A motion picture industry analyst is studying movies based on epic novels. The following data were obtained for 10 Hollywood movies made in the past 5 years. Each movie was based on an epic novel. For these data, x₁ = fi­rst-year box offi­ce receipts of the movie, x₂ = total production costs of the movie, x₃ = total promotional costs of the movie, and x₄ = total book sales prior to movie release. All units are in millions of dollars.

(a) Generate summary statistics, including the mean and standard deviation of each variable. Compute the coefficient of variation (see Section 3.2) for each variable. Relative to its mean, which variable has the largest spread of data values? Why would a variable with a large coefficient of variation be expected to change a lot relative to its average value? Although x has the largest standard deviation, it has the smallest coefficient of variation. How does the mean of x₁ help explain this?

(b) For each pair of variables, generate the sample correlation coefficient r. Compute the corresponding coefficient of determination r₂. Which of the three variables x₂, x₃, and x₄ has the least influence on box office receipts? What percent of the variation in box office receipts can be attributed to the corresponding variation in production costs?

(c) Perform a regression analysis with x₁ as the response variable. Use x₂, x₃, and x₄ as explanatory variables. Look at the coefficient of multiple determination. What percentage of the variation in x₁ can be explained by the corresponding variations in x₂, x₃, and x₄ taken together?

(d) Write out the regression equation. Explain how each coefficient can be thought of as a slope. If x₂ (production costs) and x₄ (book sales) were held fixed but x₃ (promotional costs) was increased by $1 million, what would you expect for the corresponding change in x₁ (box office receipts)?

(e) Test each coefficient in the regression equation to determine if it is zero or not zero. Use level of significance 5%. Explain why book sales x₄ probably are not contributing much information in the regression model to forecast box office receipts x₁.

(f) Find a 90% confidence interval for each coefficient.

(g) Suppose a new movie (based on an epic novel) has just been released. Production costs were x₂ = 11.4 million; promotion costs were x₃ = 4.7 million; book sales were x₄ = 8.1 million. Make a prediction for x₁ = first-year box office receipts and find an 85% confidence interval for your prediction (if your software supports prediction intervals).

(h) Construct a new regression model with x₃ as the response variable and x₁, x₂, and x₄ as explanatory variables. Suppose Hollywood is planning a new epic movie with projected box office sales x = 100 million and product ion costs x, = 12 million. The book on which the movie is based had sales of x₄ = 9.2 million. Forecast the dollar amount (in millions) that should be budgeted for promotion costs x₃ and find an 80% confidence interval for your prediction.

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