The following data shows heights in centimeters of 10 persons selected at random from a village. Find out whether, it would be reasonable or justifiable to suppose the height of that village population as 182 cm. The heights are 185, 181, 173, 177, 176, 174, 189, 185, 183, 177. [At 5% level of significance]

$n=10\\\bar x={\sum x\over n}={1800\over10}=180$

$s^2={\sum x^2-{(\sum x)^2\over n}\over n-1}={324260-324000\over9}=28.8888889$

The null and alternative hypothesis tested are,

$H_0:\mu=182\\vs\\H_1:\mu\not=182$

We shall apply the t distribution to perform this hypothesis test because the sample size $n=10\lt30$ and the population variance is unknown.

The test statistic is given as,

$t_c={\bar x-\mu\over{s\sqrt{n}} }={180-182\over {5.375\over\sqrt{10}}}={-2\over1.6997}=-1.1766968$

The critical value is the t distribution table value at $\alpha=0.05$ with $n-1=10-1=9$ degrees of freedom given as,

$t_{{0.05\over2},9}=t_{0.025,9}=2.262$

The null hypothesis is rejected if, $|t_c|\gt t_{0.025,9}$.

Since $|t_c|=1.1766968\lt t_{0.025,9}=2.262,$ we fail to reject the null hypothesis and conclude that there is sufficient evidence to show that the  height of this village population is 182 cm at 5% significance level.