Question

a researcher wishes to estimate, with 99% confidence, the population proportion of adults who support labeling legislation for genetically modified organisms (gmos). 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 70% of the respondents said they support labeling legislation for gmos. (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: Determine the confidence level and margin of error

The confidence level is \(99\%\), so the significance level \(\alpha = 1 - 0.99=0.01\). Then \(\alpha/2 = 0.005\). From the standard normal distribution table, \(z_{\alpha/2}=z_{0.005} = 2.576\). The margin of error \(E = 0.02\). When no preliminary estimate is available, we use \(p = 0.5\) (this gives the maximum value of \(p(1 - p)\)).

Step2: Use the formula for sample size

The formula for sample size when estimating a proportion is \(n=\frac{z_{\alpha/2}^{2}\times p(1 - p)}{E^{2}}\). Substitute \(z_{\alpha/2}=2.576\), \(p = 0.5\), and \(E = 0.02\) into the formula:

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Answer:

\(n = 4148\)