To conduct a hypothesis test for the proportion, we can use the following formulas:
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Sample proportion (p̂): \[ \hat{p} = \frac{x}{n} \] where \( x \) is the number of successes and \( n \) is the sample size.
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Test statistic (z): \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \] where \( p_0 \) is the hypothesized proportion.
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P-value: For a one-tailed test, the P-value is calculated from the z-score.
Step-by-step calculation:
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Identify the sample size (n):
The sample size \( n \) is not provided in your query, so let's denote \( n \) as the total sample size.
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Calculate the sample proportion (p̂):
Given \( x = 62 \) (number of successes), the sample proportion is: \[ \hat{p} = \frac{62}{n} \]
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Specify the hypothesized proportion: \[ p_0 = 0.33 \]
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Now substituting into the test statistic formula: \[ z = \frac{\frac{62}{n} - 0.33}{\sqrt{\frac{0.33(1 - 0.33)}{n}}} = \frac{\frac{62}{n} - 0.33}{\sqrt{\frac{0.33 \times 0.67}{n}}} \]
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We need to know sample size \( n \): Since we do not know \( n \), we won't be able to calculate an exact value for the test statistic and p-value.
However, if you provide the sample size \( n \), the calculations can proceed.
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Assuming we knew the sample size \( n \), you would further:
- Calculate the value of \( z \).
- Use the standard normal distribution to find the p-value corresponding to \( z \).
Please provide the sample size \( n \) to proceed with these calculations. Alternatively, I can show you how to calculate if you give a specific sample size.
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