Question

Stats teachers' cars A random sample of AP Statistics teachers were asked to report the age (in years) and mileage of their primary vehicles. A scatterplot of the data is shown below.

Computer output from a least-squares regression analysis of these data is shown below. Assume that the conditions for regression inference are met.

(a) Verify that the 95% confidence interval for the slope of the population regression line is (9016.4, 14,244.8 ).

(b) A national automotive group claims that the typical driver puts 15,000 miles per year on his or her main vehicle. We want to test whether AP Statistics teachers are typical drivers. Explain why an appropriate pair of hypotheses for this test is $H_0:\beta =15,000$versus $H_a:\beta \ne 15,000$.

(c) What conclusion would you draw for this significance test based on your interval in part (a)? Justify your answer.

Short answer

a) $(9016.443,14244.757)$

b) $H_0:\beta =15000$

$H_a:\beta \ne 15000$

c)There is sufficient evidence to reject the claim that AP Statistics teachers are typical drivers.

Step-by-step solution

Given Information(Part a)

Given:

$$\begin{gathered} n=21 \\ b=11630.6 \\ S{E}_b=1249 \end{gathered}$$

Explanation(Part a)

The degrees of freedom is the sample size decreased by 2 :

$$\begin{gathered} df=n-2 \\ =21-2 \\ =19 \end{gathered}$$

The critical t-value can be found in table B in the row of d f=19 and in the column of c=95% :

$t^{*}=2.093$

The boundaries of the confidence interval then become:

localid="1650543247381" $$\begin{gathered} b-t^{*}\times S{E}_b \\ =11630.6-2.093\times 1249 \\ =9016.443 \end{gathered}$$

localid="1650543258240" $$\begin{gathered} b+t^{*}\times S{E}_b \\ =11630.6+2.093\times 1249 \\ =14244.757 \end{gathered}$$

The slight deviation is due to rounding errors.

Given Information(Part b)

Need for an appropriate pair of hypotheses for this test.

Explanation (Part b)

Claim: the typical driver puts 15000 miles per year on his or her main vehicle.

This means that the mileage is expected to be about 15000 miles per year, which corresponds with a slope of 15000. The null hypothesis states that the population parameter is equal to the value given in the claim:

$H_0:\beta =15000$

The alternative hypothesis states the opposite of the null hypothesis:

$H_a:\beta \ne 15000$

Given Information(Part c)

Need to find this significance test based on your interval in part (a).

Explanation(Part c)

$H_0:\beta =15000$

$H_a:\beta \ne 15000$

Confidence interval found in part a:

$(9016.443,14244.757)$

The confidence interval does not contain 15000 and thus it is unlikely to obtain $\beta =15000$, which means that there is sufficient evidence to reject the claim that AP Statistics teachers are typical drivers.