Question

2. [40 points) Consider the following pairs of observations: X 5 3 2 6 6 0 1 1 7 5 у 3 3 1 4 1 (a) (15 points] Use the method of least squares to fit a straight line to the seven data points in the table. (b) [5 pointsSpecify the null and alternative hypotheses you would use to test whether the data provide sufficient evidence to indicate that x contributes infor- mation for the (linear) prediction of y. (c) (10 points] What test statistic that should be used for part (b)? Specify the number of degrees of freedom associated with the test statistic. Conduct the hypothesis test using a = = 0.05. (d) (10 points) Construct a 95% confidence interval for B1.

Answer 1

Using Excel, go to Data, select Data Analysis, choose Regression.

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.945
R Square	0.893
Adjusted R Square	0.875
Standard Error	0.611
Observations	8
ANOVA
	df	SS	MS	F	Significance F
Regression	1	18.633	18.633	49.863	0.000
Residual	6	2.242	0.374
Total	7	20.875
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	0.526	0.397	1.327	0.233	-0.444	1.497
x	0.626	0.089	7.061	0.000	0.409	0.843

a) y = 0.526 + 0.626x

b) H₀: β₁ = 0, There is not sufficient evidence that x contributes information for linear prediction of y

H₁: β₁ ≠ 0, There is sufficient evidence that x contributes information for linear prediction of y

c) Test statistic: t-test

Degrees of freedom = n-2 = 8-2 = 6

Level of significance = 0.05

Test statistic (t Stat) = 7.061

Critical value (Using Excel function T.INV.2T(probability,df)) = T.INV.2T(0.05,6) = 2.45

Since test statistic is more than 0.05, we reject the null hypothesis.

There is sufficient evidence that x contributes information for linear prediction of y.

d) 95% confidence interval for β₁: (0.409, 0.843)