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 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 34% 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 = 1692 (round up to the nearest whole number as needed.) (b) what is the minimum sample size needed using a prior study that found that 34% of the respondents said they think congress is doing a good or excellent job? n = 1519 (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: Recall the formula for sample size in proportion

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 corresponding to the confidence level, \(p\) is the estimated 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 \(E=0.02\) (since \(2\%=0.02\)).

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

When no prior estimate is available, we use \(p=0.5\) (this maximizes the value of \(p(1 - p)\)).
Substitute \(z = 1.645\), \(p = 0.5\), \(1-p=0.5\), and \(E = 0.02\) into the formula:

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Step3: Calculate sample size when \(p = 0.34\)

Here \(p = 0.34\) and \(1 - p=0.66\).
Substitute \(z = 1.645\), \(p = 0.34\), \(1 - p = 0.66\), and \(E = 0.02\) into the formula:

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Step4: Compare the two sample sizes

We have \(n_{1}=1692\) (no prior estimate) and \(n_{2}=1519\) (with prior estimate \(p = 0.34\)). Since \(1519<1692\)

Answer:

(a) \(n = 1692\)
(b) \(n = 1519\)
(c) B. Having an estimate of the population proportion reduces the minimum sample size needed.