Question

A statistical program is recommended.

The owner of a movie theater company would like to predict weekly gross revenue as a function of advertising expenditures. Historical data for a sample of eight weeks follow.

Weekly Gross Revenue ($1,000s)	Television Advertising ($1,000s)	Newspaper Advertising ($1,000s)
96	5	1.5
91	2	2
95	4	1.5
93	2.5	2.5
95	3	3.2
94	3.5	2.3
94	2.5	4.2
94	3	2.5

(a)

Use α = 0.01 to test the hypotheses

H0:	β1 = β2 = 0
Ha:	β1 and/or β2 is not equal to zero

for the model

y = β₀ + β₁x₁ + β₂x₂ + ε,

where

x1	=	television advertising ($1,000s)
x2	=	newspaper advertising ($1,000s).

Find the value of the test statistic. (Round your answer to two decimal places.)

Find the p-value. (Round your answer to three decimal places.)

p-value =

State your conclusion.

Reject H₀. There is sufficient evidence to conclude that there is a significant relationship among the variables.Do not reject H₀. There is insufficient evidence to conclude that there is a significant relationship among the variables.     Reject H₀. There is insufficient evidence to conclude that there is a significant relationship among the variables. Do not reject H₀. There is sufficient evidence to conclude that there is a significant relationship among the variables.

(b)

Use α = 0.05 to test the significance of

β₁.

State the null and alternative hypotheses.

H0: β1 = 0

Ha: β1 > 0

H0: β1 < 0

Ha: β1 = 0

     

H0: β1 = 0

Ha: β1 ≠ 0

H0: β1 = 0

Ha: β1 < 0

H0: β1 ≠ 0

Ha: β1 = 0

Find the value of the test statistic. (Round your answer to two decimal places.)

Find the p-value. (Round your answer to three decimal places.)

p-value =

State your conclusion.

Reject H₀. There is insufficient evidence to conclude that β₁ is significant.Do not reject H₀. There is insufficient evidence to conclude that β₁ is significant.     Reject H₀. There is sufficient evidence to conclude that β₁ is significant.Do not reject H₀. There is sufficient evidence to conclude that β₁ is significant.

Should

x₁

be dropped from the model?

YesNo   

(c)

Use α = 0.05 to test the significance of

β₂.

State the null and alternative hypotheses.

H0: β2 = 0

Ha: β2 ≠ 0

H0: β2 < 0

Ha: β2 = 0

     

H0: β2 = 0

Ha: β2 > 0

H0: β2 = 0

Ha: β2 < 0

H0: β2 ≠ 0

Ha: β2 = 0

Find the value of the test statistic. (Round your answer to two decimal places.)

Find the p-value. (Round your answer to three decimal places.)

p-value =

State your conclusion.

Reject H₀. There is sufficient evidence to conclude that β₂ is significant.Reject H₀. There is insufficient evidence to conclude that β₂ is significant.     Do not reject H₀. There is sufficient evidence to conclude that β₂ is significant.Do not reject H₀. There is insufficient evidence to conclude that β₂ is significant.

Should

x₂

be dropped from the model?

YesNo   

Answer 1

Perform multiple regression In Rstudio:

Use lm function in R to fit Y on x1 and x2

coeffcients function to get the coeffcients

Summary function to get t,p values

Rcode:

df1 =read.table(header = TRUE, text ="
y x1 x2
96   5   1.5
91   2   2
95   4   1.5
93   2.5   2.5
95   3   3.2
94   3.5   2.3
94   2.5   4.2
94   3   2.5
"
)
df1
linreg=lm(y~x1+x2 ,data=df1)
coefficients(linreg)
summary(linreg)

Output:

> df1
y x1 x2
1 96 5.0 1.5
2 91 2.0 2.0
3 95 4.0 1.5
4 93 2.5 2.5
5 95 3.0 3.2
6 94 3.5 2.3
7 94 2.5 4.2
8 94 3.0 2.5
> linreg=lm(y~x1+x2 ,data=df1)
> coefficients(linreg)
(Intercept) x1 x2
85.9101271 1.8090433 0.9435725
> summary(linreg)

Call:
lm(formula = y ~ x1 + x2, data = df1)

Residuals:
1 2 3 4 5 6 7
-0.3707 -0.4154 0.4383 0.2083 0.6433 -0.4120 -0.3957
8
0.3038

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 85.9101 1.2970 66.237 1.49e-08 ***
x1 1.8090 0.2493 7.256 0.000777 ***
x2 0.9436 0.2666 3.540 0.016569 *
---
Signif. codes:
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.525 on 5 degrees of freedom
Multiple R-squared: 0.9139,   Adjusted R-squared: 0.8794
F-statistic: 26.53 on 2 and 5 DF, p-value: 0.002177
ANSWER(A)

Find the value of the test statistic. (Round your answer to two decimal places.)

F=26.53

Find the p-value. (Round your answer to three decimal places.)

p-value =0.002

p<0.01

State your conclusion.

Reject H0. There is sufficient evidence to conclude that there is a significant relationship among the variables

ANSWER(B)

H0: β1 = 0

Ha: β1 ≠ 0

Find the value of the test statistic. (Round your answer to two decimal places.)

t= 7.26

Find the p-value. (Round your answer to three decimal places.)

p=0.001

p<0.05

Reject Ho

Reject H0. There is sufficient evidence to conclude that β1 is significant

NO it should not be dropped form model.

ANSWER(C)

H0: β2 = 0

Ha: β2 ≠ 0

Find the value of the test statistic. (Round your answer to two decimal places.)

t=3.54

p-value = 0.017

p<0.05

.Reject H0. There is insufficient evidence to conclude that β2 is significant.

NO it should not be dropped form model