.2927
\][/tex]
- The number of females surveyed ([tex]\( n_2 \)[/tex]) is 59.
- The number of females who responded "Yes" ([tex]\( x_2 \)[/tex]) is 21.
- The sample proportion of females ([tex]\( p_2 \)[/tex]) is calculated as:
[tex]\[
p_2 = \frac{x_2}{n_2} = \frac{21}{59} \approx 0.3559
\][/tex]
Step 2: Compute the pooled proportion
To calculate the pooled proportion ([tex]\( p_{pool} \)[/tex]), we use the combined data from both groups:
[tex]\[
p_{pool} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{12 + 21}{41 + 59} = \frac{33}{100} = 0.33
\][/tex]
Step 3: Compute the standard error
The standard error ([tex]\( SE \)[/tex]) for the difference between two proportions is given by:
[tex]\[
SE = \sqrt{p_{pool} \cdot (1 - p_{pool}) \cdot \left(\frac{1}{n_1} + \frac{1}{n_2}\right)} = \sqrt{0.33 \cdot (1 - 0.33) \cdot \left(\frac{1}{41} + \frac{1}{59}\right)}
\][/tex]
[tex]\[
SE \approx 0.0956
\][/tex]
Step 4: Compute the test statistic
The test statistic for comparing two proportions (using [tex]\( Z \)[/tex]-distribution) is calculated by:
[tex]\[
Z = \frac{p_1 - p_2}{SE} = \frac{0.2927 - 0.3559}{0.0956} \approx -0.6616
\][/tex]
Step 5: Round the test statistic
Round the test statistic to two decimal places:
[tex]\[
Z \approx -0.66
\][/tex]
### Final Answer:
The test statistic for the hypothesis test, rounded to two decimal places, is:
[tex]\[
-0.66
\][/tex]
