Questions: Two popular brands of tires for tractor-trailers are the Puma and the Eternal. Lashonda is a buyer for a major shipping company and wants to determine if there is any difference between the two brands of tire in the mean distance (in thousands of km) driven on them before they need to be replaced. In the company's testing lab, Lashonda tests a random sample of 13 Puma tires and a random sample of 15 Eternal tires. (These samples are chosen independently.) For the Puma tires, the sample mean distance (in thousands of km) until they would need to be replaced is 55.19 with a sample variance of 10.58. For the Eternal tires, the sample mean distance (in km) until they would need to be replaced is 51.37 with a sample variance of 94.24. Assume that the two populations of distances driven are approximately normally distributed. Can Lashonda 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 H0 and the alternate hypothesis H1.

Two popular brands of tires for tractor-trailers are the Puma and the Eternal. Lashonda is a buyer for a major shipping company and wants to determine if there is any difference between the two brands of tire in the mean distance (in thousands of km) driven on them before they need to be replaced. In the company's testing lab, Lashonda tests a random sample of 13 Puma tires and a random sample of 15 Eternal tires. (These samples are chosen independently.) For the Puma tires, the sample mean distance (in thousands of km) until they would need to be replaced is 55.19 with a sample variance of 10.58. For the Eternal tires, the sample mean distance (in km) until they would need to be replaced is 51.37 with a sample variance of 94.24. Assume that the two populations of distances driven are approximately normally distributed. Can Lashonda 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 H0 and the alternate hypothesis H1.
Transcript text: Two popular brands of tires for tractor-trailers are the Puma and the Eternal. Lashonda is a buyer for a major shipping company and wants to determine if there is any difference between the two brands of tire in the mean distance (in thousands of km) driven on them before they need to be replaced. In the company's testing lab, Lashonda tests a random sample of 13 Puma tires and a random sample of 15 Eternal tires. (These samples are chosen independently.) For the Puma tires, the sample mean distance (in thousands of km) until they would need to be replaced is 55.19 with a sample variance of 10.58. For the Eternal tires, the sample mean distance (in km) until they would need to be replaced is 51.37 with a sample variance of 94.24. Assume that the two populations of distances driven are approximately normally distributed. Can Lashonda 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 $H_{0}$ and the alternate hypothesis $H_{1}$.
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Solution

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Solution Steps

Step 1: State the Hypotheses

We are testing whether there is a significant difference between the mean distances driven on Puma tires and Eternal tires. The hypotheses are stated as follows:

  • Null Hypothesis \( H_0: \mu_1 = \mu_2 \)
  • Alternative Hypothesis \( H_1: \mu_1 \neq \mu_2 \)
Step 2: Calculate the Standard Error

The standard error \( SE \) is calculated using the formula:

\[ SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} = \sqrt{\frac{0.0}{13} + \frac{0.0}{15}} = 0.0 \]

Step 3: Calculate the Test Statistic

The test statistic \( t \) is calculated using the formula:

\[ t = \frac{\bar{x}_1 - \bar{x}_2}{SE} = \frac{55.19 - 51.37}{0.0} = 1862360629958512.2 \]

Step 4: Calculate the Degrees of Freedom

The degrees of freedom \( df \) are calculated using the formula:

\[ df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2 - 1}} = \frac{0.0}{0.0} = 12.0 \]

Step 5: Calculate the P-value

The P-value is calculated as follows:

\[ P = 2(1 - T(|t|)) = 2(1 - T(1862360629958512.2)) = 0.0 \]

Step 6: Determine the Critical Value

For a significance level \( \alpha = 0.05 \) and \( df = 12 \), the critical value from the t-distribution is:

\[ \text{Critical Value} = 2.179 \]

Step 7: Conclusion

Since the calculated P-value \( 0.0 \) is less than the significance level \( \alpha = 0.05 \), we reject the null hypothesis \( H_0 \). This indicates that there is a statistically significant difference between the mean distances driven on Puma tires and Eternal tires.

Final Answer

\[ \boxed{H_0 \text{ is rejected, indicating a significant difference in mean distances.}} \]

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