To perform the hypothesis test, we need to calculate the test statistic and then the p-value for the sample data provided.
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Calculating the Sample Proportion (\( \hat{p} \)): \[ \hat{p} = \frac{x}{n} = \frac{76}{212} \approx 0.3585 \]
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Calculating the Test Statistic (z): The formula for the z-test statistic for a proportion is: \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \] Where:
- \( \hat{p} \) is the sample proportion.
- \( p_0 \) is the null hypothesis proportion (0.4 in this case).
- \( n \) is the sample size (212).
Substituting in the values: \[ z = \frac{0.3585 - 0.4}{\sqrt{\frac{0.4(1 - 0.4)}{212}}} = \frac{-0.0415}{\sqrt{\frac{0.4 \times 0.6}{212}}} = \frac{-0.0415}{\sqrt{\frac{0.24}{212}}} = \frac{-0.0415}{\sqrt{0.00113208}} = \frac{-0.0415}{0.0336} \approx -1.237 \]
Test Statistic \( z \) (accurate to three decimal places) = -1.237
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Calculating the p-value: Since this is a one-tailed test (we are looking for \( H_a: p < 0.4 \)), we find the p-value associated with \( z = -1.237 \).
Using standard normal distribution tables or a calculator, we get: \[ p\text{-value} = P(Z < -1.237) \approx 0.1087 \]
p-value (accurate to four decimal places) = 0.1087
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Decision Making: We compare the p-value to the significance level \( \alpha = 0.01 \). Since \( 0.1087 > 0.01 \), we do not reject the null hypothesis.
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Final Conclusion:
- The decision is to fail to reject the null.
- Thus, the final conclusion is:
- There is not sufficient evidence to warrant rejection of the claim that the population proportion is less than 0.4.
Summary of Results:
- Test Statistic (z) = -1.237
- p-value = 0.1087
- Decision: fail to reject the null
- Conclusion: There is not sufficient evidence to warrant rejection of the claim that the population proportion is less than 0.4.
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