Question

96. Some manufacturers claim that non-hybrid sedan cars have a lower mean miles-per-gallon (mpg) than hybrid ones. Suppose that consumers test 21 hybrid sedans and get a mean of 31 mpg with a standard deviation of seven mpg. Thirty one non-hybrid sedans get a mean of $22\mathrm{mpg}$ with a standard deviation of four mpg. Suppose that the population standard deviations are known to be six and three, respectively. Conduct a hypothesis test to evaluate the manufacturers claim.

Short answer

(a) The null hypothesis: $H_0:{\mu }_1\ge {\mu }_2$
(b) The alternate hypothesis: $H_a:{\mu }_1<{\mu }_2$
(c) The difference in the mean miles per gallon of non-hybrid and hybrid sedans is the random variable.
(d) The distribution is normal.
(e) The test statistics: $-6.356$

(f) The $p$-value is $0$.
(g)(i) $\alpha =0.05$
(ii) Decision: reject the null hypothesis.
(iii) Reason for Decision: $p-$value$<\alpha$.
(iv) As a result: At the $5\%$ level of significance, there is sufficient data to establish that non-hybrid cars get less miles per gallon than hybrid sedans.

Step-by-step solution

Given information

To conduct a hypothesis test to evaluate the manufacturers claim. Let, the population standard deviations are known to be six and three, respectively.

Explanation

(a) The null hypothesis is described as follows:
$H_0:{\mu }_1\ge {\mu }_2$
(b) The alternate hypothesis is described as follows:
$H_a:{\mu }_1<{\mu }_2$
(c) The difference in the mean miles per gallon of non-hybrid and hybrid sedans is the random variable.
(d) The distribution is normal.
(e) To find test static as follows:
To access the stat list editor, click STAT and then 1.

Then ENTER all values the OUTPUT will be:

Hence, the test statistics is$-6.356$.

Explanation

(f) The $p$-value from the output is determined as $0$.
(g) Obtain a clear picture of the situation using the information from the last task. The horizontal axis should be clearly labelled and scaled, and the region(s) corresponding to the p-value should be shaded on the graph.

(i) $\alpha =0.05$
(ii) Decision: reject the null hypothesis.
(iii) Reason for Decision: $p$-value$<\alpha$.
(iv) As a result:At the $5\%$ level of significance, there is sufficient data to establish that non-hybrid cars get less miles per gallon than hybrid sedans.