Gm533 week 6 lecture april 2012
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  • 1. Week 6 Checkpoint Help GM533B Heard(Don’t copy or post without mypermission, students maydownload a copy for personal use)
  • 2. Week 6 Checkpoint Help GM533• Always be on the lookout for typos…..• Remember I usually do these charts “on the fly” ▫ Do not rely solely on me or these charts to do well in this class! ▫ The live lectures are only part of what you have available to you ▫ Good Luck!
  • 3. Week 6 Checkpoint Help GM53313.8 THE REAL ESTATE SALES PRICE CASE• A real estate agency collects data concerning y = the sales price of a house (in thousands of dollars), and x = the home size (in hundreds of square feet). The data are given in the table below. The MINITAB output from fitting a least squares regression line to the data is on the next page.
  • 4. Week 6 Checkpoint Help GM533 a) By using the formulas illustrated in Example 13.2 (see page 497) and the data provided, verify that (within rounding) b0 = 48.02 and b1 = 5.700, as shown on the MINITAB output. I put the data in Excel and did the math there. The formulas are provided in the text.
  • 5. Week 6 Checkpoint Help GM533 Size (x) Price (y) 23 180 11 98.1 20 173.1 17 136.5 15 141 21 165.9 24 193.5 13 127.8 19 163.5 25 172.5
  • 6. Week 6 Checkpoint Help GM533 Size (x) Price (y) xy x^2 23 180 4140 529 11 98.1 1079.1 121 20 173.1 3462 400 17 136.5 2320.5 289 15 141 2115 225 21 165.9 3483.9 441 24 193.5 4644 576 13 127.8 1661.4 169 19 163.5 3106.5 361 25 172.5 4312.5 625 Sum of xs Sum of ys sum of xys sum of x^2s (sum of xs)^2 188 1551.9 30324.9 3736 35344 n 10 SSxy SSxx 1149.18 201.6 b1 y bar x bar 5.700298 155.19 18.8 b0 48.0244 y hat = b0 + b1*x y hat = 48.0244 plus 5.70029762 x
  • 7. Week 6 Checkpoint Help GM533b) Interpret the meanings of b0 and b1. Does the interpretation of b0 make practical sense?The b1 is 5.70 which basically is saying for every 100 square feet the average sales price increases that muchb0 is the y intercept when x is zero. In other words, it says that a house with zero square feet would cost about 48 thousand dollars. No, this doesn’t make sense (I will talk about this).
  • 8. Week 6 Checkpoint Help GM533c) Write the least squares prediction equation.y hat = 48.02 + 5.7x y hat = b0 + b1*x y hat = 48.0244 plus 5.70029762 x
  • 9. Week 6 Checkpoint Help GM533d) Use the least squares line to obtain a point estimate of the mean sales price of all houses having 2,000 square feet and a point prediction of the sales price of an individual house having 2,000 square feet.Plug and chug (insert 20 for x remember the size was in 100’s of square feet)y hat = 48.02 + 5.7(20) = 162.02 (in thousands)So the predicted price would be $162,020
  • 10. Week 6 Checkpoint Help GM53313.21 THE STARTING SALARY CASEThe MINITAB output of a simple linear regression analysis of the data set for this case (see Exercise 13.4 on page 501) is given in Figure 13.11. Recall that a labeled MINITAB regression output is on page 509.
  • 11. Week 6 Checkpoint Help GM533
  • 12. Week 6 Checkpoint Help GM533 bo (Part a) b1
  • 13. Week 6 Checkpoint Help GM533 SSE (Part b) s^2 s
  • 14. Week 6 Checkpoint Help GM533 sb1 (Part c) t t = b1/sb1 (show)
  • 15. Week 6 Checkpoint Help GM533 df (Part d)
  • 16. Week 6 Checkpoint Help GM533 (Part d continued) t.025 = 2.57 compared to 14.44 ? Reject because it’s way out there in the rejection region  Reject H0, there is strong evidence of something going on Table from with x and y http://www.statsoft.com/textbook/distribution-tables/#t (I just search on Internet, you have one in text)
  • 17. Week 6 Checkpoint Help GM533 (Part e) t.005 = 4.03 compared to 14.44 ? Reject because it’s still way out there in the rejection region  Reject H0, there is strong evidence of something going on Table from with x and y (Very http://www.statsoft.com/textbook/distribution-tables/#t strong relationship) (I just search on Internet, you have one in text)
  • 18. Week 6 Checkpoint Help GM533f) p value was .000 agrees with previous two to reject at all alphas. Very very strong evidence of an x and y relationshipg) 95% CI Just use what you have found aboveThe interval is b1 +/- t.025 sb1h) 99% CI Just use what you have found aboveThe interval is b1 +/- t.005 sb1
  • 19. Week 6 Checkpoint Help GM533 bo sbo (Part i) t t = b0/sb0 (show)
  • 20. Week 6 Checkpoint Help GM533j) p value was .000, reject at all alphas. Very very strong evidence of an x and y relationshipk) Use the formulas and the data! (should give you the same answer you got in part c for sb1 and in part i for sbo.
