A political pollster is conducting an analysis of sample results in order to make predictions on election night.Assuming a two-candidate election,if a specific candidate receives at least 54%of the vote in the sample,that candidate will be forecast as the winner of the election.You select a random sample of 100 voters. Suppose that the sample size was increased to 400.Repeat process(a)through(c),using this new sample size.Comment on the difference. The probability isthat a candidate will be forecast as the winner when the popuio percentage of her vote is50.1%. Choose the correct answer below. A. Increasing the sample size by a factor of 4 increases the standard error by a factor of 2. Changing the standard error doubles the magnitude of the standardized Z-value. B. Increasing the sample size by a factor of 4 decreases the standard error by a factor of 2. Changing the standard error doubles the magnitude of the standardized Z-value. C. Increasing the sample size by a factor of 4 increases the standard error by a factor of 2. Changing the standard error decreases the standardized Z-value to half of its original value. D. Increasing the sample size by a factor of 4 decreases the standard error by a factor of 2. Changing the standard error decreases the standardized Z-value to half of its original value.
Question
A political pollster is conducting an analysis of sample results in order to make predictions on election night.Assuming a two-candidate election,if a specific candidate receives at least 54%of the vote in the sample,that candidate will be forecast as the winner of the election.You select a random sample of 100 voters. Suppose that the sample size was increased to 400.Repeat process(a)through(c),using this new sample size.Comment on the difference. The probability isthat a candidate will be forecast as the winner when the popuio percentage of her vote is50.1%. Choose the correct answer below. A. Increasing the sample size by a factor of 4 increases the standard error by a factor of 2. Changing the standard error doubles the magnitude of the standardized Z-value. B. Increasing the sample size by a factor of 4 decreases the standard error by a factor of 2. Changing the standard error doubles the magnitude of the standardized Z-value. C. Increasing the sample size by a factor of 4 increases the standard error by a factor of 2. Changing the standard error decreases the standardized Z-value to half of its original value. D. Increasing the sample size by a factor of 4 decreases the standard error by a factor of 2. Changing the standard error decreases the standardized Z-value to half of its original value.
Solution
The correct answer is D.
Increasing the sample size by a factor of 4 decreases the standard error by a factor of 2. This is because the standard error is inversely proportional to the square root of the sample size. Therefore, if you quadruple the sample size, the standard error will be halved.
Changing the standard error in this way decreases the standardized Z-value to half of its original value. This is because the Z-value is calculated by dividing the difference between the sample mean and the population mean by the standard error. If the standard error is halved, the Z-value will also be halved, assuming the difference between the sample mean and the population mean remains constant.
Therefore, with a larger sample size, the Z-value decreases, making it less likely that a candidate with 50.1% of the vote in the sample will be forecast as the winner. This is because a smaller Z-value indicates that the sample mean is closer to the population mean, suggesting less evidence to reject the null hypothesis that the two candidates are equally popular.
Similar Questions
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