Question

Choose one correct answer from the four options below. A large, national grocery retailer tracks productivity and costs of its facilities closely. . Each data point for each variable represents one week of activity. The variables included are the number of cases shipped (X1), the indirect costs of the total labor hours as a percentage (X2), a qualitative predictor called holiday that is coded 1 if the week has a holiday and 0 otherwise (X3), and the total labor hours (Y) . Assume that the regression model Y = B0 + B1X1 + B2X2 + B3X3 + e with the e’s being independent N(0, sigma²) is appropriate. Find joint interval estimates of B1, B2 and B3, using a 90% percent family confidence coefficient. Choose one correct answer from the four options below.

1. 0.000117 ≤ B1≤ 0.016; 13.7616 ≤ B2≤ 37.4296; 38.3037 ≤ B3≤ 76.8053
2. -0.00117 ≤ B1≤ 0.016; -65.7616 ≤ B2≤ 73.4296; 586.3037 ≤ B3≤ 870.8053
3. -0.0000117 ≤ B1≤ 0.0016; -63.7616 ≤ B2≤ 37.4296; 486.3037 ≤ B3≤ 760.8053
4. 0.0117 ≤ B1≤ 0.16; -63.7616 ≤ B2≤ 15.4296; 486.3037 ≤ B3≤ 760.8053

Y   X1   X2   X3
4264   305657   7.17   0
4496   328476   6.20   0
4317   317164   4.61   0
4292   366745   7.02   0
4945   265518   8.61   1
4325   301995   6.88   0
4110   269334   7.23   0
4111   267631   6.27   0
4161   296350   6.49   0
4560   277223   6.37   0
4401   269189   7.05   0
4251   277133   6.34   0
4222   282892   6.94   0
4063   306639   8.56   0
4343   328405   6.71   0
4833   321773   5.82   1
4453   272319   6.82   0
4195   293880   8.38   0
4394   300867   7.72   0
4099   296872   7.67   0
4816   245674   7.72   1
4867   211944   6.45   1
4114   227996   7.22   0
4314   248328   8.50   0
4289   249894   8.08   0
4269   302660   7.26   0
4347   273848   7.39   0
4178   245743   8.12   0
4333   267673   6.75   0
4226   256506   7.79   0
4121   271854   7.89   0
3998   293225   9.01   0
4475   269121   8.01   0
4545   322812   7.21   0
4016   252225   7.85   0
4207   261365   6.14   0
4148   287645   6.76   0
4562   289666   7.92   0
4146   270051   8.19   0
4555   265239   7.55   0
4365   352466   6.94   0
4471   426908   7.25   0
5045   369989   9.65   1
4469   472476   8.20   0
4408   414102   8.02   0
4219   302507   6.72   0
4211   382686   7.23   0
4993   442782   7.61   1
4309   322303   7.39   0
4499   290455   7.99   0
4186   411750   7.83   0
4342   292087   7.77   0

Answer 1

Regressing Y against X1, X2 and X3 in Excel (Go to Data tab -> Data Analysis -> Regression, choose X1, X2, X3 as the X-column, and Y as the Y-column) at a confidence level of 90%, we get the following output

	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 90.0%	Upper 90.0%
Intercept	4149.887212	195.5654055	21.21995	4.9E-26	3756.677	4543.098	3821.880182	4477.894242
X1	0.00078708	0.000364554	2.159023	0.035877	5.41E-05	0.00152	0.000175642	0.001398519
X2	-13.16601919	23.0917315	-0.57016	0.571227	-59.5951	33.26302	-51.89602999	25.56399162
X3	623.5544807	62.64094656	9.954423	2.94E-13	497.6064	749.5025	518.4915695	728.617392

The Lower and Upper limits for 90% confidence interval for all the regression coefficients are shown in bold in the last column above (these are accurate as they're calculated in Excel as explained above)