Question

Consider the following data for a dependent variable y and two independent variables, x₁ and x₂.

x1	x2	y
30	12	94
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	209

The estimated regression equation for these data is ŷ = −17.02 + 1.99x₁ + 4.70x₂.

Here, SST = 14,902.9, SSR = 13,773.1, s_b1 = 0.2470, and s_b2 = 0.9480.

(1a) Test for a significant relationship among x₁, x₂, and y. Use α = 0.05. State the null and alternative hypotheses.

(1b)Find the value of the test statistic. (Round your answer to two decimal places.)

(1c)Find the p-value. (Round your answer to three decimal places.)

(1d) State your conclusion.

Do not 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.

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

(2a) Is β₁ significant? Use α = 0.05. State the null and alternative hypotheses.

(2b)Find the value of the test statistic. (Round your answer to two decimal places.)

(2c)Find the p-value. (Round your answer to three decimal places.)

(2d) State your conclusion.

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.   

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.

(3a) Is β₂ significant? Use α = 0.05. State the null and alternative hypotheses.

(3b)Find the value of the test statistic. (Round your answer to two decimal places.)

(3c)Find the p-value. (Round your answer to three decimal places.)

(3d) State your conclusion.

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

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.

Answer 1

I used Excel to solve this question.

Step.1 Enter data in excel sheet.

Step.2 Go to 'Data' menu ---> 'Data Analysis' ---> Select 'Regression'.

Step.3 New window will pop-up on screen. Refer following screen shot and enter information accordingly.

Excel output:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.96134835
R Square	0.92419066
Adjusted R Square	0.90253084
Standard Error	12.7042123
Observations	10
ANOVA
	df	SS	MS	F	Significance F
Regression	2	13773.1209	6886.56047	42.6684514	0.00011996
Residual	7	1129.77907	161.39701
Total	9	14902.9
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	-17.020369	17.9638254	-0.9474802	0.3749464	-59.498066	25.4573284	-59.498066	25.4573284
x1	1.98917833	0.24701342	8.0529161	8.739E-05	1.40508441	2.57327225	1.40508441	2.57327225
x2	4.69681483	0.94803447	4.95426588	0.00164826	2.45506953	6.93856013	2.45506953	6.93856013

Que.1a

Hypothesis:

H_a : At least one is not zero.

1b.

Test statistic, F = 42.67

1c.

P-value = 0.000

1d.

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

Que. 2

a.

Hypothesis:

b.

Test statistic, t = 8.05

c.

p-value = 0.000

d.

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

Que.3

a.

Hypothesis:

b.

Test statistic, t = 4.95

c.

p-value = 0.002

d.

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