Question

We are interested in predicting the fuel efficiency (gasoline mileage in mpg) using the weight of a vehicle in pounds. A random sample of 12 vehicles is taken. The Excel output is given below. Use the output and the fact that R2 = 0.640 to answer the following questions. Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 48.409 5.978 8.098 0.000 35.089 61.728 Weight -0.006 0.002 -4.214 0.002 -0.010 -0.002 Interpret the slope in context. Find and interpret the correlation coefficient. For a 3100-pound vehicle, the Fuel efficiency is 22 mpg, find the residual.

Answer 1

The regression equation is
fuel efficieny = 48.409 - 0.006 weight

Interpret the slope in context.
One unit increase in the weight decreases the fuel efficiency by 0.006

Let us consider the regression equation
y = a + bx
In this equation
y is the dependent variable.(the one we are trying to predict)
x is the independent variable( or the predictor variable)
a is the y intercept (The point on the y axis, where the regression line cuts it in the graph)
b is the coefficient or the slope of the variable.
We can have many variable and each variable will have a coefficient.

Interpreting the meaning of the coefficient
check two thing from the regression output
= the sign of the coefficient
- the value of the coefficient.

The sign (positive or negative) indicate whether the predictor variable increase or decrease the dependent variable.
The value indicates the value or magnitude of the change.
We state it as follows : one unit increase in x (independent variable) causes an increase/decrease (depends on the sign) of (value of the coefficient of the variable) in y (dependent variable)
For example : one unit increase in x , cause an increase of b units in y.

Find and interpret the correlation coefficient.

r^2 = 0.640
correlation coefficeint =

But we have negative sign for the slope, hence the coerrlation coefficient is -0.8

Weight and fuel efficiency has a strong negative correlation.

Understanding correlation
Correlation between two variables define the strength and the direction of the linear relationship between two variable.
By strength we mean, how strong or weak is the association between the two variables.

The correlation coefficient takes a value between 0 and 1 and it can have a positive or negative sign depending on the relationship.

Higher the value, stronger is the relationship.
A positive sign indicates that as one variables increase or decreases, the other variable also increase or decreases in the same proportion.

A negative sign indicates that as one variables increases the other decreases and vice versa.

Furthermore, the strength of the relationship depends on how the points are scatter, it there is a linear pattern between the two points, the correlation is strong.

For a 3100-pound vehicle, the Fuel efficiency is 22 mpg, find the residual.

We are given weight = 3100, we put this value in regression equation and find the predicted fuel efficieny.

fuel efficieny = 48.409 - 0.006 weight
fuel efficieny = 48.409 - 0.006*(3100)=29.809

Residual = Actual - predicted = 22- 29.809 = -7.809