Question

15-6: Consider the following set of data:

x₁              10        8          11        7          10        11            6

x₂         50        45        37        32        44        51            42

y          103      85        115      73        97        102            65

1. Obtain the estimate regression equation.

2. Examine the coefficient of determination and the adjusted of determination. Does it seem that either of the independent variables’ addition to R² does not justify the reduction in degrees of freedom that results from its addition to the regression model? Support your assertions.

3. Conduct a hypothesis test to determine if the dependent variable increases when x₂ increases. Use a significance level of 0.025 and the p-value approach.

4. Construct a 95% confidence interval for the coefficient of x_1.

Answer 1

(a) The estimate regression equation is:

y = 21.0426 + 9.0383*x1 - 0.2549*x2

(b) The coefficient of determination and the adjusted of determination are 0.951 and 0.926 respectively. Yes, independent variables does not justify the reduction in degrees of freedom that results from its addition to the regression model becuase we take the pair of data in regression.

(c) The hypothesis being tested is:

H₀: β₂ = 0

H₁: β₂ ≠ 0

The p-value from the output is 0.4754.

Since the p-value (0.4754) is greater than the significance level (0.05), we cannot reject the null hypothesis.

Therefore, we cannot conclude that the dependent variable increases when x₂ increases.

(d) The 95% confidence interval for the coefficient of x₁ is between 5.9882 and 12.0885.

	R²	0.951
	Adjusted R²	0.926
	R	0.975
	Std. Error	4.859
	n	7
	k	2
	Dep. Var.	y
ANOVA table
Source	SS	df	MS	F	p-value
Regression	1,817.2788	2	908.6394	38.49	.0024
Residual	94.4355	4	23.6089
Total	1,911.7143	6
Regression output					confidence interval
variables	coefficients	std. error	t (df=4)	p-value	95% lower	95% upper
Intercept	21.0426
x1	9.0383	1.0986	8.227	.0012	5.9882	12.0885
x2	-0.2549	0.3240	-0.787	.4754	-1.1543	0.6446