Question

a researcher wishes to estimate, with 99% confidence, the population proportion of families who eat fast food at least once per week. her estimate must be accurate within 4% of the population 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 40% of the respondents said they eat fast food four to six times per week. (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 = 1037 (round up to the nearest whole number as needed.) (b) what is the minimum sample size needed using a prior study that found that 40% of the respondents eat fast food at least once per week? n = 996 (round up to the nearest whole number as needed.) (c) how do the results from (a) and (b) compare? a. having an estimate of the population proportion reduces the minimum sample size needed. b. having an estimate of the population proportion raises the minimum sample size needed. c. having an estimate of the population proportion has no effect on the minimum sample size needed.

Explanation:

Step1: Analyze sample size formula for proportion

The formula for sample size \(n\) when estimating a proportion is \(n=\frac{z^{2}\cdot p(1 - p)}{E^{2}}\). For a \(99\%\) confidence level, the critical value \(z\) (from standard normal distribution tables) is \(z = 2.576\), and the margin of error \(E=0.04\).

Step2: Case (a) - No prior estimate

When no prior estimate is available, we use \(p = 0.5\) (maximizes \(p(1 - p)\)). Substitute \(p = 0.5\), \(z = 2.576\), \(E=0.04\) into the formula:

$$n=\frac{(2.576)^{2}\times0.5\times(1 - 0.5)}{(0.04)^{2}}=\frac{6.635776\times0.25}{0.0016}=\frac{1.658944}{0.0016}=1036.84\approx1037$$

Step3: Case (b) - Prior estimate \(p = 0.4\)

Substitute \(p = 0.4\), \(z = 2.576\), \(E = 0.04\) into the formula:

$$n=\frac{(2.576)^{2}\times0.4\times(1 - 0.4)}{(0.04)^{2}}=\frac{6.635776\times0.24}{0.0016}=\frac{1.59258624}{0.0016}=995.3664\approx996$$

Step4: Compare (a) and (b)

Since \(1037>996\), having an estimate of the population proportion (\(p = 0.4\) in part (b)) reduces the minimum sample size needed compared to using \(p=0.5\) (no prior estimate in part (a)).

Answer:

A. Having an estimate of the population proportion reduces the minimum sample size needed.