Please help mee

Question

Please An IQ test is designed so that the mean is 100 and the standard deviation is 10 for the population of normal adults. Find the sample size necessary to estimate the mean IQ score of statistics students such that it can be said with 95% confidence that the sample mean is within 8 IQ points of the true mean. Assume that σ =10 and determine the required sample size using technology.

Then determine if this is a reasonable sample size for a real world calculation.

The required sample size is ?. (Round up to the nearest integer.)

Answer 1

Because you are given the standard deviation, σ, of the population, you can use z rather than t distribution.

For 95% CI, you need a z of 1.96. The sample deviation is estimated as σ/sqrt(n).

We want a 95% confidence interval for xbar - μ = ±8

1.96 = (xbar - μ)/(σ/sqrt(n)) = 8sqrt(n)/10

n =(1.96*10/8)² = 6.0025 (I expect that if you use the calculator, you will get closer to 6. It seems 6 is a better answer than going up to 7. I ran a hypothesis test and with sample of 6 and Wolfram said it passed at the 5% level two-tailed.

This number for the sample size is very small - usually 30+ for the central limit theorem to be valid. The CLT allows you to treat the sample distribution as normal, even if the original isn't.