Changing from a 95% confidence interval estimate for a population proportion to a 99% confidence interval estimate, with all other things being equalSelect one:a. Decreases the interval size by 4%b. Increases the interval size by 4%c. Increases the interval size by 31%d. Decreases the interval size by 31%e. This question cannot be answered without knowing the sample size
Question
Changing from a 95% confidence interval estimate for a population proportion to a 99% confidence interval estimate, with all other things being equalSelect one:a. Decreases the interval size by 4%b. Increases the interval size by 4%c. Increases the interval size by 31%d. Decreases the interval size by 31%e. This question cannot be answered without knowing the sample size
Solution
The correct answer is c. Increases the interval size by 31%.
Here's why:
The confidence interval for a population proportion is calculated using the formula: p̂ ± Z*(√((p̂*(1-p̂))/n)), where p̂ is the sample proportion, Z* is the Z-score, and n is the sample size.
The Z-score is a measure of how many standard deviations an element is from the mean. For a 95% confidence interval, the Z-score is approximately 1.96. For a 99% confidence interval, the Z-score is approximately 2.58.
Increasing the confidence level from 95% to 99% means increasing the Z-score from 1.96 to 2.58. This increase in the Z-score widens the confidence interval.
The percentage increase in the interval size can be calculated as ((2.58-1.96)/1.96)*100% = 31.63%, which is approximately 31%.
Therefore, changing from a 95% confidence interval estimate for a population proportion to a 99% confidence interval estimate, with all other things being equal, increases the interval size by approximately 31%.
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