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 = 4148 (round up to the nearest whole number as needed.) (b) what is the minimum sample size needed using a prior study that found that 70% of the respondents support labeling legislation? n = (round up to the nearest whole number as needed.)

Explanation:

Step1: Determine the z - value

For a 99% confidence level, the significance level \(\alpha=1 - 0.99 = 0.01\). Then \(\alpha/2=0.005\). Using the standard normal distribution table or a calculator, \(z_{\alpha/2}=z_{0.005} = 2.576\). The margin of error \(E = 0.02\).

Step2: Use the formula for sample size when proportion is known

The formula for the sample size \(n\) when estimating a proportion is \(n=\dfrac{z_{\alpha/2}^{2}\cdot p(1 - p)}{E^{2}}\). Given \(p = 0.7\) (from the prior study), \(1-p=0.3\), \(z_{\alpha/2}=2.576\), and \(E = 0.02\).

Substitute the values into the formula:

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Answer:

\(n = 3484\)