Question

Question: Revenues of popular movies. The Internet Movie Database (www.imdb.com) monitors the gross revenues for all major motion pictures. The table on the next page gives both the domestic (United States and Canada) and international gross revenues for a sample of 25 popular movies.

1. Write a first-order model for foreign gross revenues (y) as a function of domestic gross revenues (x).
2. Write a second-order model for international gross revenues y as a function of domestic gross revenues x.
3. Construct a scatterplot for these data. Which of the models from parts a and b appears to be the better choice for explaining the variation in foreign gross revenues?
4. Fit the model of part b to the data and investigate its usefulness. Is there evidence of a curvilinear relationship between international and domestic gross revenues? Try using$\mathit{\alpha }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}$.
5. Based on your analysis in part d, which of the models from parts a and b better explains the variation in international gross revenues? Compare your answer with your preliminary conclusion from part c.

Short answer

Answer:

1. The first-order model equation for foreign gross revenues (y) as a function of domestic gross revenues (x) can be written as$\mathit{y}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}\mathit{x}\mathbf{+}\mathit{\epsilon }$.
2. The second-order model equation for international gross revenues as a function of domestic gross revenue (x) can be written as$\mathit{y}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}\mathit{x}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathit{\epsilon }$.
3. Looking at the graph, it is visible that there is a curvilinear relation between foreign gross revenues (y) and domestic gross revenues (x). Hence, it would be appropriate to fit a second-order model equation to the data.
4. The model equation here becomes$\mathit{y}\mathbf{=}\mathbf{-}\mathbf{322}\mathbf{.}\mathbf{95}\mathbf{+}\mathbf{2}\mathbf{.}\mathbf{8874}\mathit{x}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{000988}{\mathit{x}}^{\mathbf{2}}$and at 95% confidence level,${\mathit{\beta }}_{\mathbf{2}}\mathbf{=}\mathbf{0}$this means that the parabola doesn’t have a curvature and it essentially is a straight line.
5. From part d, it is concluded that the curvilinear relation between x and y does not exist. Hence, the model the part a where a first-order model equation is fitted to the data is better to explain the variation in the international gross revenue. In part c, it was concluded from the graph that I curvilinear relationship between x and y might exist however, a linear relationship is a better option.

Step-by-step solution

First-order model equation

The first-order model equation for foreign gross revenues (y) as a function of domestic gross revenues (x) can be written as$\mathit{y}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}\mathit{x}\mathbf{+}\mathit{\epsilon }$ .

Second-order model equation

The second-order model equation for international gross revenues as a function of domestic gross revenue (x) can be written as $\mathit{y}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}\mathit{x}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathit{\epsilon }$.

Scatterplot and model fit for the data

Movie Title (year)	International Gross	Domestic Gross	${\mathit{x}}^{\mathbf{2}}$
Star Wars VII (2015)	1122	936.4	876845
Avatar (2009)	2023.4	760.5	578360.3
Jurassic World (2015)	1018.1	652.2	425364.8
Titanic (1997)	1548.9	658.7	433885.7
The Dark Knight (2008)	469.5	533.3	284408.9
Pirates of the Caribbean (2006)	642.9	423.3	179182.9
E.T.(1982)	357.9	434.9	189138
Spider-Man (2002)	418	403.7	162973.7
Frozen (2013)	873.5	400.7	160560.5
Jurassic Park (1993)	643.1	395.7	156578.5
Furious 7 (2015)	1163	351	123201
Lion King (1994)	564.7	422.8	178759.8
Harry Potter and the Sorcerer's Stone (2001)	657.2	317.6	100869.8
Inception (2010)	540	292.6	85614.76
Sixth Sense (1999)	379.3	293.5	86142.25
The jungle book (2016)	491.2	292.6	85614.76
The Hangover (2009)	201.6	277.3	76895.29
Jaws (1975)	210.6	260	67600
Ghost (1990)	300	217.6	47349.76
Saving Private Ryan (1998)	263.2	216.1	46699.21
Gladiator (2000)	268.6	187.7	35231.29
Dances with wolves (1990)	240	184.2	33929.64
The Exorcist (1973)	153	204.6	41861.16
My Big Fat Greek Wedding (2002)	115.1	241.4	58273.96
Rocky IV (1985)	172.6	127.9	16358.41

Looking at the graph, it is visible that there is a curvilinear relation between foreign gross revenues (y) and domestic gross revenues (x). Hence, it would be appropriate to fit a second-order model equation to the data.

Model equation and significance of β2

The excel output is attached here

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.802142835
R Square	0.643433127
Adjusted R Square	0.611017957
Standard Error	293.541447
Observations	25
ANOVA
	df	SS	MS	F	Significance F
Regression	2	3420771	1710385	19.84975	1.18E-05
Residual	22	1895665	86166.58
Total	24	5316436
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	-322.9500969	285.3873	-1.13162	0.269979	-914.807	268.907	-914.807	268.907
Domestic Gross	2.887488271	1.321995	2.184189	0.039893	0.145838	5.629139	0.145838	5.629139
	-0.000988686	0.00129	-0.76639	0.451591	-0.00366	0.001687	-0.00366	0.001687

The model equation here becomes$\mathit{y}\mathbf{=}\mathbf{-}\mathbf{322}\mathbf{.}\mathbf{95}\mathbf{+}\mathbf{2}\mathbf{.}\mathbf{8874}\mathit{x}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{000988}{\mathit{x}}^{\mathbf{2}}$

To check for the curvilinear relationship between variable x and y

$$\begin{gathered} {\mathit{H}}_{\mathbf{0}}\mathbf{}\mathbf{:}{\mathit{\beta }}_{\mathbf{2}}\mathbf{=}\mathbf{0} \\ {\mathit{H}}_{\mathbf{a}}\mathbf{}\mathbf{:}{\mathit{\beta }}_{\mathbf{2}}\mathbf{\ne }\mathbf{0} \end{gathered}$$

Here, t-test statistic$\mathbf{=}\frac{\widehat{{\mathbf{\beta }}_{\mathbf{2}}}}{\mathbf{s}{\mathbf{\beta }}_{\mathbf{2}}}\mathbf{=}\frac{\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{000988}}{\mathbf{0}\mathbf{.}\mathbf{00129}}\mathbf{=}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{7658}$

Value of${\mathit{t}}_{\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{,}\mathbf{385}}$is 2.069

is rejected if t statistic > . For$\mathit{\alpha }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}$, since t <${\mathit{t}}_{\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{,}\mathbf{385}}$

Not sufficient evidence to reject at 95% confidence interval.

Therefore, ${\mathit{\beta }}_{\mathbf{2}}\mathbf{=}\mathbf{0}$

This means that the parabola doesn’t have a curvature and it essentially is a straight line.

Interpretation of variations in international gross revenue

From part d, it is concluded that the curvilinear relation between x and y does not exist. Hence, the model the part a where a first-order model equation is fitted to the data is better to explain the variation in the international gross revenue. In part c, it was concluded from the graph that I curvilinear relationship between x and y might exist however, a linear relationship is a better option.