Question

For a random sample of 10 men, the mean head circumference is = 57.3 cm and the sample standard deviation xis s = 2 cm. What is the standard error of the mean?

Answer 1

Let $X=$ the mean head circumference. If $X\sim B(n, p),$ then

$$E(X)=\mu_X=np, \sigma_{X}^2 =np(1-p)$$

$$Var({1 \over n}\displaystyle\sum_{i=1}^nX_i)={1 \over n^2}\bigg(\displaystyle\sum_{i=1}^nVar(X_i)\bigg)=$$

$$={1 \over n^2}(n\sigma_{X}^2)={\sigma_X^2 \over n} =p(1-p)$$

The standard error of the mean

$$SE=\sqrt{{\sigma_X^2 \over n}}={\sigma_X \over \sqrt{n}}$$

Given that

$$E(X)=\mu_X=np=57.3\ cm,\sigma_X=2\ cm, n=10$$

Then

$$SE={\sigma_X \over \sqrt{n}}={2\ cm \over \sqrt{10}}\approx0.6325 cm$$

$$SE=0.6325\ cm$$