Question

9. An analyst believes that long-distance running reduces muscle mass. She collects data on subjects (pounds of muscle mass, hours of running per week, and calories consumed per week) and performs a regression. Here are the results:

	Coefficient	Standard Error
Intercept	20	2.23
Running	-3.5	0.56
Calories	0.009	0.44

Interpret the RUNNING coefficient:

a. Running one more hour per week will reduce calories by 3.5

b. People with greater muscle mass run than

3.5 less miles per week

c. Running more than 3.5 miles per week will reduce muscle mass

d. Running one more hour per week will reduce muscle mass by 3.5 pounds

Is the Running variable statistically significant?

a. Yes

b. No

If Frank runs 3 hours per week and consumes 12,000 calories per week, estimate his weight with the regression model..?

If the coefficient of determination (R2) is 0.76 for this regression, this means that

a.24 percent of the variation in the dependent variable is explained by the regression.

b.24 percent of the variation in the independent variable is explained by the regression.

c.76 percent of the variation in the dependent variable is explained by the regression.

d.76 percent of the variation in the independent variable is explained by the regression

Answer 1

Solution:

In the model, we wish to see how running and calory consumption affects the muscle mass, so the dependent variable is muscle mass and independent variables are number of running per week and calory consumption per week.

1. A regression coefficient tells how an independent variable marginally affects the dependent variable, and in what direction. Since, we are talking about the running coefficient, it will tell us how running impacts the muscle mass. Notice that running coefficient = -3.5, since it is negative, it means that running affects the muscle mass negatively (so more running means less muscle mass). So, complete interpretation of this coefficient can be stated as:

As the running hour is increased by 1, the muscle mass reduces by 3.5 pounds. Thus, correct option is (d).

2. The critical t-value for running coefficient = running coefficient/standard error

T-value = -3.5/0.56 = -6.25

This t value, in absolute terms, is quite high (bigger than approximately 3), or in other words the p-value for this test statistic is very low. So, yes, the running variable can be said as statistically significant.

3. Denoting muscle mass by M, number of running hours by R and calory consumption by C. Given the regression results, the regression model/line can be written as:

M^ = 20 - 3.5*R + 0.009*C ; M^ is for estimated muscle mass level

So, with R = 3 and C = 12000, we get

M^ = 20 - 3.5*3 + 0.009*12000

M^ = 20 - 10.5 + 108 = 117.5

So, estimated weight of Frank is 117.5 pounds

4. Coefficient of determination, R2, tells the goodness of fit of a model, that is how well a regression line explain the actual model. So, with R2 = 0.76, it means that 76% of model is explained by the regression model, while remaining (1 - 0

76 =) 0.24, or 24% of model cannot be explained (random errors). Also, a model is simply how independent variables affect dependent variable.

With 76% of goodness of fit, it means that 76% of variations in the dependent variable can be explained through regression. Thus, correct option is (c).