  • 21. Week 6 Checkpoint Help GM53313.30 THE FUEL CONSUMPTION CASEThe following partial MINITAB regression output for the fuel consumption data relates to predicting the city’s fuel consumption (in MMcf of natural gas) in a week that has an average hourly temperature of 40°F.
  • 22. Week 6 Checkpoint Help GM533
  • 23. Week 6 Checkpoint Help GM533 Part a
  • 24. Week 6 Checkpoint Help GM533 Part b
  • 25. Week 6 Checkpoint Help GM533c) Remembering that s = .6542; SSxx = 1,404.355; n = 8, hand calculate the distance value when x0 = 40. Remembering that the distance value equals , use s and from the computer output to calculate (within rounding) the distance value using this formula. Note that, because MINITAB rounds sy, the first hand calculation is the more accurate calculation of the distance value.
  • 26. Week 6 Checkpoint Help GM533Distance Value (dv) = 1/8 + (40-43.98)2 / 1404.355 = 0.1363AndDistance Value (dv) = (0.241 / 0.6542)2 = 0.1357
  • 27. Week 6 Checkpoint Help GM533d) Remembering that for the fuel consumption data b0 = 15.84 and b1 = -.1279, calculate (within rounding) the confidence interval of part a and the prediction interval of part b.CI: 15.84 - 0.1279 (40) ± 2.447*0.6542*√(0.1363) = [10.13299, 11.31501]For PI, just substitute 1.1363 for 0.1363
  • 28. Week 6 Checkpoint Help GM533e) Remember you are predicting for one day, so use prediction interval.Since 9.01 < 9.595 and 12.43 > 11.847 thecity cannot be ____ % confident it won’t pay a fine. (Fill in the blank)
  • 29. Week 6 Checkpoint Help GM533THE FRESH DETERGENT CASEIn Exercises 13.50 through 13.55, we give MINITAB and Excel outputs of simple linear regression analyses of the data sets related to six previously discussed case studies. Using the appropriate computer output,a Use the explained variation and the unexplained variation as given on the computer output to calculate (within rounding) the F (model) statistic.b Utilize the F (model) statistic and the appropriate critical value to test H0 : β1 = 0 versus Hα : α1 ≠ 0 by setting a equal to .05. What do you conclude about the regression relationship between y and x?c Utilize the F (model) statistic and the appropriate critical value to test H0 : β1 = 0 versus Hα : β1 ≠ 0 by setting a equal to .01. What do you conclude about the regression relationship between y and x?d Find the p -value related to F (model) on the computer output and report its value. Using the p -value, test the significance of the regression model at the .10, .05, .01, and .001 levels of significance. What do you conclude?e Show that the F (model) statistic is (within rounding) the square of the t statistic for testing H0 : β1 = 0 versus Hα : b1 ≠ 0. Also, show that the F.05 critical value is the square of the t025 critical value.Note that in the lower right hand corner of each output we give (in parentheses) the number of observations, n, used to perform the regression analysis and the t statistic for testing H0 : β1 = 0 versus Hα : β1 ≠ 0.
  • 30. Week 6 Checkpoint Help GM533 Part a
  • 31. Used a table at http://www.statsoft.com/textbook/distribution-tables/#f (I was lazy)
  • 32. Week 6 Checkpoint Help GM533F.05 =4.196, reject H0 (df1 (top) = 1, df2 (left) = 28). Looks like there is STRONG evidence of a significant relationship between x and y.
  • 33. c) F.01 =7.636, reject H0 (df1 (top) = 1, df2 (left) = 28). Looks like there is STRONG (Very because of .01) evidence of a significant relationship between x and y.Used a table at http://www.statsoft.com/textbook/distribution-tables/#f (I
  • 34. Week 6 Checkpoint Help GM533d) p-value is ______ (smaller than all levels of significance), reject H0 . Again, pick your “ly” ending word that means there is definitely strong evidence of a significant relationship between x and y.
  • 35. Week 6 Checkpoint Help GM533 Part e Square this number, you should see that it gives you a result within rounding error. Then get your table value for t.025 and verify (t.025)2 = 4.19 = F.05
  • 36. Week 6 Checkpoint Help GM533I will post these in the Stat Cave atwww.facebook.com/statcave