Question

a researcher wishes to estimate, with 90% confidence, the population proportion of likely u.s. voters who think congress is doing a good or excellent job. her estimate must be accurate within 1% of the true proportion. (a) no preliminary estimate is available. find the minimum sample size needed. (b) find the minimum sample size needed, using a prior study that found that 40% of the respondents said they think congress is doing a good or excellent job. (c) compare the results from parts (a) and (b). (a) what is the minimum sample size needed assuming that no prior information is available? n = 6766 (round up to the nearest whole number as needed.) (b) what is the minimum sample size needed using a prior study that found that 40% of the respondents said they think congress is doing a good or excellent job? n = 6495 (round up to the nearest whole number as needed.) (c) how do the results from (a) and (b) compare? a. having an estimate of the population proportion has no effect on the minimum sample size needed. b. having an estimate of the population proportion reduces the minimum sample size needed. c. having an estimate of the population proportion raises the minimum sample size needed.

Explanation:

Step1: Recall the formula for sample size

The formula for sample size \(n\) when estimating a population proportion is \(n=\frac{z^{2}\cdot p(1 - p)}{E^{2}}\). For a \(90\%\) confidence level, the critical value \(z\) (from the standard normal distribution) is \(z = 1.645\) and the margin of error \(E=0.01\).

Step2: Analyze part (a)

When no preliminary estimate is available, we use \(p = 0.5\) (this gives the maximum value of \(p(1 - p)\)). Then \(n=\frac{(1.645)^{2}\times0.5\times(1 - 0.5)}{(0.01)^{2}}=\frac{2.706025\times0.25}{0.0001}=6765.0625\approx6766\) (rounded up).

Step3: Analyze part (b)

When \(p = 0.4\) (from the prior study), then \(n=\frac{(1.645)^{2}\times0.4\times(1 - 0.4)}{(0.01)^{2}}=\frac{2.706025\times0.24}{0.0001}=6494.46\approx6495\) (rounded up).

Step4: Compare (a) and (b)

Since \(6495<6766\), having an estimate of the population proportion (\(p = 0.4\) in part (b)) reduces the minimum sample size needed compared to using \(p = 0.5\) (when no estimate is available in part (a)).

Answer:

B. Having an estimate of the population proportion reduces the minimum sample size needed.