Chapter 8, Problem 42E

(a)

To determine

To explain do you think a linear model is appropriate.

Answer to Problem 42E

Yes, the linear model is appropriate.

Explanation of Solution

In the question, we have given the information about the percentage of teenagers who had used marijuana and other drugs. The scatterplot of the percentage of other drugs and marijuana is also given. As we see in the scatterplot that the relationship between the percentage of other drugs and the marijuana is positive, linear and fairly strong relationship, thus we think that the linear model is appropriate.

(b)

To determine

To interpret the statistic in this context.

Explanation of Solution

In the question, we have given the information about the percentage of teenagers who had used marijuana and other drugs. The scatterplot of the percentage of other drugs and marijuana is also given. Thus it is given to us that $R^2=87.3\%$ . And it represents how much the linear model is able to account for variation. Therefore, marijuana use accounts for $87.3\%$ of variation in use of other drugs.

(c)

To determine

To write the equation you would use to estimate the percentage of teens who use other drugs from the percentage who have used marijuana.

Answer to Problem 42E

  $\text{<mover accent="true"><mo>D</mo><mo>^</mo></mover> rugs}=-3.001+0.611(\text{Marijuana)}$ .

Explanation of Solution

In the question, we have given the information about the percentage of teenagers who had used marijuana and other drugs. The scatterplot of the percentage of other drugs and marijuana is also given. It is given to us:

  $\begin{array}{l}r=0.934\\ {\mu }_1=11.6\\ {\mu }_2=23.9\\ {\sigma }_1=10.2\\ {\sigma }_2=15.6\end{array}$

Thus, the regression equation will be calculated as:

  $\begin{array}{l}\beta =r\times \frac{{\sigma }_1}{{\sigma }_2}\\ =0.934\times \frac{10.2}{15.6}\\ =0.611\\ \alpha ={\mu }_1-\beta \times {\mu }_2\\ =11.6-0.611\times 23.9\\ =-3.001\end{array}$

The equation you would use to estimate the percentage of teens who use other drugs from the percentage who have used marijuana is as:

  $\begin{array}{l}\text{<mover accent="true"><mo>D</mo><mo>^</mo></mover> rugs}=\alpha +\beta (\text{Marijuana)}\\ =-3.001+0.611(\text{Marijuana)}\end{array}$

(d)

To determine

To explain in context what the slope of this line means.

Explanation of Solution

In the question, we have given the information about the percentage of teenagers who had used marijuana and other drugs. The scatterplot of the percentage of other drugs and marijuana is also given. It is given to us:

  $\begin{array}{l}r=0.934\\ {\mu }_1=11.6\\ {\mu }_2=23.9\\ {\sigma }_1=10.2\\ {\sigma }_2=15.6\end{array}$

And the regression equation is:

  $\text{<mover accent="true"><mo>D</mo><mo>^</mo></mover> rugs}=-3.001+0.611(\text{Marijuana)}$

The slope of the regression line use to estimate the percentage of teens who use other drugs from the percentage who have used marijuana interprets that as marijuana use increases by $1\%$ , the use of other drugs increases by $0.611\%$ .

(e)

To determine

To explain do these results confirm that marijuana is a “gateway drug” that is, that marijuana use leads to use leads to the use of other drugs.

Answer to Problem 42E

No, these results do not confirm that marijuana is a “gateway drug”.

Explanation of Solution

In the question, we have given the information about the percentage of teenagers who had used marijuana and other drugs. The scatterplot of the percentage of other drugs and marijuana is also given. It is given to us:

  $\begin{array}{l}r=0.934\\ {\mu }_1=11.6\\ {\mu }_2=23.9\\ {\sigma }_1=10.2\\ {\sigma }_2=15.6\end{array}$

And the regression equation is:

  $\text{<mover accent="true"><mo>D</mo><mo>^</mo></mover> rugs}=-3.001+0.611(\text{Marijuana)}$

Thus, we can say that these results do not confirm that marijuana is a “gateway drug” that is, that marijuana use leads to use leads to the use of other drugs because we are unable to confirm that marijuana is a gateway drug. We only know that this data indicates that there is an association between marijuana and other drug use.