Question

The standard deviation of heights for students in a school is 0.81. A random sample of 50 students is taken, and the standard deviation of heights of the sample is 0.96. A researcher in charge of the study believes the standard deviation of heights for the school is greater than 0.81.

a) State the null and alternative hypotheses.

b) State the degree of freedom.

c) What can you conclude at the 5% significance level?

Answer 1

a)

The following null and alternative hypotheses need to be tested:

$H_0:\sigma^2\leq 0.81^2$

$H_1:\sigma^2>0.81^2$

This corresponds to a right-tailed test, for which a Chi-Square test for a single population variance will be used.

$$\chi^2=\dfrac{(n-1)s^2 }{\sigma^2}$$

b) Test of a single variance statistic where:

$n=50$ ​is sample size

$s=0.96$ ​ is sample standard deviation

$\sigma=0.81$ ​ is population standard deviation

$df=30-1$ ​degrees of freedom

$$\chi^2=\dfrac{(50-1)(0.96)^2 }{(0.81)^2}\approx68.8285$$

The p-value for right-tailed test, $df=50-1=49$ degrees of freedom,

$\chi^2=68.8285$ ​is $p= 0.032287.$

c) The significance level is $\alpha = 0.05.$

Since p-value is $p= 0.032287<0.05=\alpha,$ it is then concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population variance $\sigma^2$ is greater than $0.81^2,$ at the $\alpha=0.05$ significance level.