Question

A random sample of 10 college students was drawn from a large university. Their ages are 22, 17, 27, 20, 23, 19, 24, 18, 19, and 24 years, with a sample standard deviation of 3.2. Suppose that you want to test whether the population mean age differs from 20. What is your decision based on a test at a 1% significance level

Answer 1

Solution :

Given :

The sample mean = 21.3

Standard deviation = 3.2

The null hypothesis is : $H_0: \mu =20$

The alternate hypothesis : $H_a:\mu \neq20$

This is a two tailed test, for which a $\text{t-test for one mean}$ with an unknown population of a standard deviation is being used.

Now the significance level, $\alpha = 0.1$, as well as the critical value for a two tailed test is $t_c = 1.833$

The rejection region is $R = \{ t:|t| > 1.833 \}$

The t-statistic is computed as follows :

$t=\frac{\overline x - \mu_0}{s/ \sqrt n}$

$t=\frac{21.3-20}{3.2/ \sqrt{10}}$

 = 1.285

Since it is observed that $|t| = 1.285 \leq t_c=1.833$, it is then concluded that  $\text{the null hypothesis is not rejected.}$

The p-value is p=0.231 and since p 0.231 ≥ 0.1, it is concluded that $\text{the null hypothesis is not rejected.}$

Conclusion

Thus we concluded that $\text{null hypothesis}$ $H_0$ is not rejected. Therefore, the population mean $\mu$ is different than 20, at the 0.1 significance level.