Question

the population standard deviation is estimated by the sample standard deviation of eight hours what is the probability that the difference between the sample mean and the population mean

Answer 1

For example:

The estimated difference between the sample mean and the population mean:

$\Delta=3$

Sample:

$n=100$

The sample standard deviation:

$\sigma=8$

The sample mean:

$\overline{x}=20$

The population mean:

$\mu=15$

Significance level:

$\alpha=5\ \%$

$H_0:$ $\Delta>3$

$H_a\le3$

Then:

$z=\frac{\overline{x}-\mu-\Delta}{\sigma/\sqrt{n}}=\frac{20-15-3}{8/\sqrt{100}}=0.03$

p-value:

$P(\Delta>3)=1-P(z<0.03)=1-0.51197=0.48803$

Since p-value > $\alpha$ , we accept the null hypothesis.