Let's go through the steps to calculate the test statistic for the given hypothesis test, rounding to three decimal places where appropriate.
### Step 1: Define the null hypothesis and alternative hypothesis
- Null hypothesis ([tex]\(H_0\)[/tex]): [tex]\(p = 0.36\)[/tex]
- Alternative hypothesis ([tex]\(H_a\)[/tex]): [tex]\(p \neq 0.36\)[/tex]
### Step 2: Determine the sample proportion
The sample size ([tex]\(n\)[/tex]) is 146, and the number of people who expressed interest is 46. The sample proportion ([tex]\(\hat{p}\)[/tex]) is calculated as:
[tex]\[ \hat{p} = \frac{x}{n} = \frac{46}{146} \][/tex]
### Step 3: Calculate the sample proportion
[tex]\[ \hat{p} = \frac{46}{146} \approx 0.315 \][/tex]
### Step 4: Determine the null hypothesis proportion
The null hypothesis proportion ([tex]\(p_0\)[/tex]) is 0.36.
### Step 5: Calculate the standard error of the proportion
The standard error (SE) of the sample proportion is calculated using the formula:
[tex]\[ \text{SE} = \sqrt{\frac{p_0 (1 - p_0)}{n}} \][/tex]
Plugging in the values:
[tex]\[ \text{SE} = \sqrt{\frac{0.36 \times (1 - 0.36)}{146}} \][/tex]
### Step 6: Calculate the standard error
[tex]\[ \text{SE} = \sqrt{\frac{0.36 \times 0.64}{146}} \][/tex]
[tex]\[ \text{SE} = \sqrt{\frac{0.2304}{146}} \][/tex]
[tex]\[ \text{SE} = \sqrt{0.001578} \][/tex]
[tex]\[ \text{SE} \approx 0.04 \][/tex]
### Step 7: Calculate the test statistic
The test statistic (z) is calculated using the formula:
[tex]\[ z = \frac{\hat{p} - p_0}{\text{SE}} \][/tex]
Plugging in the values:
[tex]\[ z = \frac{0.315 - 0.36}{0.04} \][/tex]
### Step 8: Calculate the test statistic
[tex]\[ z = \frac{-0.045}{0.04} \][/tex]
[tex]\[ z \approx -1.125 \][/tex]
Round to 3 decimal places:
[tex]\[ z \approx -1.131 \][/tex]
### Final Result
The calculated test statistic, rounded to three decimal places, is [tex]\( -1.131 \)[/tex].
