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A well-known brokerage firm executive claimed that at least [tex]$47\%$[/tex] of investors are currently confident of meeting their investment goals. An XYZ Investor Optimism Survey, conducted over a two-week period, found that out of 89 randomly selected people, 40 of them said they are confident of meeting their goals.

Suppose you have the following null and alternative hypotheses for a test you are running:
[tex]\[
\begin{array}{l}
H_0: p = 0.47 \\
H_a: p \neq 0.47
\end{array}
\][/tex]

Calculate the test statistic by hand, rounded to 3 decimal places.



Answer :

GinnyAnswer
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].

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