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 36% 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 36% of the respondents said they think congress is doing a good or excellent job? n = □ (round up to the nearest whole number as needed.)

Explanation:

Step1: Determine the formula for sample size

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

Step2: Substitute values for part (b)

Given \(p = 0.36\) (from the prior study), then \(1-p=1 - 0.36=0.64\).
Substitute into the formula:

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Answer:

\(n = 6235\) (since we round up to the nearest whole number)