Question

Box Office Mojo collects and posts data on movie grosses. For a random sample of \(50\) movies, we obtained both domestic (U.S) and overseas grosses, in millions of dollars. The data are presented on the WeissStats site.

a. obtain and interpret the standard error of the estimate.

b. obtain a residual plot and a normal probability plot of the residuals.

c. decide whether you can reasonably consider Assumptions \(1-3\) for regression inferences met by the two variables under considerations.

Short answer

Part a. From the MINITAB output, the standard error of the estimate is \(77.8401\).

Part b. The residual plot and normal probability plot are obtained.

Part c. The assumptions \((1\) and \(3)\) for the regression inferences are violated for the variables overseas and domestic.

Step-by-step solution

Part a. Step 1. Given information

Given,

DOMESTIC	OVERSEAS		44.3	10.5
49	28.1		2	1.7
70	11.3		4.2	2.3
16	13.6		1.5	6
125	71.1		36.9	20.8
64.6	62.1		51.8	80.8
56	51.1		79.7	165.8
9	25.5		41.2	43.2
25.9	8.7		12.5	21.6
162.8	262.6		408	283.2
73.1	211.1		1.8	7.7
2.1	17.4		12.8	56.1
31.1	92		216.4	530.5
182.2	93.1		6	3.1
83.7	221.6		1.6	3.7
54.3	73.4		18.6	27.6
17.7	15.6		448.1	636.3
93.8	68		13.4	22.2
31.6	7.9		45.5	22.8
148.8	293		19.2	126.4
218.8	330.6		262	490.2
18.5	3.6		65.4	237.6
1	3.1		35.3	46.2
5.7	4		25.3	7.4
126.5	276.9		27.1	103.4
			41.1	31

Part a. Step 2. Calculation

Find the standard error of the estimate by using MINITAB.

MINITAB procedure:

Step 1: Choose Stat > Regression > Regression.

Step 2: In Response, enter the column OVERSEAS.

Step 3: In Predictors, enter the columns DOMESTIC.

Step 4: Click OK.

MINITAB output:

From the MINITAB output, the standard error of the estimate is \(77.8401\).

Interpretation:

The predicted overseas in the sample differ on average from the observed overseas by \(77.8401\).

Part b. Step 1. Calculation

Construct the residual plot by using MINITAB.

MINITAB procedure:

Step 1: Choose Stat > Regression > Regression.

Step 2: In Response, enter the column OVERSEAS.

Step 3: In Predictors, enter the columns DOMESTIC.

Step 4: In Graphs, enter the columns DOMESTIC under Residuals versus the variables.

Step 5: Click OK.

MINITAB output:

Construct the normal probability plot of residuals by using MINITAB.

MINITAB procedure:

Step 1: Choose Stat > Regression > Regression.

Step 2: In Response, enter the column OVERSEAS.

Step 3: In Predictors, enter the columns DOMESTIC.

Step 4: In Graphs, select Normal probability plot of residuals.

Step 5: Click OK.

MINITAB output:

Thus, the residual plot and normal probability plot are obtained.

Part c. Step 1. Calculation

The assumption for regression inferences is given below:

Population regression line:

The conditional mean of the response variable \((Y)\) is \(\beta _{0}+\beta _{1}X\), for each value \(X\) of predictor variable.

Equal standard deviation:

The standard deviation for the response variable \((Y)\) is same for the standard deviation for the explanatory variable \((X)\). The standard deviation is denoted as \(\sigma\).

Normal populations:

The distribution of the response variable follows normal.

Independent observations:

The observations of the response variable are independent of each other.

Check whether the graph suggests violation of one or more of the assumptions for the regression inferences.

• From the residual plot, it is clear that the residuals are fall in the horizontal band.
• From the normal probability plot of residuals, it is clear that the residuals are in the linear pattern.

Hence, the assumptions \((1\) and \(3)\) for the regression inferences are violated for the variables overseas and domestic. That is, the model is not good for the given data. Moreover, the residuals are not normally distributed.