The confidence interval formula for estimating a population proportion p is p^±E, where The number zα∕2 is determined by the desired level of confidence. To say that we wish to estimate the population proportion to within a certain number of percentage points means that we want the margin of error E to be no larger than that number (expressed as a proportion). Thus we obtain the minimum sample size needed by solving the displayed equation for n.
The estimated minimum sample size n needed to estimate a population proportion p to within E at 100(1−α)% confidence is n=(zα∕2)2p^(1−p^)E2 (rounded up)There is a dilemma here: the formula for estimating how large a sample to take contains the number p^, which we know only after we have taken the sample. There are two ways out of this dilemma. Typically the researcher will have some idea as to the value of the population proportion p, hence of what the sample proportion p^ is likely to be. For example, if last month 37% of all voters thought that state taxes are too high, then it is likely that the proportion with that opinion this month will not be dramatically different, and we would use the value 0.37 for p^ in the formula. The second approach to resolving the dilemma is simply to replace p^ in the formula by 0.5. This is because if p^ is large then 1−p^ is small, and vice versa, which limits their product to a maximum value of 0.25, which occurs when p^=0.5. This is called the most conservative estimateThe estimate obtained using p^=0.5, which gives the largest estimate of n., since it gives the largest possible estimate of n.
Find the necessary minimum sample size to construct a 98% confidence interval for p with a margin of error E = 0.05,
Solution: Confidence level 98% means that α=1−0.98=0.02 so α∕2=0.01. From the last line of Figure 12.3 "Critical Values of " we obtain z0.01=2.326.
A dermatologist wishes to estimate the proportion of young adults who apply sunscreen regularly before going out in the sun in the summer. Find the minimum sample size required to estimate the proportion to within three percentage points, at 90% confidence. Solution: Confidence level 90% means that α=1−0.90=0.10 so α∕2=0.05. From the last line of Figure 12.3 "Critical Values of " we obtain z0.05=1.645. Since there is no prior knowledge of p we make the most conservative estimate that p^=0.5. To estimate “to within three percentage points” means that E = 0.03. Then n=(zα∕2)2p^(1−p^)E2=(1.645)2(0.5)(1−0.5)0.032=751.6736111which we round up to 752.
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