Chapter 15.2, Problem 11E

a.

To determine

To find:The hypothesis for the test statistic.

Answer to Problem 11E

The hypothesis are $H_0\text{:}{m}_1=m_2$ and $H_1\text{:}{m}_1<m_2$ .

Explanation of Solution

Given information:

The level of significance is $0.05$ and the given data is,

Compact
34.5	33.7	26.1	28.5	27.4
30.6	31.1	28.0	33.0	33.0
32.8	28.5	25.5	32.1	34.9
Midsize
21.6	21.1	29. 1	24.8	28.5
28.1	21.9	22.5	20.5	26.1

Calculations:

The hypothesis are,

  $H_0\text{:}$ The median mileage is not less for compact car than for the midsize car.

  $H_1\text{:}$ The median mileage is less for compact car than for the midsize car.

Therefore, the hypothesis are $H_0\text{:}{m}_1=m_2$ and $H_1\text{:}{m}_1<m_2$ .

b.

To determine

To find:The value of test statistic.

Answer to Problem 11E

The value of test statistic is $-3.24$ .

Explanation of Solution

Given information:

The level of significance is $0.05$ and the given data is,

Compact
34.5	33.7	26.1	28.5	27.4
30.6	31.1	28.0	33.0	33.0
32.8	28.5	25.5	32.1	34.9
Midsize
21.6	21.1	29. 1	24.8	28.5
28.1	21.9	22.5	20.5	26.1

Calculations:

The rank table is shown below.

Gas mileages	Sample	Rank
20.5	Midsize	1
21.1	Midsize	2
21.6	Midsize	3
21.9	Midsize	4
22.5	Compact	5
24.8	Midsize	6
25.5	Compact	7
26.1	Midsize	8.5
26.1	Compact	8.5
27.4	Compact	10
28	Compact	11
28.1	Midsize	12
28.5	Midsize	14
28.5	Compact	14
28.5	Compact	14
29.1	Midsize	16
30.6	Compact	17
31.1	Compact	18
32.1	Compact	19
32.8	Compact	20
33	Compact	21.5
33	Compact	21.5
33.7	Compact	23
34.5	Compact	24
34.9	Compact	25

The sum of rank of compact car is $253.5$ and the rank of midsize car is $71.5$ .

The sum of smaller rank is $S=71.5$ and the sample size are $n_1=10$ and $n_2=15$ .

The value of ${\mu }_s$ is,

  ${\mu }_s=\frac{n_1\left(n_1+n_2+1\right)}{2}$

Substitute the values in above equation.

  $\begin{array}{c}{\mu }_s=\frac{\left(10\right)\left(10+15+1\right)}{2}\\ =\frac{260}{2}\\ =130\end{array}$

The value of ${\mu }_s$ is,

  ${\sigma }_s=\sqrt{\frac{n_1{n}_2\left(n_1+n_2+1\right)}{12}}$

Substitute the values in above equation.

  $\begin{array}{c}{\sigma }_s=\sqrt{\frac{\left(15\right)\left(10\right)\left(10+15+1\right)}{12}}\\ =\sqrt{\frac{3900}{12}}\\ =18.028\end{array}$

The z- value is,

  $z=\frac{S-{\mu }_s}{{\sigma }_s}$

Substitute the values in above equation.

  $\begin{array}{c}z=\frac{71.5-130}{18.028}\\ =-3.24\end{array}$

Therefore, the value of test statistics is $-3.24$ .

c.

To determine

To find:The p-value.

Answer to Problem 11E

The p-valueis $0.0006$ .

Explanation of Solution

Given information:

The level of significance is $0.05$ and the given data is,

Compact
34.5	33.7	26.1	28.5	27.4
30.6	31.1	28.0	33.0	33.0
32.8	28.5	25.5	32.1	34.9
Midsize
21.6	21.1	29. 1	24.8	28.5
28.1	21.9	22.5	20.5	26.1

Calculations:

From the table of cumulative normal distribution the p-value corresponding to z-value is $0.0006$ .

Therefore, the p-value is $0.0006$ .

d.

To determine

To find:The conclusion for the test.

Answer to Problem 11E

The median mileage is less for compact car than for the midsize car.

Explanation of Solution

Given information:

The level of significance is $0.05$ and the given data is,

Compact
34.5	33.7	26.1	28.5	27.4
30.6	31.1	28.0	33.0	33.0
32.8	28.5	25.5	32.1	34.9
Midsize
21.6	21.1	29. 1	24.8	28.5
28.1	21.9	22.5	20.5	26.1

Calculations:

Since, the p-value is $0.0006$ which less than $0.05$ .

Thus, the hypothesis $H_0$ is rejected.

Therefore, the median mileage is less for compact car than for the midsize car.