Question

A random sample of 11 observations was taken from normal population. The sample mean and standard deviation are 74.5 and 9 accordingly. Can we infer at 5% significance level that the population mean is greater than 70?

Answer 1

$Let~\overline{x}=74.5,~\sigma=9,~n=11\\ \mu=70,~\alpha=0.05$

$z=\frac{\overline{x}-\mu}{\sigma}\sqrt{n}=\frac{74.5-70}{9}\sqrt{11}\approx1.658$

$P(z>1.658)=1-P(z<1.658)\\ P(z<1.658)>P(z<1.65)$

From z-score table: $P(z<1.65)=0.9505$

$P(z<1.658)>0.9505\\ P(z>1.658)<1-0.9505=0.0495<0.05=\alpha$

Since $P(z>1.658)<\alpha,$ we can infer that the population mean is greater than 70.