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 = 664 (round up to the nearest whole number as needed.)
(b) what is 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?
n = 511 (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 has no effect on 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 raises the minimum sample size needed.

Explanation:

Step1: Analyze the sample size formula for proportion

The formula for sample size \(n\) when estimating a proportion is \(n=\frac{z^{2}\cdot p(1 - p)}{E^{2}}\). For a \(99\%\) confidence level, the critical value \(z\) (from standard normal distribution tables) is \(z = 2.576\), and the margin of error \(E=0.05\).

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

When no prior estimate is available, we use \(p=0.5\) (this maximizes \(p(1 - p)\)). Substitute into the formula: \(n=\frac{(2.576)^{2}\times0.5\times(1 - 0.5)}{(0.05)^{2}}=\frac{6.635776\times0.25}{0.0025}=663.5776\approx664\) (rounded up).

Step3: Calculate sample size with prior estimate (\(p = 0.26\))

Substitute \(p = 0.26\) into the formula: \(n=\frac{(2.576)^{2}\times0.26\times(1 - 0.26)}{(0.05)^{2}}=\frac{6.635776\times0.1924}{0.0025}=\frac{1.277723}{0.0025}=511.0892\approx511\) (rounded up).

Step4: Compare the two sample sizes

We have \(n_{1}=664\) (no prior estimate) and \(n_{2}=511\) (with prior estimate \(p = 0.26\)). Since \(511<664\), having an estimate of the population proportion reduces the minimum sample size needed.

Answer:

A. \(n = 664\)
B. \(n = 511\)
C. B. Having an estimate of the population proportion reduces the minimum sample size needed.