To test the claim, we will conduct a hypothesis test for a proportion.
Step 1: Define the Hypotheses
- Null Hypothesis \( H_0 \): The proportion of investors who are confident in meeting their goals is equal to 10%, \( p = 0.10 \).
- Alternative Hypothesis \( H_a \): The proportion of investors who are confident in meeting their goals is smaller than 10%, \( p < 0.10 \).
This is a left-tailed test.
Step 2: Determine the Test Statistic
We use the following formula for the test statistic \( z \):
\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \]
Where:
- \( \hat{p} \) = sample proportion
- \( p_0 \) = hypothesized population proportion
- \( n \) = sample size
Given data:
- \( \hat{p} = \frac{4}{100} = 0.04 \)
- \( p_0 = 0.10 \)
- \( n = 500 \)
Now, we can calculate the standard error:
\[ \text{Standard Error (SE)} = \sqrt{\frac{p_0(1 - p_0)}{n}} = \sqrt{\frac{0.10(1 - 0.10)}{500}} = \sqrt{\frac{0.10 \cdot 0.90}{500}} = \sqrt{\frac{0.09}{500}} = \sqrt{0.00018} \approx 0.013416 \]
Now, substituting the values into the formula for \( z \):
\[ z = \frac{0.04 - 0.10}{0.013416} \approx \frac{-0.06}{0.013416} \approx -4.466 \]
Step 3: Calculate the p-value
To find the p-value for a left-tailed test, we can use the standard normal (Z) distribution.
We can look up the value of \( z = -4.466 \) in the Z-table or use statistical software/calculator. The p-value is the probability of observing a value as extreme or more extreme than the test statistic under the null hypothesis:
\[ \text{p-value} \approx 0.00000425 \text{ (about 0.0000043)} \]
Step 4: Make a Decision
Now, we compare the p-value with the significance level \( \alpha = 0.05 \):
- \( \text{p-value} \approx 0.0000043 < 0.05 \)
This indicates that we can reject the null hypothesis.
Summary of Results
- The test is: left-tailed
- The test statistic is: -4.466
- The p-value is: 0.0000 (to 4 decimal places: 0.0000)
- Based on this we: Reject the null hypothesis.
Thus, there is sufficient evidence to suggest that the proportion of investors who are confident in meeting their goals is smaller than 10%.