Question

Answer each of the following questions by typing the answers into the Assignment text box below. For the predicted value of Y we will use Y-hat.

1. Sample data on exam grades (Y), hours studied (X₁) and homework average (X₂) was used to estimate the following regression equation: Y-hat = 60 + 5X₁ + .1 X_2.

The mean grade was 78.5. The standard error of estimate is Se = 5.25 and the coefficient of determination is R²=.873.

a. Interpret the standard error of the estimate.

b. Interpret the coefficient of determination.

c. Is this estimated regression equation a good fit? EXPLAIN.

Please show work.

Thanks

Answer 1

exam grades (Y), hours studied (X1) and homework average (X2)

regression equation: Y = 60 + 5X1 + .1 X2

The mean grade was 78.5.

The standard error of the estimate is Se = 5.25

the coefficient of determination is R2=.873.

a. Interpret the standard error of the estimate.

Answer:- The standard error of the estimate is Se = 5.25

The standard error of the estimate represents the average distance that the observed values fall from the regression line.

Conveniently, it tells you how wrong the regression model is on average using the units of the response variable.

Smaller values are better because it indicates that the observations are closer to the fitted line.

Se is 5.25, which tells us that the average distance of the data points from the fitted line is about 5.25% exam grades.

b. Interpret the coefficient of determination.

The coefficient of determination is R2=.873

The coefficient of determination is a statistical measurement that examines how differences in one variable can be explained by the difference in a second variable when predicting the outcome of a given event.

In other words, this coefficient, which is more commonly known as R-squared (or R2), assesses how strong the linear relationship is between two variables.

It is interpreted as the proportion of the variance in the dependent variable that is predictable from the independent variable.

An R2 between 0 and 1 indicates the extent to which the dependent variable is predictable.

For our example :

  The coefficient of determination is R2=.873

87.3 % of the variance in the exam grades that is predictable from the hours studied & homework average.

or

87.3 percentage of the exam grades variation that is explained by a linear model.

c. Is this estimated regression equation a good fit? EXPLAIN.

Here we can say that the regression equation a good fit because the Coefficient of determination is quite large so we can say that the model is good fit.

& standard error estimate is small & small estmate denotes the good fit of the model.