Week 7 GM 533 Live Lecture

B Heard(Do not post these, steal them etc. without my permission. Students can download a

copy for personal use)

Week 7 GM 533

• THE HOSPITAL LABOR NEEDS CASE 14.6• Table 14.6 presents data concerning the need for labor in

16 U.S. Navy hospitals. Here, y = monthly labor hours required; x1 = monthly X-ray exposures; x2 = monthly occupied bed days (a hospital has one occupied bed day if one bed is occupied for an entire day); and x3 = 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 x1, x2, and x3 are denoted as XRay, BedDays, and LengthStay on the output.

Week 7 GM 533

a) Find (on the output) and report the values of b1, b2, and b3, the least squares point estimates of β1, β2, and β3. Interpret b1, b2, and b3 .

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?

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a) b1 b2 b3

For b1, As the x ray exposures go up by 1, monthly labor hours go up by ______For b2 “ “ “For b3

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b) Just plug values into the equation ŷ = 1946.802 + 0.3858x1 + 1.0394x2 - 413.7578x3

c) How much less or greater was the value calculated in part than 17207.31. That is all they are asking.

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• THE FRESH DETERGENT CASE 14.12• Model: y = β0 + β1 x1 + β2 x2 + β3 x3 + ε Sample

size: n = 30

Week 7 GM 5331 Report SSE, s2, and s as shown on the output. Calculate s2 from SSE and other

numbers. 2 Report the total variation, unexplained variation, and explained variation as shown

on the output. 3 Report R2 and (R Bar)2 as shown on the output. Interpret R2 and (R Bar) 2. Show how

(R Bar)2 has been calculated from R2 and other numbers. 4 Calculate the F (model) statistic by using the explained variation, the unexplained

variation, and other relevant quantities. Find F (model) on the output to check your answer (within rounding).

5 Use the F (model) statistic and the appropriate critical value to test the significance of the linear regression model under consideration by setting α equal to .05.

6 Use the F (model) statistic and the appropriate critical value to test the significance of the linear regression model under consideration by setting α equal to .01.

7 Find the p-value related to F (model) on the output. Using the p-value, test the significance of the linear regression model by setting α = .10, .05, .01, and .001. What do you conclude?

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1) SSE s2 sTo calculate s2 use the formula s2 = SSE/(n-(k+1)) where n is the sample size (30)

and k is 3. Take square root of s2 to get s.

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2) Total variation, Unexplained variation, Explained variation

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3) R2 (R Bar)2

Use equation and note if R2 and (R Bar)2 are close or not.

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• On part 4, use the equation

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• On part 5, compare your calculated value of F(model) from part 4 to the F.05 value from the table for 3 and 26 degrees of freedom. If your calculated value is greater than the table value, we would reject H0: β1 = β2 = β3 = 0

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• On part 6, do just like you did part 5 except compare your calculated value of F(model) from part 4 to the F.01 value from the table

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• Compare your p-value to alpha’s of 0.10, 0.05, 0.01 and 0.001 and note what it means. For example if your p-value was less than all of those alpha’s you would have very strong evidence that your null hypothesis was incorrect and very strong evidence that at least one of your x1, x2 or x3 is significantly related to y.

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• THE FUEL CONSUMPTION CASE 14.16• Use the MINITAB output in Figure 14.9(a) to

do (1) through (6) for each of β0, β1, and β2 .

Week 7 GM 5331 Find bj, sbj, and the t statistic for testing H0 : βj = 0 on the output and report their

values. Show how t has been calculated by using bj and sbj .

2 Using the t statistic and appropriate critical values, test H0 : βj = 0 versus Ha : ≠ 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 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?

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 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 βj. Discuss one practical application of this interval.

6 Calculate the 99 percent confidence interval for βj .

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• I will get you started on doing this for β0, you will also need to do it for β1 and β2 .

b0

sb0

tWheret = b0/sb0

This is part 1

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Part 2) We would reject H0: β0 = 0 (and conclude that the intercept is significant) with a = .05 if |t| > t.05/2 = t.025

Since t.025 = 2.571 (with n – (k + 1) = 8 – (2 + 1) = 5 degrees of freedom), we have t = 15.32 > t.025 = 2.571.

We reject H0: β0 = 0 with a = .05 and conclude that the intercept is significant at the .05 level.

Week 7 GM 533

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• Part 3 is the same as the previous one, but use a = .01 . Remember t.01/2 = t.005

(Use table value for t.005)

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• Part 4 • The Minitab output (given to us) tells us that

the p-value for testing H0: β0 = 0 is 0.000. Since it is less than each given value of a, we reject H0: β0 = 0 at each of these values of a. We can conclude that the intercept β0 is significant at the .10, .05, .01, and .001 levels of significance.

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• Part 5, use the formula for a 95% confidence interval for β0

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• Part 6, use the formula for a 99% confidence interval for β0 (similar to previous part)

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• Now you have to go through ALL 6 parts again for β1 and β2 . So this problem actually has 18 parts.

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THE REAL ESTATE SALES PRICE CASE 14.23The following MINITAB output relates to a house

having 2,000 square feet and a rating of 8.

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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.

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a) ŷ = remember this is in 1000s of dollars

The 95% CI is also given

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b) ŷ = remember this is in 1000s of dollars

The 95% PI is also given

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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.

So we know

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• So the previous chart gives us the Distance Value

• The 99% confidence interval for mean sales price is

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We calculated our Distance Value and we were given s. You can now get the lower and upper for the 99% CI

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• For the 99% prediction interval, it would be very similar but under the square root you now have “1 + Distance Value”