Question

Most individuals are aware of the fact that the average annual repair cost for an automobile depends on the age of the automobile. A researcher is interested in finding out whether the variance of the annual repair costs also increases with the age of the automobile. A sample of 26 automobiles 4 years old showed a sample standard deviation for annual repair costs of $170 and a sample of 25 automobiles 2 years old showed a sample standard deviation for the annual repair cost of $100.
a. State the null and alternative versions of the research hypothesis that the variance in annual repair costs is larger for older automobiles.
b. At a .01 level of significance, what is your conclusion? What is the p-value? Discuss the reasonableness of your findings.

Answer 1

Answer:

(a) Null Hypothesis, $H_0$ : $\sigma_1^{2} = \sigma_2^{2}$

    Alternate Hypothesis, $H_1$ : $\sigma_1^{2} > \sigma_2^{2}$

(b) At a .01 level of significance, we conclude that the variance in annual repair costs is larger for older automobiles.

Step-by-step explanation:

We are given that a researcher is interested in finding out whether the variance of the annual repair costs also increases with the age of the automobile.

For this, a sample of 26 automobiles 4 years old showed a sample standard deviation for annual repair costs of $170 and a sample of 25 automobiles 2 years old showed a sample standard deviation for the annual repair cost of $100.

Let the population and sample variance for 4 years old automobiles be $\sigma_1^{2}$ and $s_1^{2}$ respectively;

And the population and sample variance for 2 years old automobiles be $\sigma_2^{2}$ and $s_2^{2}$  respectively.

(a) Null Hypothesis, $H_0$ : $\sigma_1^{2} = \sigma_2^{2}$ {means that the variance in annual repair costs is same for both type of automobiles}

Alternate Hypothesis, $H_1$ : $\sigma_1^{2} > \sigma_2^{2}$ {means that the variance in annual repair costs is larger for older automobiles}

(b) The test statistics that will be used here is;

                 $\frac{s_1^{2} }{s_2^{2} } * \frac{\sigma_1^{2} }{\sigma_2^{2} }$ ~ $F_n__1-1, n_2-1$  where, $n_1$ = 26  and  $n_2$ = 25

Test Statistics =  $\frac{170^{2} }{100^{2} } * 1$ ~ $F_2_5_,_2_4$

                       = 2.89

Now, at 1% level of significance, F table gives the critical value of between 0.411 and 2.435 at degree of freedom 25 , 24. since our test statistics is higher than both the critical value and it will fall in rejection region so we conclude that null hypothesis will be rejected .

And we conclude that the variance in annual repair costs is larger for older automobiles.

P-value is the exact % where our test statistics lie.

         If P-value > Significance level   -  Accept null hypothesis

         If P-value < Significance level   -  Reject null hypothesis  

So, here since our test statistics gets rejected so we conclude that p-value is less than 0.01 .