To test the claim that the proportion of people who are confident is larger than 80%, we can set up our hypothesis testing as follows:
Null and Alternative Hypotheses:
- Null Hypothesis (H0): \( p \leq 0.80 \) (The proportion of confident investors is less than or equal to 80%)
- Alternative Hypothesis (H1): \( p > 0.80 \) (The proportion of confident investors is greater than 80%)
Type of Test:
Since we are checking if the proportion of confident investors is greater than 80%, this is a right-tailed test.
Test Statistic Calculation:
To calculate the test statistic for a proportion, we use the formula for the z-test statistic:
\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \]
Where:
- \( \hat{p} \): Sample proportion (86% or 0.86)
- \( p_0 \): Hypothesized population proportion (80% or 0.80)
- \( n \): Sample size (800)
Substituting the values into the equation:
\[ z = \frac{0.86 - 0.80}{\sqrt{\frac{0.80(1 - 0.80)}{800}}} \]
Calculating the standard error:
\[ \text{Standard Error} = \sqrt{\frac{0.80 \times 0.20}{800}} = \sqrt{\frac{0.16}{800}} = \sqrt{0.0002} \approx 0.0141421 \]
Now, substituting back to find \( z \):
\[ z \approx \frac{0.06}{0.0141421} \approx 4.243 \]
So, the test statistic is: 4.243 (to three decimals).
P-value Calculation:
To find the p-value for a right-tailed test with \( z = 4.243 \), we can use a z-table or a statistical software. Looking up \( z = 4.243 \):
The p-value is extremely small; for practical purposes, it can be considered close to 0.0000 (to four decimals).
Conclusion Based on P-value:
Given that the p-value is significantly smaller than the significance level of \( 0.005 \):
- We reject the null hypothesis.
Summary of Results:
-
Null and Alternative Hypotheses:
- \( H_0: p \leq 0.80 \)
- \( H_1: p > 0.80 \)
-
Test Type: Right-tailed
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Test Statistic: \( z \approx 4.243 \)
-
P-value: \( \approx 0.0000 \)
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Conclusion: Reject the null hypothesis.
Thus, the claim that the proportion of people who are confident is larger than 80% is supported by the data.
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