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Chapter 7: Problem 100

Find the sample size required for a margin of error of 3 percentage points, and then find one for a margin of error of \(1.5\) percentage points; for both, use a \(95 \%\) confidence level. Find the ratio of the larger sample size to the smaller sample size. To reduce the margin of error to half, by what do you need to multiply the sample size?

Short Answer

Expert verified
The sample sizes needed for a 3% and 1.5% margin of error are approximately 1068 and 4269, respectively. The larger sample size is four times the smaller one. Therefore, to halve the margin of error, the sample size must be multiplied by 4.

Step by step solution

01

Understand and use the formula for sample size

The formula for sample size is \(n = (Z^2 * P*(1-P)) / E^2\), where 'Z' is the Z-score, 'P' is the population proportion (unknown in our case and often set to 0.5 to provide the maximum sample size), and 'E' is the desired margin of error. Given 'E' is 3% or 0.03, 'P' is 0.5, and 'Z' is roughly 1.96 (which matches a 95% confidence level), substitute these values into the formula to calculate the required sample size.
02

Calculating for 3 percentage point margin of error

Substituting 'Z' = 1.96, 'P' = 0.5, and 'E' = 0.03 into the formula for sample size, gives \(n = (1.96^2 * 0.5 * (1-0.5)) / 0.03^2 = 1067.1\). Since sample size cannot be in decimals, round this up to the nearest whole number, which is 1068.
03

Calculating for 1.5 percentage point margin of error

Next, compute for a 1.5% or 0.015 margin of error. Applying 'Z' = 1.96, 'P' = 0.5, 'E' = 0.015 into the formula results in \(n = (1.96^2 * 0.5 * (1-0.5)) / 0.015^2 = 4268.4\). Once again, round up to the nearest whole number, which yields 4269.
04

Finding the ratio

The ratio of the larger sample size to the smaller one is simply \(4269 / 1068 = 4\). Thus, the larger sample size is 4 times the smaller one.
05

Determining the needed multiple

Since the sample size quadrupled when the margin of error was halved, this means that to reduce the margin of error to half, the sample size needs to be multiplied by 4.

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