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)Do not rejectReject H0
H0: β1 = 0;	(Click to select)Do not rejectReject H0
H0: β2 = 0;	(Click to select)Do not rejectReject H0
H0: β3 = 0;	(Click to select)Do not rejectReject 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)Do not rejectReject H0
H0: β2 = 0;	(Click to select)Do not rejectReject H0
H0: β3 = 0;	(Click to select)Do not rejectReject 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 β ₂ = 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)0.010.0010.05
H0: β2 = 0 is	; Reject H0at α = (Click to select)0.0010.000050.00001
H0: β3 = 0 is	; Reject H0at α = (Click to select)0.050.0010.01

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

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

(6) Calculate the 99 percent confidence interval for β_j. (Round your answers to 3 decimal places.)

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

Answer 1

1)

	bj	sbj	t
intercept	1946.8020	504.1819	3.861
XRay (x1)	0.0386	0.0130	2.958
BedDays(x2)	1.0394	0.0676	15.386
LengthSt(x3)	-413.7578	98.5983	-4.196

b0=1946.8020:

The mean of Response Y is 1946.8020 when all the values of the independent variables XRay, BedDays, and lengthST are equal to zero.

b1= 0.0386:

The mean of Response Y is increased by 0.0386 when the independent variable XRay is increased by 1 unit and keeping the other variables fixed.

b2=1.0394:

The mean of Response Y is increased by 1.0394 when the independent variable BedDays is increased by 1 unit and keeping the other variables fixed.

b3=-413.7578:

The mean of Response Y is decreased by 413.7578 when the independent variable LengthSt is increased by 1 unit and keeping the other variables fixed.

2)

The critical value of t when alpha=0.05 at df=12 is 2.179.

The absolute values of the test statistics of the independent variables XRay, BedDay, and LengthSt are 2.958 15.386 and 4.196 respectively, and greater than 2.179 critical value. Hence,

H0: β1 = 0;  Reject H0.

H0: β2 = 0;  Reject H0.

H0: β3 = 0;  Reject H0.

The independent variables XRay, BedDay, and LengthSt are significantly related to y in the model with α = .05.

3)

The critical value of t when alpha=0.01 at df=12 is 3.055.

The absolute values of the test statistics of the independent variables XRay, BedDay, and LengthSt are 2.958 15.386 and 4.196 respectively. Hence,

H0: β1 = 0; Do not Reject H0.

H0: β2 = 0;  Reject H0.

H0: β3 = 0;  Reject H0.

The independent variables BedDay, and LengthSt are significantly related to y in the model with α = .01.

4)

	βj	p-vlaue
XRay (x1)	β1	0.0120
BedDays(x2)	β2	0.0000
LengthSt(x3)	β3	0.0012

If the p-value is less than α, we reject the null hypothesis.

H0: β1 = 0; Reject H0 at α = 0.10, 0.05.

H0: β2 = 0;  Reject H0 at α = 0.10, 0.05, 0.01, 0.001.

H0: β3 = 0;  Reject H0 at α = 0.10, 0.05, 0.01.

5)

		95% confidence interval
	βj	95% Lower	95% Upper
	β0	848.284	3045.3201
XRay (x1)	β1	0.0102	0.067
BedDays(x2)	β2	0.8922	1.1866
LengthSt(x3)	β3	-628.585	-198.9306

6)

				99% confidence interval
	βj	bj	sbj	99% Lower	99% Upper
	β0	1946.8020	504.1819	406.5263	3487.0777
XRay (x1)	β1	0.0386	0.0130	-0.0011	0.0783
BedDays(x2)	β2	1.0394	0.0676	0.8329	1.2459
LengthSt(x3)	β3	-413.7578	98.5983	-714.9756	-112.5400