Question

# Use the Excel output in the below table to do (1) through (6) for each ofβ0,...

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 bj, sbj, and the t statistic for testing H0: βj = 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: βj = 0 versus Ha: βj ≠ 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: βj = 0 versus Ha: βj ≠ 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: βj = 0 versus Ha: βj ≠ 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 βj. (Round your answers to 3 decimal places.)

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

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

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