Chapter 6, Problem 1SE

Exercises 1 to 4 describe experiments that require a hypothesis test. For each experiment, describe the appropriate test. State the appropriate null and alternate hypotheses, describe the test statistic, and specify which table should be used to find the P-value. If relevant, state the number of degrees of freedom for the test statistic.

1.    A fleet of 100 taxis is divided into two groups of 50 cars each to see whether premium gasoline reduces maintenance costs. Premium unleaded fuel is used in group A, while regular unleaded fuel is used in group B. The total maintenance cost for each vehicle during a one-year period is recorded. Premium fuel will be used if it is shown to reduce maintenance costs.

To determine

Describe the appropriate test.

State the appropriate null and alternate hypotheses and specify which table should be used to find the P-value.

State the number of degrees of freedom for the test statistic if relevant.

Answer to Problem 1SE

The appropriate test is “difference between two means”.

Null hypothesis: $H_0:{\mu }_1-{\mu }_2\ge 0$

Alternative hypothesis: $H_1:{\mu }_1-{\mu }_2<0$

The z-table should be used to find the P-value.

Explanation of Solution

Justification:

A fleet of 100 taxis is divided into two groups of 50 cars. For group A, Premium unleaded fuel is used and for group B, regular unleaded fuel is used.

The number of samples in group A is 50 and the number of samples in group B is 50.

Here, the samples are independent.

Therefore, the appropriate test is “difference between two means”.

State the test hypotheses:

Let ${\mu }_1$ be the mean annual cost for cars using regular fuel and ${\mu }_2$ be the mean annual cost for cars using premium fuel.

Null hypothesis:

$H_0:{\mu }_1-{\mu }_2\ge 0$

Alternative hypothesis:

$H_1:{\mu }_1-{\mu }_2<0$

Test statistic:

$t=\frac{\left(\overline{X}-\overline{Y}\right)-0}{\sqrt{\frac{s_X^2}{n_X}+\frac{s_Y^2}{n_Y}}}$

The test statistic is the difference between the mean cost between group A and group B.

The z-table should be used to find the P-value.