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 5% 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 26% 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.)

Explanation:

Step1: Determine the z - value

For a 99% confidence level, the significance level \(\alpha=1 - 0.99 = 0.01\). Then \(\alpha/2=0.005\). Looking up in the standard normal distribution table, \(z_{\alpha/2}=z_{0.005} = 2.576\). The margin of error \(E = 0.05\). When no prior estimate is available, we use \(p=0.5\) (this maximizes the value of \(p(1 - p)\)).

Step2: Use the formula for sample size in proportion

The formula for sample size \(n=\frac{z_{\alpha/2}^{2}\times p(1 - p)}{E^{2}}\). Substitute \(z_{\alpha/2}=2.576\), \(p = 0.5\), \(1-p=0.5\), and \(E = 0.05\) into the formula:

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Answer:

\(n = 664\)