Question

The standard IQ test has a mean of 101 and a standard deviation of 16. We want to be 98% certain that we are within 5 IQ points of the true mean. Determine the required sample size​

Answer 1

We want to find the sample size n such that

Pr[|µ - 101| < 5] = 0.98

where µ is ... some mean. It can be the mean for either the true population or the sample. It's unclear which is which from the given info, but that's not actually important.

In the inequality, divide both sides by the standard error, σ/√n = 16/√n :

Pr[|µ - 101|/(16/√n) < 5/(16/√n)] = 0.98

Use a lookup table to find the z-score corresponding to 0.98 :

⇒   5/(16/√n) ≈ 2.05375

Solve for n :

⇒   n ≈ 43.1911 ≈ 44