A random sample of 100 people was taken. Eighty of the people in the sample favored Candidate A. We are interested in determining whether or not the proportion of the population in favor of Candidate A is significantly more than 75%.The test statistic is
Question
A random sample of 100 people was taken. Eighty of the people in the sample favored Candidate A. We are interested in determining whether or not the proportion of the population in favor of Candidate A is significantly more than 75%.The test statistic is
Solution
The test statistic in this case can be calculated using the formula for a one-sample z-test for proportions. Here are the steps:
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Identify a sample proportion: Since 80 out of 100 people favored Candidate A, the sample proportion (p̂) is 0.80.
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Identify a population proportion: We are testing whether the population proportion (p) is significantly more than 0.75.
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Calculate the standard error (SE): The formula for the standard error of a proportion is sqrt [ p(1 - p) / n ] , where n is the size of the sample. In this case, we are testing the hypothesis that p > 0.75, so we use 0.75 for p in the formula. So, SE = sqrt [ 0.75 * (1 - 0.75) / 100 ] = 0.0433.
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Calculate the z-score: The formula for the z-score is (p̂ - p) / SE. So, z = (0.80 - 0.75) / 0.0433 = 1.155.
This z-score tells us how many standard errors our sample proportion is from the population proportion. A z-score of 1.155 is 1.155 standard errors above the population proportion.
- Determine the significance: The significance of this z-score can be determined by looking it up in a standard normal (z) table or using a z-score calculator. If the p-value associated with this z-score is less than the significance level (typically 0.05), then we can reject the null hypothesis and conclude that the proportion of the population in favor of Candidate A is significantly more than 75%.
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