Question

A report stated that the average age of commercial jets in country X is 14 years. An executive of a large airline company selects a sample of 36 planes and finds that the average age of the plane is 11.8 years. The standard deviation of the sample is 2.7 years. At 0.01 level of significance can it be concluded that the average age of the planes in his company is less than the national average. 

Answer 1

$n=36\\\bar{x}=11.8\\s=2.7\\\alpha=0.01$

The hypotheses tested are,

$H_0:\mu=14\\vs\\H_1:\mu\lt14$

To perform this test, we shall apply the student's t distribution since the population variance is unknown.

The test statistic is given as,

$t={\bar x-\mu\over{s\over\sqrt{n}}}={11.8-14\over{2.7\over\sqrt{36}}}={-2.2\over{2.7\over6}}=-4.89$

$t$ is compared with the table value at $\alpha=0.01$ with $n-1=36-1=35$ degrees of freedom. The table value is given as,

$t_{0.01,35}=-2.437723$

The null hypothesis is rejected if $t\lt t_{0.01,35}$

Since $t=-4.89\lt t_{0.01,35}=-2.437723,$ we reject the null hypothesis and conclude that there is sufficient evidence to show that the average age of the planes in his company is less than the national average at 1% level of significance.