Final answer:
The question addresses the use of linear regression to predict the relationship between exam scores, with the third exam score as the independent variable x, and the final exam score as the dependent variable y. By using the provided regression equation ŷ = -173.51 + 4.83x, predictions for specified x values can be made, and the significance of the linear relationship is determined by the correlation coefficient r.
Step-by-step explanation:
The question involves a mathematical concept known as linear regression, which is used to predict the relationship between an independent variable (x) and a dependent variable (y). In the case given, the independent variable is the third exam score, and the dependent variable is the final exam score. To perform a linear regression and predict the y value, one must use the least-squares regression line formula provided: ŷ = -173.51 + 4.83x. For predicting values, we simply substitute x with the given third exam scores.
a. For a textbook with 400 pages, predict the cost.
Using the regression equation to predict cost:
ŷ for 400 pages = -173.51 + 4.83(400)
ŷ = -173.51 + 1932
ŷ = 1758.49
b. For a textbook with 600 pages, predict the cost.
Using the same regression equation:
ŷ for 600 pages = -173.51 + 4.83(600)
ŷ = -173.51 + 2898
ŷ = 2724.49
To evaluate the significance of the regression, we can use the correlation coefficient (r). If r = .6631 as given, we can square it to obtain the coefficient of determination (r²), which indicates the proportion of the variance for the dependent variable that's explained by the independent variable.
c. Is there a linear relationship between the variables?
Yes, based on the positive correlation coefficient r = .6631, which suggests a moderate positive linear relationship between the third exam score and the final exam score.