Question

15 mark for review
a researcher surveyed a random sample of 1,851 people from a certain state to estimate the percentage of people who support the governors recent approval of a bill. from the survey, the researcher estimated that 71% of people from the state support the governors approval of the bill, with an associated margin of error of 2.07%. the researcher repeated the survey with a random sample of people from the state that was double the original sample size. assuming the margins of error were calculated in the same way, which of the following is true about the margin of error associated with the estimate from the larger sample?
a the margin of error is between 2.07% and 3.07%.
b the margin of error is between 4.07% and 5.07%.
c the margin of error is greater than 5.07%.
d the margin of error is less than 2.07%.

Explanation:

Step1: Recall Margin of Error Relationship

The margin of error (ME) for a proportion is related to sample size \( n \) by the formula \( ME \propto \frac{1}{\sqrt{n}} \) (assuming same confidence level and population proportion).

Step2: Analyze Effect of Doubling Sample Size

If the new sample size \( n_2 = 2n_1 \) (where \( n_1 \) is original, \( n_2 \) is new), then the new margin of error \( ME_2 \) and original \( ME_1 \) satisfy \( \frac{ME_2}{ME_1}=\frac{\sqrt{n_1}}{\sqrt{n_2}}=\frac{\sqrt{n_1}}{\sqrt{2n_1}}=\frac{1}{\sqrt{2}}\approx0.707 \).

Step3: Compare New ME to Original

Since \( ME_2 = ME_1\times\frac{1}{\sqrt{2}} \), and \( ME_1 = 2.07\% \), then \( ME_2\approx2.07\%\times0.707\approx1.46\% \), which is less than \( 2.07\% \).

Answer:

D. The margin of error is less than 2.07%.