A researcher wishes to estimate, with $90 \%$ confidence, the population proportion of adults who support labeling legislation for genetically modified organisms (GMOs). Her estimate must be accurate within $5 \%$ of the true proportion.
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(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 $69 \%$ 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?
$\mathrm{n}=\square$ (Round up to the nearest whole number as needed.)
(b) What is the minimum sample size needed using a prior study that found that $69 \%$ of the respondents support labeling legislation?
$n=\square$ (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 raises the minimum sample size needed.
B. Having an estimate of the population proportion reduces the minimum sample size needed.
C. Having an estimate of the population proportion has no effect on the minimum sample size needed.
Answer
Comparing the results from parts (a) and (b), we see that having an estimate of the population proportion raises the minimum sample size needed. So, the answer is A.
Step 1 ：Given that the Z-score for a 90% confidence level is approximately 1.645 and the desired margin of error is 5% or 0.05, we can use the formula for the minimum sample size needed: \(n = \frac{Z^2 \cdot p \cdot (1-p)}{E^2}\)
Step 2 ：When no preliminary estimate is available, we use the conservative estimate of 0.5 for the population proportion. Substituting these values into the formula, we get: \(n = \frac{(1.645)^2 \cdot 0.5 \cdot (1-0.5)}{(0.05)^2} = 270.6025\)
Step 3 ：Since we can't have a fraction of a person, we round up to the nearest whole number. So, the minimum sample size needed is \(\boxed{271}\)
Step 4 ：If a prior study found that 69% of the respondents support labeling legislation, we use this as our estimate for the population proportion. Substituting p = 0.69 into the formula, we get: \(n = \frac{(1.645)^2 \cdot 0.69 \cdot (1-0.69)}{(0.05)^2} = 323.9824\)
Step 5 ：Again, we round up to the nearest whole number. So, the minimum sample size needed is \(\boxed{324}\)
Step 6 ：Comparing the results from parts (a) and (b), we see that having an estimate of the population proportion raises the minimum sample size needed. So, the answer is A.
