Question

In a random sample of 500 observations drawn from a normal population, the sample mean and sample standard deviation were calculated as X =350 and s = 100.

a) Estimate the population mean with 99% confidence.

b) Repeat part (a) assuming that you know that the population standard deviation is  = 100.

c) Explain why the interval estimates were virtually identical. 

Answer 1

a) 99% interval estimate of $\mu$ :

$\overline{x}\pm z_{\alpha/2}s/\sqrt{n}=350\pm 2.586\cdot100/\sqrt{500}=$

$=350\pm 11.564=(338.436,361.564)$

b)

$\overline{x}\pm z_{\alpha/2}s=350\pm 2.586\cdot100=350\pm 258.6=(91.4,608.6)$

c) The interval estimates were virtually identical because the variable is normally distributed and the sample size is large (more than 30).