Sure, let's go through the steps to calculate the test statistic for this hypothesis test step by step.
### Step 1: Sample Proportion
First, we need to determine the sample proportion ([tex]\(\hat{p}\)[/tex]). This is done by dividing the number of successes (people confident of meeting their goals) by the total sample size.
[tex]\[
\hat{p} = \frac{40}{89} \approx 0.449
\][/tex]
### Step 2: Null Hypothesis Proportion
The null hypothesis states that the proportion of all investors confident of meeting their goals is [tex]\(0.47\)[/tex].
[tex]\[
p_0 = 0.47
\][/tex]
### Step 3: Standard Error (SE)
Next, we calculate the standard error of the sampling distribution of the sample proportion. The formula for the standard error when dealing with proportions is:
[tex]\[
SE = \sqrt{\frac{p_0(1 - p_0)}{n}}
\][/tex]
Where:
- [tex]\(p_0\)[/tex] is the null hypothesis proportion
- [tex]\(n\)[/tex] is the sample size
Substitute the given values into the formula:
[tex]\[
SE = \sqrt{\frac{0.47 \times (1 - 0.47)}{89}} \approx 0.053
\][/tex]
### Step 4: Calculate the Test Statistic (Z)
The test statistic (Z) is calculated using the formula:
[tex]\[
Z = \frac{\hat{p} - p_0}{SE}
\][/tex]
Putting in the known values:
[tex]\[
Z = \frac{0.449 - 0.47}{0.053} \approx -0.389
\][/tex]
### Summary
- Sample Proportion ([tex]\(\hat{p}\)[/tex]): 0.449
- Null Hypothesis Proportion ([tex]\(p_0\)[/tex]): 0.47
- Standard Error (SE): 0.053
- Test Statistic (Z): -0.389
Thus, the test statistic rounded to three decimal places is approximately [tex]\(-0.389\)[/tex].
