You are sampling from a population with a known standard deviation of 20 and want to construct a 95%
confidence interval for the mean with a margin of error of no more than 4. What is the smallest sample size
that will produce such an interval?
To find the smallest sample size that will produce a confidence interval with a margin of error no more than 4, we can use the following formula:
n = (Z * σ / E)^2
where:
- n is the sample size
- Z is the Z-score corresponding to the desired confidence level (in this case, 95%)
- σ is the known population standard deviation
- E is the desired margin of error
First, let's find the Z-score for a 95% confidence level. The Z-score can be looked up in a Z-table or calculated using statistical software. For a 95% confidence level, the Z-score is approximately 1.96.
Now, let's substitute the known values into the formula:
n = (1.96 * 20 / 4)^2
n = 9.8^2
n ≈ 96.04
Since the sample size should be a whole number, we will round up to the next integer.
Therefore, the smallest sample size that will produce a 95% confidence interval for the mean with a margin of error no more than 4 is 97.