Let's work through the problem step by step.

Given:

• Pacer model:
  $\bar{x}_1 = 27.3$, $s_1 = 6.2$, $n_1 = 9$
• Road Runner model:
  $\bar{x}_2 = 22.5$, $s_2 = 6.8$, $n_2 = 14$
• Level of significance $\alpha = 0.01$
• Hypothesis: Test if the Pacer’s mean gasoline consumption is higher than the Road Runner's (right-tailed test).

(a) State the null and alternate hypotheses:

Null Hypothesis ($H_0$):
$H_0: \mu_1 \leq \mu_2$
(Pacer's mean gasoline consumption is less than or equal to the Road Runner's)

Alternate Hypothesis ($H_a$):
$H_a: \mu_1 > \mu_2$
(Pacer's mean gasoline consumption is higher than the Road Runner's)

This is a right-tailed test.

(b) Identify the appropriate sampling distribution and calculate the test statistic:

The sample sizes are small ($n_1 = 9$, $n_2 = 14$), so we use the Student's $t$-distribution.

The formula for the test statistic is:
$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{s_p^2 \left( \frac{1}{n_1} + \frac{1}{n_2} \right)}}$

Where $s_p^2$ is the pooled variance:
$s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}$

Step 1: Calculate $s_p^2$

$s_p^2 = \frac{(9 - 1)(6.2)^2 + (14 - 1)(6.8)^2}{9 + 14 - 2}$
$s_p^2 = \frac{8(38.44) + 13(46.24)}{21} = \frac{307.52 + 601.12}{21} = \frac{908.64}{21} \approx 43.27$

Step 2: Calculate the test statistic $t$

$t = \frac{27.3 - 22.5}{\sqrt{43.27 \left( \frac{1}{9} + \frac{1}{14} \right)}}$ $t = \frac{4.8}{\sqrt{43.27 \left( 0.1111 + 0.0714 \right)}} = \frac{4.8}{\sqrt{43.27 \cdot 0.1825}} = \frac{4.8}{\sqrt{7.894}} = \frac{4.8}{2.81} \approx 1.71$

The test statistic is $t \approx 1.71$.

(c) Find or estimate the $P$-value:

Degrees of freedom ($df$):
$df = n_1 + n_2 - 2 = 9 + 14 - 2 = 21$

Using a $t$-distribution table or software for $t = 1.71$ with $df = 21$, we find:
$P \text{-value} \approx 0.05$

(d) Conclusion:

Compare the $P$-value to the significance level ($\alpha = 0.01$):
$P \text{-value} (0.05) > \alpha (0.01)$

Since the $P$-value is greater than the significance level, we fail to reject the null hypothesis.

Conclusion: There is insufficient evidence to conclude that the Pacer’s mean gasoline consumption is higher than the Road Runner’s.

Would you like more explanation or calculations?

5 Related Questions:

1. How is the pooled variance calculated, and why is it necessary?
2. What are the assumptions of a $t$-test, and are they satisfied here?
3. What would happen if the test were two-tailed instead of one-tailed?
4. How does increasing the sample size impact the results?
5. How can software tools like Python or R be used to calculate the $P$-value?

Tip:

Always check assumptions of normality and equal variances before performing a pooled $t$-test.