Question

Assume that you observe (0,1),(1,2),(3,1),(4,3) from a simple linear model.

a). Construct a 95% confidence interval for β₁ and determine its margin error.

b). Test H₀ : β₀ = 2 versus H₁ : β₀ < 2 at α = 0.01 level of significance.

c). Find the power of the test from part b) at β₀ = 1.

Answer 1

a). Construct a 95% confidence interval for β1 and determine its margin error.

The 95% confidence interval for β1 is between -1.0086 and 1.6086.

The margin error is 0.3041.

b). Test H0 : β0 = 2 versus H1 : β0 < 2 at α = 0.01 level of significance.

t = 2/0.775403 = 2.58

p-value = 0.0616

Since the p-value (0.0616) is greater than the significance level (0.01), we fail to reject the null hypothesis.

Therefore, we can conclude that β0 = 2.

c). Find the power of the test from part b) at β0 = 1.

Power = 0.1287

	r²	0.327
	r	0.572
	Std. Error	0.962
	n	4
	k	1
	Dep. Var.	y
ANOVA table
Source	SS	df	MS	F	p-value
Regression	0.9000	1	0.9000	0.97	.4279
Residual	1.8500	2	0.9250
Total	2.7500	3
Regression output				confidence interval
variables	coefficients	std. error	t (df=2)	p-value	95% lower	95% upper
Intercept	1.1500	0.775403
x	0.3000	0.3041	0.986	.4279	-1.0086	1.6086