Chapter 27, Problem 17E

a)

To determine

To perform the hypothesis test.

Answer to Problem 17E

There is sufficient evidence that the slope is non zero and there is association between weight of a car and its gas mileage.

Explanation of Solution

Given:

  

Formula:

Test statistic:

  $t=\frac{b-{\beta }_0}{S{E}_b}$

The null and alternative hypotheses:

  $\begin{array}{l}{H}_0:{\beta }_1=0\\ H_a:{\beta }_1\ne 0\end{array}$

Test statistic:

  $t=\frac{-8.21362-0}{0.6738}=-12.190$

The degrees of freedom = df = 48

Therefore, p-value would be,

P-value = 0 …Using excel formula, =TDIST(12.190,48,2)

Decision: P-value < 0.05, reject H₀.

Conclusion: There is sufficient evidence that the slope is non zero and there is association between weight of a car and its gas mileage.

b)

To determine

To check the assumptions for regression.

Explanation of Solution

Given:

  

The conditions are: straight enough, Independence, Randomization, Random residual, Does plot thicken? and Nearly normal condition.

We will check it one by one.

Straight enough condition: Satisfied, because no curvature is present in the scatterplot.

Independence assumption: Satisfied, because the 50 car models are less than 10% of the population.

Randomization condition: Satisfied, assuming car models were randomly selected.

Random residuals condition: Satisfied, because there is no obvious pattern in the residual plot.

Does the plot thicken? condition: Satisfied, because the vertical spread in the residual plot is not increasing or decreasing.

Nearly normal condition: Satisfied, because the histogram of the residual is roughly symmetric and unimodal.

c)

To determine

To state the hypothesis and conclusion.

Answer to Problem 17E

There is sufficient evidence that the slope is non zero and there is association between weight of a car and its gas mileage.

Explanation of Solution

Given:

  

The null and alternative hypotheses:

  $\begin{array}{l}{H}_0:{\beta }_1=0\\ H_a:{\beta }_1\ne 0\end{array}$

That means,

  $H_0:$ There is no association between weight of a car and its gas mileage.

  $H_a:$ There is association between weight of a car and its gas mileage.

Conclusion: There is sufficient evidence that the slope is non zero and there is association between weight of a car and its gas mileage.