Question

check your understanding
michael uses his apple watch to keep track of his distance and the number of calories he burns for 20 training runs. a scatterplot of y = calories burned and x = distance (in miles) shows a fairly strong, positive linear relationship. the regression equation $hat{y}=20 + 160x$ is a good model for the data.
a. interpret the slope of the regression line.
b. does the value of the y - intercept have meaning in this context? if so, interpret the y intercept. if not, explain why.
c. predict the number of calories michael burns if he runs 5 miles.
d. calculate and interpret the residual if his apple watch said that he burned 760 calories on a 5 - mile run.
e. michael is thinking about signing up for his first marathon. so far, his longest run has only been 10 miles. should he use the regression equation to predict how many calories he would burn if he runs a marathon (26.2 miles)? explain.
math medic

Explanation:

Step1: Interpret slope

The regression equation is $\hat{y}=20 + 160x$. The slope is 160. For every 1 - mile increase in distance, the number of calories burned increases by 160.

Step2: Analyze y - intercept

The y - intercept is 20. In this context, it represents the number of calories Michael burns even when he runs 0 miles. It could be the calories burned due to his basal metabolic rate and non - running activity during the time he is tracking.

Step3: Predict calories for 5 miles

Substitute $x = 5$ into the regression equation $\hat{y}=20+160x$. Then $\hat{y}=20 + 160\times5=20+800 = 820$ calories.

Step4: Calculate and interpret residual

The residual $e=y-\hat{y}$. Given $y = 760$ (actual calories burned) and $\hat{y}=820$ (predicted calories burned for 5 miles), $e=760 - 820=- 60$. This means that the model over - predicted the number of calories burned by 60 calories for a 5 - mile run.

Step5: Evaluate using regression for marathon

The model is based on data from runs up to 10 miles. A marathon is 26.2 miles, which is outside the range of the data used to create the regression model. Extrapolating this far beyond the observed data range may not be accurate as the relationship between distance and calories burned may change for longer runs. So, he should not use the regression equation to predict calories burned for a marathon.

Answer:

a. For every 1 - mile increase in distance, the number of calories burned increases by 160.
b. Yes, it represents the number of calories Michael burns when he runs 0 miles, likely due to basal metabolic rate and non - running activity.
c. 820 calories
d. Residual is - 60. The model over - predicted the number of calories burned by 60 calories for a 5 - mile run.
e. No. The model is based on runs up to 10 miles, and a marathon (26.2 miles) is outside the range of the data used to create the regression model, so extrapolation may not be accurate.