Answer to Question #316515 in Statistics and Probability for secret

Question

It is claimed that a car is driven on the average less than 28700 kilometers per year with a population standard deviation of 2100 kilometers. To test this claim, a random sample of 58 car owners are asked to keep a record of the kilometers they travelled. Would you agree with this claim if the random sample showed an average of 22500 kilometers? Use a 0.01 level of significance

Answer 1

$\mu=28700, \sigma=2100,n=58,\bar{x}=22500,\alpha=0.01.$

Null and alternative hypotheses:

$H_0:\mu\geq28700;\\ H_1:\mu<28700.$

Because $\sigma$ is known and $n=58>30,$ we can use the z-test.

$z=\cfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\cfrac{22500-28700}{2100/\sqrt{58}}=-22.48.$

In z-table, the area corresponding to $z=-22.48$ is 0. Because the test is a left-tailed test, the P-value is equal to the area to the left of $z=-22.48,$ so, $P=0.$

Because the P-value is less than $\alpha$ =0.01, we reject the null hypothesis, there is enough evidence at the 1% level of significance to support the claim that the mean run is less than 28700 kilometers per year.