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 = (round up to the nearest whole number as needed.)

Explanation:

Step1: Determine the z - value

For a 95% confidence level, the z - value \(z_{\alpha/2}\) is 1.96. When no preliminary estimate is available, we use \(p = 0.5\) (this gives the maximum value of \(p(1 - p)\)). The margin of error \(E=0.02\).

Step2: Use the formula for sample size

The formula for the sample size \(n\) when estimating a proportion is \(n=\frac{z_{\alpha/2}^{2}\times p(1 - p)}{E^{2}}\).
Substitute \(z_{\alpha/2}=1.96\), \(p = 0.5\), and \(E = 0.02\) into the formula:

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Answer:

\(2401\)