To test the claim that the mean speed of all cars is greater than the posted speed limit of 65 mi/h, we can conduct a one-sample t-test. The hypothesis statements are formulated as follows:
- Null hypothesis (H0): μ ≤ 65 (The mean speed is less than or equal to 65 mi/h)
- Alternative hypothesis (H1): μ > 65 (The mean speed is greater than 65 mi/h)
Step 1: Determine the test statistic
Since we know the sample mean, sample standard deviation, and the sample size, we can calculate the t-statistic using the formula:
\[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \]
Where:
- \( \bar{x} \) is the sample mean (68.4 mi/h)
- \( \mu_0 \) is the hypothesized population mean (65 mi/h)
- \( s \) is the sample standard deviation (5.7 mi/h)
- \( n \) is the sample size (40)
Plugging in the values:
\[ t = \frac{68.4 - 65}{5.7 / \sqrt{40}} \]
Calculate the denominator:
\[ s / \sqrt{n} = \frac{5.7}{\sqrt{40}} \approx \frac{5.7}{6.3246} \approx 0.901 \]
Now substitute this back into the t statistic formula:
\[ t \approx \frac{3.4}{0.901} \approx 3.77 \]
Step 2: Determine the critical value
Using a significance level of 0.05 for a one-tailed test and degrees of freedom (df = n - 1 = 39), we can find the critical t-value from the t-distribution table.
Using a t-table or calculator:
- Critical t-value (one-tailed, df = 39) at α = 0.05 is approximately 1.685.
Step 3: Make the decision
Compare the calculated t-statistic to the critical t-value:
- Calculated t = 3.77
- Critical t ≈ 1.685
Since \( 3.77 > 1.685 \), we reject the null hypothesis.
Step 4: Conclusion
The conclusion is as follows:
- We reject the null hypothesis, which indicates that there is sufficient evidence to support the claim that the mean speed of all cars is greater than 65 mi/h.
Thus, the final conclusion corresponds to option D:
D) Reject the null hypothesis; there is sufficient evidence to support the claim that the mean speed is greater than 65 mi/h.