Question

Use the Excel output in the below table to do (1) through (6) for each ofβ₀, β₁, β₂, and β₃.

y = β₀ + β₁x₁ + β₂x₂ + β₃x₃ + ε     df = n – (k + 1) = 16 – (3 + 1) = 12

Excel output for the hospital labor needs case (sample size: n = 16)

	Coefficients	Standard Error	t Stat	p-value	Lower 95%	Upper 95%
Intercept	1946.8020	504.1819	3.8613	0.0023	848.2840	3045.3201
XRay (x1)	0.0386	0.0130	2.9579	0.0120	0.0102	0.0670
BedDays(x2)	1.0394	0.0676	15.3857	2.91E-09	0.8922	1.1866
LengthSt(x3)	-413.7578	98.5983	-4.1964	0.0012	-628.5850	-198.9306

(1) Find b_j, sb_j, and the t statistic for testing H₀: β_j = 0 on the output and report their values. (Round your t value answers to 3 decimal places and other answers to 4 decimal places.)

	bj	sbj	t
H0: β0 = 0
H0: β1 = 0
H0: β2 = 0
H0: β3 = 0

(2) Using the t statistic and appropriate critical values, test H₀: β_j = 0 versus H_a: β_j ≠ 0 by setting α equal to .05. Which independent variables are significantly related to y in the model with α = .05? (Round your answer to 3 decimal places.)

t_{.025 ( )}   

H0: β0 =0;	(Click to select)RejectDo not reject H0
H0: β1 =0;	(Click to select)Do not rejectReject H0
H0: β2 =0;	(Click to select)RejectDo not reject H0
H0: β3 =0;	(Click to select)RejectDo not reject H0

(3) Using the t statistic and appropriate critical values, test H₀: β_j = 0 versus H_a: β_j ≠ 0 by setting α equal to .01. Which independent variables are significantly related to y in the model with α = .01? (Round your answer to 3 decimal places.)

t_{.005 ( )}

H0: β0 =0;	(Click to select)Do not rejectReject H0
H0: β1 =0;	(Click to select)RejectDo not reject H0
H0: β2 =0;	(Click to select)Do not rejectReject H0
H0: β3 =0;	(Click to select)RejectDo not reject H0

(4) Find the p-value for testing H₀: β_j = 0 versus H_a: β_j ≠ 0 on the output. Using the p-value, determine whether we can reject H₀ by setting α equal to .10, .05, .01, and .001. What do you conclude about the significance of the independent variables in the model? (Round your answers to p-value at β₂ = 0 and β₃ = 0 to 4 decimal places. Round other answers to 3 decimal places.)

H0: β1 = 0 is____	; Reject H0at α = (Click to select)0.010.050.001
H0: β3 = 0 is____	; Reject H0at α = (Click to select)0.050.010.001

(5) Calculate the 95 percent confidence interval for β_j. (Round your answers to 3 decimal places.)

	95% C.I.
β0	[, ]
β1	[, ]
β2	[, ]
β3	[, ]

Answer 1

given:

	Coefficients	Standard Error	t Stat	p-value	Lower 95%	Upper 95%
Intercept	1946.802	504.1819	3.8613	0.0023	848.284	3045.3201
XRay (x1)	0.0386	0.013	2.9579	0.012	0.0102	0.067
BedDays(x2)	1.0394	0.0676	15.3857	2.91E-09	0.8922	1.1866
LengthSt(x3)	-413.7578	98.5983	-4.1964	0.0012	-628.585	-198.9306

1)

	bj	sbj	t= bj/se_bj
H0: β0 = 0	1946.8020	504.1819	3.861
H0: β1 = 0	0.0386	0.0130	2.958
H0: β2 = 0	1.0394	0.0676	15.386
H0: β3 = 0	-413.7578	98.5983	-4.196

2)

alpha = 5%
	t(a/2,n-2) = t.025 (14)	Decision (Reject Ho if |t|>t(a/2,n-2) )
H0:β0 =0;	2.145	Reject Ho
H0: β1=0;	2.145	Reject Ho
H0: β2=0;	2.145	Reject Ho
H0: β3=0;	2.145	Reject Ho

3)

alpha = 1%
	t(a/2,n-2) = t.005 (14)	Decision (Reject Ho if |t|>t(a/2,n-2) )
H0:β0 =0;	2.977	Reject Ho
H0: β1=0;	2.977	Do not reject Ho
H0: β2=0;	2.977	Reject Ho
H0: β3=0;	2.977	Reject Ho

4)

	P-value = 2*(1-P(T<|t|) with (n-2) d.f.		Decision (Reject Ho if p-value < alpha )
H0:β0 =0;	0.002	= T.DIST.2T(ABS(3.8613),16-2)	Reject H0 at α = (0.10, .05, .01)
H0: β1=0;	0.010	= T.DIST.2T(ABS(2.9579),16-2)	Reject H0 at α = (0.10, .05)
H0: β2=0;	0.0000	= T.DIST.2T(ABS(15.3857),16-2)	Reject H0 at α = (0.10, .05, .01, and .001)
H0: β3=0;	0.0009	= T.DIST.2T(ABS(-4.1964),16-2)	Reject H0 at α = (0.10, .05, .01, and .001)

5)

95% C.I.	Lower = bj - t(a/2,n-2)*SE_bj		Upper = bj + t(a/2,n-2)*SE_bj
β0	865.332	1946.802-2.145*504.1819	3028.272	1946.802+2.145*504.1819
β1	0.011	0.0386-2.145*0.013	0.066	0.0386+2.145*0.013
β2	0.894	1.0394-2.145*0.0676	1.184	1.0394+2.145*0.0676
β3	-625.251	-413.7578-2.145*98.5983	-202.264	-413.7578+2.145*98.5983