Answer to Question #274617 in Statistics and Probability for dindin

Question

A report in LTO stated that the average age of taxis in the Philippines is 9 years. An operations manager of a large taxi company selects a sample of 40 taxis and finds the average age of the taxis is 8.2 years. The standard deviation of the population is 2.3 years. At 0.05 level of significance, can it be concluded that the average age of taxis in his company is less than the national average? GIve the full solution also, Thank you!

Answer 1

$H_0:u=9$

$H_1:u<9$ , where u - company average age

In this case we will conclude that company average is less than national if null hypothesis will be rejected

Test statistic calculated the following way

$T={\frac {(x-a)*\sqrt{n}} {\sigma}}$ , where x - sample mean, a - assumed population mean, n - sample size, $\sigma$ - sample standard deviation(if unknown sample standard deviation is used)

So, in the given case we have

$T={\frac {(8.2-9)*\sqrt{40}} {2.3}}=-2.2$

According to the form of alternative hypothesis, we should run left-tailed test, so if T<Cr, then null hypothesis is rejected in favor of alternative one. Cr - critical value. Since population stabdard deviation is known then it is appropriate to use z-score as critical value, so

$P(Z<Cr)=a=0.05\implies Cr=-1.65$

We obtained that T<Cr, so we should admit that there is enough statistical evidence based on the given data to reject the null hypothesis in the 0.05 level of significance and conclued that average company age is indeed less than national average