Question

A well-known brokerage firm executive claimed that 20% of investors are currently confident of meeting their investment goals. An XYZ Investor Optimism Survey, conducted over a two week period, found that in a sample of 700 people, 25% of them said they are confident of meeting their goals.

Test the claim that the proportion of people who are confident is larger than 20% at the 0.025 significance level.


The test is: (pick one)

right-tailed

two-tailed


The test statistic is:
(to 3 decimals)

The p-value is:
(to 4 decimals)

Based on this we:
Reject the null hypothesis
Fail to reject the null hypothesis

Answer 1

To test the claim that the proportion of people who are confident is larger than 20%, we can conduct a hypothesis test.

### Step 1: Define the Hypotheses

- Null Hypothesis (\(H_0\)): \(p \leq 0.20\)
- Alternative Hypothesis (\(H_a\)): \(p > 0.20\)

This is a **right-tailed test**.

### Step 2: Calculate the Test Statistic

We can use the z-test for proportions. The formula for the z-test is:

\[
z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}
\]

Where:
- \(\hat{p} = \frac{x}{n} = \frac{0.25 \times 700}{700} = 0.25\)
- \(p_0 = 0.20\)
- \(n = 700\)

Using this information:
1. Calculate \(\hat{p}\):
\(\hat{p} = 0.25\)

2. Calculate the standard error:
\[
SE = \sqrt{\frac{0.20(1 - 0.20)}{700}} = \sqrt{\frac{0.20 \times 0.80}{700}} \approx \sqrt{\frac{0.16}{700}} \approx \sqrt{0.00022857} \approx 0.0151
\]

3. Calculate the z-statistic:
\[
z = \frac{0.25 - 0.20}{0.0151} \approx \frac{0.05}{0.0151} \approx 3.311
\]

### Step 3: Calculate the p-value

The p-value for a right-tailed test can be found using the standard normal distribution:

To find the p-value:
1. Look up the z-score in the standard normal distribution table or use a calculator. For \(z = 3.311\), you can use a Z-table or an online calculator.

The corresponding p-value can be approximated as:
\[
p \approx 1 - P(Z < 3.311) \approx 1 - 0.9996 \approx 0.0004
\]

### Step 4: Conclusion

**Test Statistic:**
- \(z \approx 3.311\) (to 3 decimals)

**p-value:**
- \(p \approx 0.0004\) (to 4 decimals)

### Decision

- Since the p-value \(0.0004\) is less than the significance level \(\alpha = 0.025\), we reject the null hypothesis.

The final responses are:

**The test is:** right-tailed

**The test statistic is:** 3.311

**The p-value is:** 0.0004

**Based on this we:** Reject the null hypothesis