To test your suspicion that the roof rack lowered the car’s fuel efficiency, we can perform a hypothesis test. Since we're dealing with a sample and we know the population standard deviation, we will use a one-sample t-test.

Hypothesis Setup

• Null Hypothesis (H₀): The mean efficiency after adding the roof rack is the same as the original mean, i.e., no effect.
  $H_0: \mu = 32 \, \text{mpg}$

• Alternative Hypothesis (H₁): The mean efficiency after adding the roof rack is less than the original mean, i.e., the roof rack lowers the efficiency.
  $H_1: \mu < 32 \, \text{mpg}$

Given Data

• Population mean efficiency before the roof rack: $\mu_0 = 32 \, \text{mpg}$
• Sample mean efficiency after the roof rack: $\bar{x} = 29 \, \text{mpg}$
• Sample standard deviation: $s = 8 \, \text{mpg}$
• Sample size: $n = 31$
• Confidence level: 96% → significance level $\alpha = 1 - 0.96 = 0.04$
• Degrees of freedom for the t-test: $df = n - 1 = 31 - 1 = 30$

Test Statistic (t)

The formula for the t-statistic is:

$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$

Substitute the values:

$t = \frac{29 - 32}{8 / \sqrt{31}} = \frac{-3}{8 / 5.568} = \frac{-3}{1.437} \approx -2.09$

Critical Value

For a one-tailed t-test with a confidence level of 96%, the critical value at $\alpha = 0.04$ and 30 degrees of freedom can be found using a t-distribution table or a calculator. From the t-table or software, the critical value for $t_{0.04, 30}$ is approximately:

$t_{\text{critical}} \approx -1.697$

Decision Rule

• If $t$ is less than $t_{\text{critical}}$, we reject the null hypothesis.
• If $t$ is greater than $t_{\text{critical}}$, we fail to reject the null hypothesis.

Since our calculated $t$-value of $-2.09$ is less than the critical value of $-1.697$, we reject the null hypothesis.

Conclusion

There is sufficient evidence at the 96% confidence level to support the suspicion that the roof rack lowers the fuel efficiency of the car.

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Would you like further details on how the test was performed or explanations on hypothesis testing? Here are some related questions:

1. How does a one-tailed t-test differ from a two-tailed t-test?
2. What does the p-value represent in hypothesis testing?
3. How would you interpret the results if the test statistic were close to the critical value?
4. What are the assumptions made in performing a t-test?
5. What impact would increasing the sample size have on the test?

Tip: When performing hypothesis tests with smaller sample sizes (less than 30), always use the t-distribution, as the sample distribution of the test statistic may not follow a normal distribution.