Two popular brands of tires for tractor-trailers are the Puma and the Eternal. Debra is a buyer for a major shipping company and wants to determine if there is any difference between the two brands of tires in the mean distance (in thousands of km) driven on them before they need to be replaced.

In the company's testing lab, Debra tests a random sample of 13 Puma tires and a random sample of 15 Eternal tires. (These samples are chosen independently.) For each tire, she logs the distance driven (in thousands of km) before the tire would need to be replaced. These data are shown in the table below:

[tex]
\[
\begin{tabular}{|c|l|}
\hline
& \multicolumn{1}{c|}{Distances (in thousands of km)} \\
\hline
Puma & $56.5, 54.9, 51.7, 61.8, 53.9, 53.7, 49.4, 57.2, 54.9, 52.9, 58.4, 58.4, 53.8$ \\
\hline
Eternal & \begin{tabular}{l}
$55.8, 61.8, 38.7, 58.5, 60.3, 64.5, 39.4, 52.4, 33.5, 48.3, 44.4, 53.1, 42.0, 59.2, 58.7$
\end{tabular} \\
\hline
\end{tabular}
\]
[/tex]

Assume that the two populations of distances driven are approximately normally distributed. Can Debra conclude, at the 0.05 level of significance, that there is a difference between the population mean of the distances (in thousands of km) driven on Puma tires before they need to be replaced and the population mean of the distances (in thousands of km) driven on Eternal tires before they need to be replaced?

Perform a two-tailed test. Then complete the parts below. Carry your intermediate computations to three or more decimal places. (If necessary, consult a list of formulas.)

(a) State the null hypothesis [tex]$H_0$[/tex] and the alternate hypothesis [tex]$H_1$[/tex].

[tex]
\[
\begin{array}{l}
H_0: \square \\
H_1: \square
\end{array}
\]
[/tex]

(b) Determine the type of test statistic to use.
(Choose one) [tex]$\nabla$[/tex]

(c) Find the value of the test statistic. (Round to three or more decimal places.)
[tex]$\square$[/tex]

(d) Find the two critical values. (Round to three or more decimal places.)
[tex]$\square$[/tex] and [tex]$\square$[/tex]

(e) At the 0.05 level of significance, can Debra conclude that there is a difference between the mean distance (in thousands of km) driven on Puma tires before they need to be replaced and the mean distance (in thousands of km) driven on Eternal tires before they need to be replaced?
Yes [tex]$\square$[/tex] No [tex]$\square$[/tex]

Sure, let's work through the problem step by step.

### (a) State the null hypothesis [tex]$ H_0 $[/tex] and the alternate hypothesis [tex]$ H_1 $[/tex].

The null hypothesis ([tex]$ H_0 $[/tex]) and the alternative hypothesis ([tex]$ H_1 $[/tex]) can be stated as follows:

- [tex]$ H_0: \mu_1 = \mu_2 $[/tex] (There is no difference in the mean distance driven on Puma and Eternal tires)
- [tex]$ H_1: \mu_1 \neq \mu_2 $[/tex] (There is a difference in the mean distance driven on Puma and Eternal tires)

### (b) Determine the type of test statistic to use.

Since we are comparing the means of two independent samples, we will use a two-sample t-test, assuming that the populations are normally distributed, and do not assume equal variances.

### (c) Find the value of the test statistic.

The test statistic is calculated using the formula for the two-sample t-test:

[tex]\[ t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \][/tex]

Plugging in the provided numbers:

- [tex]$\bar{x}_1 = 55.192$[/tex]
- [tex]$\bar{x}_2 = 51.373$[/tex]
- [tex]$s_1 = 3.253$[/tex]
- [tex]$s_2 = 9.708$[/tex]
- [tex]$n_1 = 13$[/tex]
- [tex]$n_2 = 15$[/tex]

The calculated test statistic [tex]$ t $[/tex] is approximately [tex]$ 1.434 $[/tex].

### (d) Find the two critical values.

To determine the critical values for a two-tailed test at the 0.05 significance level, we calculate the critical t-values. With degrees of freedom of approximately 17.52, the critical values are approximately:

- Lower critical value: [tex]$-2.105$[/tex]
- Upper critical value: [tex]$2.105$[/tex]

### (e) At the 0.05 level of significance, can Debra conclude that there is a difference between the mean distance driven on Puma tires and Eternal tires?

We compare the test statistic to the critical values. If the test statistic falls outside the range of these critical values, we reject the null hypothesis.

Since [tex]$ t = 1.434 $[/tex] is between [tex]$-2.105$[/tex] and [tex]$2.105$[/tex], we do not reject the null hypothesis.

Conclusion: At the 0.05 level of significance, Debra cannot conclude that there is a difference between the mean distance driven on Puma tires before they need to be replaced and the mean distance driven on Eternal tires before they need to be replaced. Therefore, the result is "No."