Question

a researcher wishes to be 95% confident that her estimate of the true proportion of individuals who travel overseas is within 4% of the true proportion. find the sample necessary if in a prior study, a sample of 200 people showed that 40 traveled overseas last year. 385 if no estimate of the sample proportion is available, how large should the sample be? 200

Explanation:

Step1: Determine the formula for sample size when no prior estimate is available

When no estimate of the sample proportion \(p\) is available, we use \(p = 0.5\) (this gives the maximum value of \(p(1 - p)\)). The formula for sample size \(n\) in estimating a proportion is \(n=\frac{z^{2}_{\alpha/2}\times p(1 - p)}{E^{2}}\)

For a 95% confidence level, \(\alpha=1 - 0.95=0.05\), and \(\alpha/2=0.025\). The \(z\)-value \(z_{\alpha/2}=z_{0.025}\). From the standard normal table, \(z_{0.025} = 1.96\). The margin of error \(E = 0.04\)

Step2: Substitute the values into the formula

Substitute \(z_{\alpha/2}=1.96\), \(p = 0.5\), \(1-p=0.5\) and \(E = 0.04\) into the formula:

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

Answer:

\(n = 601\)