Question

a researcher wishes to estimate, with 95% 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 2% 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 = 2401 (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 population proportion is \(n=\frac{z^{2}\cdot p(1 - p)}{E^{2}}\). For a 95% confidence level, the critical value \(z = 1.96\), and the margin of error \(E=0.02\).

Step2: Substitute values for part (b)

Given \(p = 0.36\) (from the prior study), then \(1-p=1 - 0.36=0.64\).
Substitute \(z = 1.96\), \(p = 0.36\), \(1 - p=0.64\), and \(E = 0.02\) into the formula:

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Step3: Round up the value

Since the sample size \(n\) must be a whole number, we round up \(2212.75\) to \(2213\).

Answer:

\(n = 2213\)