A simple random sample of voters will be taken in a large state. Researchers will use the methods of our course to construct an approximate 95% confidence interval for the percent of the state’s voters who will vote for Candidate X. The minimum sample size needed to ensure that the width of the interval (right end minus left end) is at most 6% is __________. (Fill in the blank with a positive integer; correct to the nearest 50 is OK.)
To determine the minimum sample size needed to ensure that the width of the confidence interval is at most 6%, we need to use the formula:
Sample Size (n) = (Z * sigma / E) ^ 2
Where:
- Z is the z-score corresponding to the desired confidence level
- sigma is the estimated standard deviation of the population proportion (if unknown, it can be conservatively estimated as 0.5)
- E is the desired margin of error (expressed as a decimal, in this case, 0.06 for 6%)
Since we are constructing a 95% confidence interval, the corresponding z-score is approximately 1.96. Let's plug in the values into the formula:
n = (1.96 * sigma / 0.06) ^ 2
As we don't have the exact value of sigma, we can conservatively estimate it as 0.5:
n = (1.96 * 0.5 / 0.06) ^ 2
Simplifying the equation:
n = (32.67) ^ 2
n ≈ 1067
Therefore, the minimum sample size needed to ensure that the width of the interval is at most 6% is approximately 1067 voters.