Question

You may need to use the appropriate technology to answer this question.

Consider the following data for a dependent variable y and two independent variables,

x₁

and

x₂.

x1	x2	y
30	12	93
47	10	108
25	17	112
51	16	178
40	5	94
51	19	175
74	7	170
36	12	117
59	13	142
76	16	212

The estimated regression equation for these data is

ŷ = −19.56 + 2.03x₁ + 4.76x₂.

Here, SST = 15,418.9, SSR = 14,275.9,

s_b1 = 0.2485,

and

s_b2 = 0.9536.

(a)

Test for a significant relationship among

x₁, x₂, and y.

Use α = 0.05.

State the null and alternative hypotheses.

H₀: β₁ ≠ 0 and β₂ ≠ 0
H_a: One or more of the parameters is equal to zero.H₀: β₁ = β₂ = 0
H_a: One or more of the parameters is not equal to zero.     H₀: β₁ ≠ 0 and β₂ = 0
H_a: β₁ = 0 and β₂ ≠ 0H₀: β₁ > β₂
H_a: β₁ ≤ β₂H₀: β₁ < β₂
H_a: β₁ ≥ β₂

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.

Do not 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.     Reject H₀. There is sufficient 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.

(b)

Is

β₁

significant? Use α = 0.05.

State the null and alternative hypotheses.

H₀: β₁ ≠ 0
H_a: β₁ = 0H₀: β₁ > 0
H_a: β₁ ≤ 0     H₀: β₁ = 0
H_a: β₁ > 0H₀: β₁ < 0
H_a: β₁ ≥ 0H₀: β₁ = 0
H_a: 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 sufficient 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 insufficient evidence to conclude that β₁ is significant.

(c)

Is

β₂

significant? Use α = 0.05.

State the null and alternative hypotheses.

H₀: β₂ = 0
H_a: β₂ > 0H₀: β₂ = 0
H_a: β₂ ≠ 0     H₀: β₂ > 0
H_a: β₂ ≤ 0H₀: β₂ ≠ 0
H_a: β₂ = 0H₀: β₂ < 0
H_a: 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.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.Reject H₀. There is insufficient evidence to conclude that β₂ is significant.

Answer 1

Solution:

Performmultiple 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 ="
x1 x2 y
30   12   93
47   10   108
25   17   112
51   16   178
40   5   94
51   19   175
74   7   170
36   12   117
59   13   142
76   16   212
"
)
df1
linreg=lm(y~x1+x2 ,data=df1)
coefficients(linreg)
summary(linreg)

Output:

> coefficients(linreg)
(Intercept) x1 x2
-19.562591 2.027889 4.763688
> summary(linreg)

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

Residuals:
Min 1Q Median 3Q Max
-20.0108 -4.1081 0.9271 6.3340 17.9213

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -19.5626 18.0685 -1.083 0.31483
x1 2.0279 0.2485 8.162 8.02e-05 ***
x2 4.7637 0.9536 4.996 0.00157 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 12.78 on 7 degrees of freedom
Multiple R-squared: 0.9259,   Adjusted R-squared: 0.9047
F-statistic: 43.71 on 2 and 7 DF, p-value: 0.0001109

ANSWER(A)

H0: β1 = β2 = 0
Ha: One or more of the parameters is not equal to zero.

F=43.71

p-value: 0.000

p<0.05,Reject Ho

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

ANSWER(B)

H0: β1 = 0
Ha: β1 ≠ 0

t=8.16

p=0.000

p<0.05

Reject Ho

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

ANSWER(C)

H0: β2 = 0
Ha: β2 ≠ 0

value of the test statistic=T=5.00

p- value= 0.002

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