Question

Suppose you run a regression containing observations for each of the 74 kinds of cars released in 1978 in the United States, and you regress price (in dollars) on weight (in pounds). You get the following results:
ˆ β0 is -6.71, with SE 1174.4. ˆ β1 (slope coefficient on weight) is 2.04 with SE .377.
• (a) Say, in words, what the slope coefficient means in this case, without taking a stand on causality.

• (b) Suppose I give you the following information: the sum of squares total is roughly equal to 635 million, and the sum of squares explained is equal to 185 million. Please report the R2 and the correlation coefficient between price and weight.

• (c) Use your regression model to predict what the price would be for a car that weighs 3000 pounds.

• (d) Use the SER to add uncertainty and make this prediction an interval. • (5 points) Build a 95% confidence interval for the slope coefficient and report the p-value for comparing it to 0. Interpret your results.

Answer 1

Suppose you run a regression containing observations for each of the 74 kinds of cars released in 1978 in the United States, and you regress price (in dollars) on weight (in pounds). You get the following results:
ˆ β0 is -6.71, with SE 1174.4. ˆ β1 (slope coefficient on weight) is 2.04 with SE .377.
• (a) Say, in words, what the slope coefficient means in this case, without taking a stand on causality.

The regression equation we get from the information provided, we get

price = -6.71 + 2.04 weight

Interpretation for the slope : One pound increase in the weight increase the price by 2.04

• (b) Suppose I give you the following information: the sum of squares total is roughly equal to 635 million, and the sum of squares explained is equal to 185 million. Please report the R2 and the correlation coefficient between price and weight.

Coefficient of determination(rsqaure) = 0.2913
It is the measure of the amount of variability in y explained by x. Its value lies between 0 and 1. Greater the value, better is the model. In this case, it 29.13%, hence the model is bad

• (c) Use your regression model to predict what the price would be for a car that weighs 3000 pounds.

We are given weight = 3000 pounds.
We substitute this value in the regression

price = -6.71 + 2.04*(3000) = 6113.29

d.(5 points) Build a 95% confidence interval for the slope coefficient and report the p-value for comparing it to 0. Interpret your results.

95% confidence interval for the slope

Calculate the p-value for the slope coefficient.

Significance of the Slope coefficient

To understand if the slope coefficient is significant, we conduct the following hypothesis

We check the pvalue associated with that variable,
if the pvalue is less than 0.01(level of significance specified), then we reject the null hypothesis and conclude that the variable is significant or is statistically different from zero. Hence it is a significant predictor of y.

if the pvalue is greater than 0.01(level of significance specified), then we fail to reject the null hypothesis and conclude that the variable is not significant or is not statistically different from zero. Hence it is not a significant predictor of y and can be dropped from the regression.

We see that the pvalue for the slope in our case is 0.00, which is less than 0.05, hence we conclude that weight is an important predictor of y.