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:
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Calculate \(\hat{p}\): \(\hat{p} = 0.25\)
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Calculate the standard error: \[ SE = \sqrt{\frac{0.20(1 - 0.20)}{700}} = \sqrt{\frac{0.20 \times 0.80}{700}} \approx \sqrt{\frac{0.16}{700}} \approx \sqrt{0.00022857} \approx 0.0151 \]
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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:
- 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
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