Question

Is a movie's earnings related to its ratings? Data on the top ten grossing movies in the United States for the weekend of July 13, 2007, were extracted from the Internet Movie Database (IMDB). These data included the weekend earnings and the average rating from users of IMDB. The rating system used by IMDB is a 10-star rating system. The following table shows the data. Using these data, determine whether there is a relation between a movie's earnings and its ratings.

• Conduct the six-steps of hypothesis testing to determine if there is a relationship between the rating of a film and its gross box office earnings.
• Calculate the expected gross earnings for a movie with a rating of 9.5 and a movie with a rating of 4.8. You will need to solve for the intercept (B) by using the X and Y of any movie and the formula Y= Mx + B. Remember, M = r(Sy / Sx) where M (also called a in the slides) is the slope, r = correlation coefficient, Sy = standard deviation of Y, Sx standard deviation of X).

Table: Top Ten Movies Movie Average Rating Earnings (in millions) Harry Potter and the Order of the Phoenix 7.9 77.1 Transformers 8.0 37 Ratatouille 8.6 18 Live Free or Die Hard 8.0 11.3 License to Wed 5.2 7.31 1408 7.6 4.93 Evan Almighty 6.0 4.9 Knocked Up 8.1 3.68 Sicko 8.5 2.6 Ocean's Thirteen 7.3 1.98

Answer 1

There are n = 10 values for the variables. We shall use the t-test for the slope parameter for regression fit.

The regression output from excel is as below

Regression Statistics
Multiple R	0.223348
R Square	0.049884
Adjusted R Square	-0.06888
Standard Error	24.46995
Observations	10
ANOVA
	df	SS	MS	F	Significance F
Regression	1	251.5023	251.5023	0.420026	0.535078
Residual	8	4790.228	598.7784
Total	9	5041.73
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	-19.3823	56.4848	-0.34314	0.740335	-149.636	110.8719	-149.636	110.8719
X Variable 1	4.822115	7.440459	0.648094	0.535078	-12.3356	21.97985	-12.3356	21.97985
RESIDUAL OUTPUT
Observation	Predicted Y	Residuals
1	18.7124	58.3876
2	19.19462	17.80538
3	22.08788	-4.08788
4	19.19462	-7.89462
5	5.692692	1.617308
6	17.26577	-12.3358
7	9.550385	-4.65038
8	19.67683	-15.9968
9	21.60567	-19.0057
10	15.81913	-13.8391

Line of Best Fit Plot

Based on this regression line, the expected gross earnings for given movie ratings can be calculated as

Significance Test for Regression Fit

For the test of significance, the test statistic is

The critical value is obtained for degrees of freedom,

If we choose a significance level of 0.05, we have

Observed test statistic does not fall in the critical range, i.e. test statistic 0.6481 is less than the critical value viz. 2.306. Hence we conclude that there is no correlation between the two variables, and movie ratings cannot be used to predict its earnings.