Question 1 [20 marks] - must be completed individually on Mobius.

Question 1. To answer the following question, copy and paste the data below into RStudio and create a data frame Movies.

Budget=c(154,9,157,59,29,205,196,145,59,67,55,53,46,41,36,258,83,43,47,31,30,40,24,21,28,26,21,20,20,23,17,22,74,76,136,148,17,26,10,19,164,38,60)

Opening=c(59.65,18.11,19.91,10.19,15.11,40.13,63.26,77.64,51.56,17.69,5.93,144.26,12.97,10.84,27.60,47.91,11.34,29.40,59.29,24.01,18.49,47.28,30.67,5.04,27.66,13.1 Theatres=c(3773,2858,3190,2815,2431,3660,3392,3864,3447,3557,2109,4033,3338,2709,2738,3598,3699,3185,3685,2860,2928,3283,3137,2561,1903,2446,2958,3082,3 Ratings=c(8.0,7.3,5.3,7.4,6.4,7.2,5.3,8.6,6.5,6.0,6.7,5.0,4.7,6.0,5.2,7.7,4.5,6.8,7.1,5.8,6.0,8.4,4.9,4.3,7.8,6.4,5.8,7.2,6.4,3.9,5.3,5.4,5.8,8.4,8.3,6.5,5.2,8.0,8.1,5.6,8.1,6.7,7.  USRevenue=c(287.9,59.0,136.9,30.9,41.4,94.3,163.1,261.4,244.2,50.5,33.5,298.7,58.4,28.8,55.5,199.5,42.2,96.4,182.0,83.5,42.3,280.8,144.6,11.9,85.8,33.0,77.2,117.5,6  Movies=data.frame(USRevenue, Budget, Opening, Theatres, Ratings)

The data set contains a random sample of 43 movies collected from the Internet Movie Database (IMDb) to see if information available soon after a movie's theatrical release can successfully predict total U.S. revenue.

The response variable is a movie's total U.S. revenue (USRevenue), measured in millions of U.S. dollars. The explanatory variables are the movie's budget (Budget),

opening-weekend revenue (Opening), the number of theatres (Theatres) the movie was in for the opening weekend and the movie's IMDb rating (Ratings), which is on a 1 to 10 scale (10 being the best).

Obtain the summary and anova outputs from the multiple linear regression model fitted with all the predictors listed above and answer the following questions. (You can

simply copy and paste numerical answers from the R output, or enter your answers to at least two decimal places where appropriate. It is recommended to enter p- values as precise as possible by copying them directly from the R output.)

(a) [4 marks]  State the value of the F statistic used to test the hypothesis that β 1 = β2 = β3 = β4 = 0 versus β 1≠ 0 or β2≠ 0 or β3≠ 0 or β4≠ 0 . Observed F statistic  =

What is the conclusion from this test? Type your conclusion in plain language in the box below. Use α = 0.05 level of significance.

(b) [4 marks]  Is there evidence that a model with Opening and Budget is better than a model with just Budget? State the relevant test statistic,p-value and conclusion.

p-value (scientific notation is accepted, e.g., 8.42E-3) =

Type your conclusion in the box below. Use α = 0.05 level of significance.

(c) [4 marks] Conduct the appropriate F test to test whether a model containing all the predictors is preferred over a model with Budget as the predictor. State the relevant test statistic,p-value and conclusion.

p-value (scientific notation is accepted, e.g., 8.42E-3) =

Type your conclusion in the box below. Use α = 0.05 level of significance.

(d) [4 marks] Is there evidence that Theatres is related to the response in the presence of Budget, Opening and Ratings? State the relevant test statistic,p-value and conclusion.

p-value (scientific notation is accepted, e.g., 8.42E-3) =

Type your conclusion in the box below. Use α = 0.05 level of significance.

(e) [2 marks] A new movie was released during the same time period when these data were collected. It had budget of $30 million and was shown in 3,065 theatres,  grossing $19.83 million during the first weekend. The movie's IMDb rating was 7.6 out of 10. Obtain a 90% prediction interval for the USRevenue based on the model with all four predictors.

Prediction interval = (

) (enter lower and then upper bound of the interval).

(f) [2 mark]

Allowed Extensions: pdf

Upload one pdf file containing 1. R summary() and anova() outputs generated by fitting the linear model to the data and 2. relevant outputs for parts (b), (c), (d) and (e). Please make sure that you have pressed the upload button, and check that you have uploaded the correct file.