Question

Two variables are defined, a regression equation is given, and one data point is given. (a) Find the predicted value for the data point and compute the residual. (b) Interpret the slope in context. (c) Interpret the intercept in context, and if the intercept makes no sense in this context, explain why. \(B A C=\) blood alcohol content (\% of alcohol in the blood), Drinks \(=\) number of alcoholic drinks. \(\widehat{B A C}=-0.0127+0.018(\) Drinks \() ;\) data point is an individual who consumed 3 drinks and had a \(B A C\) of 0.08.

Short answer

The predicted BAC for an individual who consumed 3 drinks is 0.041. The residual is 0.039. The slope shows that for each additional drink consumed, the predicted BAC increases by 0.018. The intercept, -0.0127, isn't meaningful in this context because BAC can't be negative, especially when no drinks are consumed.

Step-by-step solution

Calculate the Predicted Blood Alcohol Content (BAC)

Using the given regression equation, the predicted BAC (\(\widehat{B A C}\)) for an individual who has consumed 3 drinks can be calculated as follows: \(\widehat{B A C}= -0.0127 + 0.018*3 = 0.041\)

Compute the Residual

The residual is calculated by subtracting the predicted BAC from the observed BAC. In this case, the observed BAC is 0.08, and the predicted BAC from the regression equation is 0.041. Therefore, the residual = 0.08 - 0.041 = 0.039

Interpret the Slope

In this context, the slope (0.018) of the regression equation represents the predicted increase in blood alcohol content (BAC) for each additional alcoholic drink consumed. This means that for every additional drink consumed, the BAC is predicted to increase by 0.018.

Interpret the Intercept

The intercept (-0.0127) of the regression equation represents the predicted BAC when no drinks are consumed. Since BAC cannot be less than 0 (especially when no drinks are consumed), the intercept is not meaningful in this context. If it were positive, it would indicate the predicted BAC with zero alcohol consumption, but a negative BAC doesn't make sense, showing that the regression model doesn't fit the data perfectly at zero drinks.