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	211

The estimated regression equation for these data is

ŷ = −18.89 + 2.02x₁ + 4.74x₂.

Here, SST = 15,276.0, SSR = 14,134.9,

s_b1 = 0.2482,

and

s_b2 = 0.9528.

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

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.    

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.

(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.

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.

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 insufficient evidence to conclude that β₂ is significant.

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.

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

Answer 1

Solution:

Given:

n = 10 observation

K = 2 number of independent variables .

.

. And.

SST = 15276.0

SSR = 14134.9

SST = SSR + SSE

SSE = SST - SSR

SSE = 15276.0 - 14134.9

SSE = 1141.1

a) Test for overall significance

To test the hypothesis

Vs

Ha: one or more of the parameters is not equal to zero.

Test statistic

F = 43.354790

Test statistic F = 43.35

P Value = 0.000114 from online P value calculator

P Value = 0.000

Decision : P Value

Reject Ho.

Conclusion : Reject Ho, there is sufficient evidence to conclude that there is significant relationship among the variables.

Option A is correct

b) Test for significance of

To test the hypothesis

. Vs.   

Test statistic

t = 8.1385979

Test statistic t = 8.14

P Value = 0.000082 from online P value calculator

P value = 0.000

Decision: P Value

Reject Ho.

Conclusion: Reject Ho, there is sufficient evidence to conclude that is significant.

Option B is correct.

c) Test for significance of

To test the hypothesis

. Vs.   

Test statistic

t = 4.9748110

Test statistic t = 4.97

P Value = 0.00161 from online P value calculator

P Value = 0.002

Decision : P Value

Reject Ho

Conclusion : Reject Ho, there is sufficient evidence to conclude that is significant.

Option B is correct.