Question

a researcher at a major hospital wishes to estimate the proportion of the adult population of the united states that has high blood pressure. how large a sample is needed in order to be 99% confident that the sample proportion will not differ from the true proportion by more than 2%?

a. 3,394
b. 4,145
c. 8,289
d. 33

Explanation:

Step1: Determine the confidence level and margin of error

The confidence level is \(99\%\), so \(\alpha = 1 - 0.99=0.01\). Then \(\alpha/2 = 0.005\). From the standard normal distribution table, \(z_{\alpha/2}=z_{0.005} = 2.576\). The margin of error \(E = 0.02\).

Step2: Use the formula for sample size when estimating a proportion

When no prior estimate of the proportion \(p\) is available, we use \(p = 0.5\) (this gives the maximum value of \(p(1 - p)\)). The formula for sample size \(n\) is \(n=\dfrac{z_{\alpha/2}^{2}\times p(1 - p)}{E^{2}}\).
Substitute \(z_{\alpha/2}=2.576\), \(p = 0.5\), \(1 - p=0.5\) and \(E = 0.02\) into the formula:

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Since the sample size \(n\) must be an integer, we round up to \(n = 4145\) (due to the conservative approach in sample - size calculation for proportions when no prior estimate is known).

Answer:

B. \(4,145\)