A survey asked randomly selected drivers if they would be willing to pay higher taxes for gasoline as long as all of the additional revenue went to improving roadways. Among the 41 males surveyed, 12 responded "Yes." Of the 59 females surveyed, 21 responded "Yes."

Compute the test statistic for a hypothesis test to compare the population proportions of males and females that would be willing to pay the higher tax. Round your answer to two decimal places.

Provide your answer below:



Answer :

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]