Question

a researcher wishes to estimate, with 99% 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 4% 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 42% 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 = (round up to the nearest whole number as needed.) (b) what is the minimum sample size needed using a prior study that found that 42% of the respondents said they think congress is doing a good or excellent job? n = (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 raises 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 has no effect on the minimum sample size needed.

Explanation:

Step1: Determine the z - value

For a 99% confidence level, the z - value \(z_{\alpha/2}\) is 2.576. The margin of error \(E = 0.04\).

Step2: Calculate sample size when no prior estimate (\(p = 0.5\))

The formula for sample size \(n=\frac{z_{\alpha/2}^{2}\times p\times(1 - p)}{E^{2}}\). When no prior estimate is available, we use \(p=0.5\) (this gives the maximum variance).

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Rounding up, \(n = 1037\).

Step3: Calculate sample size with prior estimate (\(p = 0.42\))

Using \(p = 0.42\) and \(1-p=0.58\) in the formula \(n=\frac{z_{\alpha/2}^{2}\times p\times(1 - p)}{E^{2}}\)

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Rounding up, \(n = 1011\).

Answer:

(a) \(n = 1037\)
(b) \(n = 1011\)
(c) B. Having an estimate of the population proportion reduces the minimum sample size needed.