Question

(1 point) Data analysts were asked to build a model to predict box office receipts for major films produced by a certain movie production company. The production company was particularly interested in predicting sales during the second week of wide release. The analysts collected weekly sales numbers for 572 movies over the course of several years, and analyzed the data. They made the following statement: Our study found that the sales during the first week of wide release can be used to predict the second week box office numbers. In fact, 68.3% of the observed differences in second week sales across all the films in our data set could be explained by considering the initial week sales. If the opening week sales were $100,000 higher in one film than another, then the expected second week sales could be expected to be about $86,000 higher, as well. As a baseline, we would expect a movie whose opening week sales were $15,000,000 to earn $9,000,000 during its second week. From the statement above we can conclude: The slope of the best-fit line for this study is m = The intercept of the best-fit line for this study is b = The correlation coefficient for this data set is R = 826438 The R2 value for this data set is .683

Answer 1

The slope of best fit line m is 0.86

The intercept of best fit line b is -3900000

Solution : Let the best-fit line equation be y=b+mx

It is given that a movie whose opening week sales were 15000000 would earn 9000000 during its second week. Hence we have the below equation 1

9000000 = b+m*15000000

We have also been given that if the movie had opening week sales of 100000 higher than another movie then the second week sales would be higher by 86000. Thus we have the below equation 2

(9000000+86000) = b+m*(15000000+100000)

9086000 = b+m*15100000

We know from equation 1 that b = 9000000-m*15000000. We substitute this vale of b in equation 2 and we have the below

9086000 = 9000000-m*15000000+m*15100000

9086000-9000000 = 100000m

86000 = 100000m

m = 86000/100000 = 0.86

Now we substitute this value of m in equation 1, we get the below

9000000 = b+0.86*15000000 = b+12900000

b = 9000000 - 12900000 = -3900000

Thus the slope of best fit line m is 0.86

The intercept of best fit line b is -3900000

Question 3 and 4) It has been given that 68.3% of observed differences in second week sales across all the films could be explained by first week sales. Hence we know that R² for this data set is 68.3% or 0.683

The correlation coefficient R is simply the square root of R². Hence R = $\sqrt{0.683}$ = 0.826438