THE HOSPITAL LABOR NEEDS CASE Table 14.6 presents data concerning the need for labor in 16 U.S. Navy hospitals.

14.6

THE HOSPITAL LABOR NEEDS CASE

Table 14.6 presents data concerning the need for labor in 16 U.S. Navy hospitals. Here, y = monthly labor hours required; x ₁ = monthly X -ray exposures; x ₂ = monthly occupied bed days (a hospital has one occupied bed day if one bed is occupied for an entire day); and x ₃ = average length of patients stay (in days). Figure 14.8 gives the Excel output of a regression analysis of the data using the model days (a hospital has one occupied bed day if one bed is occupied for an entire day); and x ₃ = average length of patients’ stay (in days). Figure 14.8 gives the Excel output of a regression analysis of the data using the model

Note that the variables x ₁ , x ₂ , and x ₃ are denoted as XRay, BedDays, and LengthStay on the output.

a Find (on the output) and report the values of b ₁ , b ₂ , and b ₃ , the least squares point estimates of β ₁ , β ₂ , and β ₃ . Interpret b ₁ , b ₂ , and b ₃ .
b Consider a questionable hospital for which XRay = 56,194, BedDays = 14,077.88, and LengthStay = 6.89. A point prediction of the labor hours corresponding to this combination of values of the independent variables is given on the Excel add-in output. Report this point prediction and show (within rounding) how it has been calculated.
c If the actual number of labor hours used by the questionable hospital was y = 17,207.31, how does this y value compare with the point prediction?

14.6

14.8

Figure 14.8: Excel Output of a Regression Analysis of the Hospital Labor Needs Data Using the Model y = β ₀ + β ₁ x ₁ + β ₂ x ₂ + β ₃ x ₃ + ε

14.12 THE FRESH DETERGENT CASE

Model: y = β ₀ + β ₁ x ₁ + β ₂ x ₂ + β ₃ x ₃ + ε Sample size: n = 30

14.16

THE FUEL CONSUMPTION CASE

Use the MINITAB output in Figure 14.9(a) to do (1) through (6) for each of β ₀ , β ₁ , and β ₂ .

Note: Hypothesis testing only for independent variable slopes (b ₁ and b ₂ ). Calculate the 95 percent Confidence intervals for b ₁ and b ₂ .

1 Find b _j , s _bj , and the t statistic for testing H ₀ : β _j = 0 on the output and report their values. Show how t has been calculated by using b _j and s _bj .
2 Using the t statistic and appropriate critical values, test H ₀ : β ₁ = 0 versus H _a : ≠ 0 by setting a equal to .05. Which independent variables are significantly related to y in the model with a = .05?
3 Using the t statistic and appropriate critical values, test H ₀ : 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?
4 Find the p -value for testing H ₀ : β _j = 0 versus H _a : β _j ≠ 0 on the output. Using the p -value, determine whether we can reject H ₀ by setting a equal to .10, .05, .01, and .001. What do you conclude about the significance of the independent variables in the model?
5 Calculate the 95 percent confidence interval for β _j . Discuss one practical application of this interval.
6 Calculate the 99 percent confidence interval for β _j .
14.23
THE REAL ESTATE SALES PRICE CASE
The following MINITAB output relates to a house having 2,000 square feet and a rating of 8.

a Report (as shown on the output) a point estimate of and a 95 percent confidence interval for the mean sales price of all houses having 2,000 square feet and a rating of 8.
b Report (as shown on the output) a point prediction of and a 95 percent prediction interval for the actual sales price of an individual house having 2,000 square feet and a rating of 8.
c Find 99 percent confidence and prediction intervals for the mean and actual sales prices referred to in parts a and b . Hint: n = 10 and s = 3.24164. Optional technical note needed.