To conduct the hypothesis test and calculate the test statistic and p-value, we proceed as follows:
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Identify the sample proportion (p̂): \[ \hat{p} = \frac{x}{n} = \frac{135}{217} \approx 0.6212 \]
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Identify the null hypothesis proportion (p_0): \[ p_0 = 0.59 \]
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Calculate the standard deviation of the sampling distribution under the null hypothesis: \[ \sigma_{\hat{p}} = \sqrt{\frac{p_0(1 - p_0)}{n}} = \sqrt{\frac{0.59(1 - 0.59)}{217}} = \sqrt{\frac{0.59 \times 0.41}{217}} \approx \sqrt{\frac{0.2419}{217}} \approx \sqrt{0.001113} \approx 0.0333 \]
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Calculate the test statistic (z): \[ z = \frac{\hat{p} - p_0}{\sigma_{\hat{p}}} = \frac{0.6212 - 0.59}{0.0333} \approx \frac{0.0312}{0.0333} \approx 0.9364 \] (rounded to three decimal places: \( z \approx 0.936 \))
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Find the p-value: The p-value is calculated using the standard normal distribution. Since this is a one-tailed test, we find \( P(Z > z) \): \[ p\text{-value} = P(Z > 0.936) \approx 1 - P(Z \leq 0.936) \approx 1 - 0.8258 \approx 0.1742 \] (rounded to four decimal places: \( p\text{-value} \approx 0.1742 \))
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Compare the p-value with alpha:
- \( \alpha = 0.10 \)
- Since \( 0.1742 > 0.10 \), we conclude that the p-value is greater than alpha.
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Decision based on p-value: Since the p-value is greater than alpha, we fail to reject the null hypothesis.
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Final conclusion: The conclusion is that: There is not sufficient evidence to warrant rejection of the claim that the population proportion is greater than 0.59.
Final Answers:
- Test statistic = 0.936
- p-value = 0.1742
- The p-value is: B. greater than alpha
- This test statistic leads to a decision to: C. fail to reject the null
- As such, the final conclusion is that: D. There is not sufficient sample evidence to support the claim that the population proportion is greater than 0.59.