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 1% 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: Find the value of \(z_{\alpha/2}\)

For a 90% confidence level, \(\alpha = 1 - 0.90=0.10\), and \(\alpha/2 = 0.05\). Using the standard normal distribution table or a calculator, \(z_{\alpha/2}=z_{0.05} = 1.645\)

Step2: Use the formula for sample size when no prior estimate is available

The formula for sample size \(n\) when estimating a proportion is \(n=\frac{z_{\alpha/2}^{2}\cdot p(1 - p)}{E^{2}}\). When no prior estimate \(p\) is available, we use \(p = 0.5\) (this maximizes the value of \(p(1 - p)\)). The margin of error \(E=0.01\)
Substitute the values into the formula:

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Step3: Round up the value of \(n\)

Since the sample size \(n\) must be an integer, we round up \(6765.0625\) to \(6766\)

Answer:

\(6766\)