You want to obtain a sample to estimate a population mean. Based on previous evidence, you believe the population standard deviation is approximately σ = 44.6 .

Question

You would like to be 95% confident that your estimate is within 2.5 of the true population mean. How large of a sample size is required? 

Answer 1

This question is dealing with confidence intervals. The first thing I always ask myself when given a confidence interval problem is... are we dealing with means or proportions? One sample or two samples? In this scenario, there is clearly one sample and we are working with a population MEAN. There are slightly different formulas we follow depending on those two questions: So our formula for a confidence interval is the following...

CI = Point Estimate ± Margin of Error

CI = sample mean ± critical value x standard error

In this problem, we are only working with the margin of error part of the formula, as we are told the margin of error should be no more than 2.5.

Margin of Error <= 2.5 or critical value x standard error <= 2.5.

Most times we don't know the population standard deviation in a problem like this, so we would use a sample standard deviation and use t-procedures moving forward. Because we do assume to know the population standard deviation, we can actually use z-procedures here.

So our formula would be z* x (pop. SD/sqrt(sample size)) <= 2.5

Our critical value z* for a 95% confidence interval would be 1.96, as for a normal distribution, 95% of the data falls within 1.96 SD's of the mean. (this can be found at the bottom of table B).

1.96(44.6/sqrt(n)) <= 2.5

n is the sample size. We would use algebra to multiply both sides of the inequality by the square root of n, then divide both sides by 2.5, and finish the equation by squaring both sides.

n >= 1222.6, but we can't sample 0.6 people, so we should round this up (always round samples sizes up in these scenarios). So to be within 2.5 or our population mean, our sample size would need to be at least 1,223.