Question

A random sample of 40 bys showed a mean weight of 120 lbs with a standard deviation of 12 lbs. The standard weight of their average age is about 130 lbs. Test the hypothesis that the selected boys are significantly lighter that the standard weight at 0.01 level of significance.

Answer 1

We are given that,

$n=40,\space \bar{x}=120,\space s=12, \mu=130$

The hypotheses tested are,

$H_0:\mu=130\space vs\space H_1:\mu\lt130$

We shall apply the student's t distribution to perform this test as follows,

The test statistic is given as,

$t^*=(\bar{x}-\mu)/(s/\sqrt{n})$

Now,

$t^*=(120-130)/(12/\sqrt{40})=-10/1.8973666=-5.2704628$

The critical value $t_{\alpha, (n-1)}$ has $n-1=40-1=39$ degrees of freedom at $\alpha=0.01$ and can be determined using the $R$ command given below.

> qt(0.01,39)

[1] -2.425841

Therefore, the critical value, $t_{0.01,39}= -2.425841$

$t^*$ is compared with $t_{0.01,39}$ and the null hypothesis is rejected if $t^*\lt t_{0.01,39}.$

Since $t^*=-5.2704628\lt t_{0.01,39}= -2.425841$, we reject the null hypothesis and conclude that there is sufficient evidence to show that the selected boys are significantly lighter than the standard weight at 1% level of significance.