R-SQUARED RESID. MEAN SQUARE (MSE) 1.885E+07 ADJUSTED R-SQUARED STANDARD ERROR OF ESTIMATE

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1 These are additional sample problems for the exam of 2012.APR.11. The problems have numbers 6, 7, and 8. This is not Minitab output, so you ll find a few extra challenges. 6. The following information concerns a regression of Y on the variables SS, TT, UU, VV, WW, and XX. Details of a best subset and a stepwise run are included. Questions follow. STATISTIX 4.0 CORRELATIONS (PEARSON) Y SS TT UU VV WW SS TT UU VV WW XX CASES INCLUDED 102 MISSING CASES 0 UNWEIGHTED LEAST SQUARES LINEAR REGRESSION OF Y PREDICTOR VARIABLES COEFFICIENT STD ERROR STUDENT S T P VIF CONSTANT SS TT UU VV WW XX R-SQUARED RESID. MEAN SQUARE (MSE) 1.885E+07 ADJUSTED R-SQUARED STANDARD ERROR OF ESTIMATE SOURCE DF SS MS F P REGRESSION E E RESIDUAL E E+07 TOTAL E+11 CASES INCLUDED 102 MISSING CASES 0 1

2 BEST SUBSET REGRESSION MODELS FOR Y UNFORCED INDEPENDENT VARIABLES: (A)SS (B)TT (C)UU (D)VV (E)WW (F)XX 2 BEST MODELS FROM EACH SUBSET SIZE LISTED. ADJUSTED P CP R SQUARE R SQUARE RESID SS MODEL VARIABLES E+11 INTERCEPT ONLY E+10 B E+11 A E+09 D F E+10 C F E+09 A D F E+09 B D F E+09 A B D F E+09 A C D F E+09 A B C D F E+09 A B D E F E+09 A B C D E F STEPWISE REGRESSION OF Y UNFORCED VARIABLES: SS TT UU VV WW XX F TO ENTER 4.00 F TO EXIT 4.00 S T U V W X STEP R SQ MSE T S T U V W X E E B E B. D E B. D. F E A B. D. F E A.. D. F RESULTING STEPWISE MODEL VARIABLE COEFFICIENT STD ERROR STUDENT S T P VIF CONSTANT SS VV XX CASES INCLUDED 102 R SQUARED MSE 1.925E+07 MISSING CASES 0 ADJ R SQ SD VARIABLES NOT IN THE MODEL CORRELATIONS VARIABLE MULTIPLE PARTIAL T TT UU WW

3 In fitting the full model, using all six independent variables, which variables have non-significant coefficients? (b) Would the regression be regarded as useful in predicting Y? (c) Based on the BEST SUBSET REGRESSION MODELS FOR Y section, which set of independent variables would you use? Based on the STEPWISE REGRESSION OF Y section, which set of independent variables would you use? 7. The section of information that follows contains information about five variables: Y, CHIPS, RAISINS, MARSHM, REPACK, FLAKE. A regression is also involved in this. Some positions are marked with letters and will be involved in the questions. Questions follow. DESCRIPTIVE STATISTICS Y CHIPS RAISINS MARSHM REPACK FLAKE N MEAN SD MINIMUM MEDIAN MAXIMUM 1.050E SKEW CORRELATIONS (PEARSON) Y CHIPS RAISINS MARSHM REPACK CHIPS RAISINS MARSHM REPACK FLAKE CASES INCLUDED 59 MISSING CASES 0 UNWEIGHTED LEAST SQUARES LINEAR REGRESSION OF Y PREDICTOR VARIABLES COEFFICIENT STD ERROR STUDENT S T P VIF CONSTANT CHIPS RAISINS (f) MARSHM REPACK FLAKE R-SQUARED RESID. MEAN SQUARE (MSE) ADJUSTED R-SQUARED STANDARD ERROR OF ESTIMATE SOURCE DF SS MS F P REGRESSION E E RESIDUAL 53 --(h) TOTAL E+07 CASES INCLUDED 59 MISSING CASES 0 3

4 What is the correlation coefficient between CHIPS and REPACK? (b) In the regression, which variable was designated as the dependent variable? (c) What is the maximum value of Y in the data base? How many data points are used in this problem? (e) What is the value of s ε, the standard error of regression? (f) What value belongs in the position marked --(f)--? (g) For which of the independent variables would you accept the null hypothesis that the true coefficient value is zero? (h) What value belongs in the position marked --(h)--? (i) The value of the F statistic was given as What null hypothesis is tested by this statistic? Do you accept or reject the null hypothesis? (j) Suppose that you wanted to run the regression of REPACK on FLAKE. That is, you want to treat REPACK as the dependent variable and FLAKE as the independent variable. What is the estimated slope in this regression? 8. A regression of PROFIT on SIZE and TRAFFIC was performed for a set of 140 randomly-selected gasoline stations. PROFIT was measured in thousands of dollars, SIZE was number of gasoline pumps, and TRAFFIC was measured in thousands of cars per hour passing the station (measured over several business days). The fitted regression was PRÔFIT = SIZE TRAFFIC (3.04) (9.18) (8.15) The figures in ( ) are standard errors. (b) (c) State the regression model on which this is based. What profit would you predict for a station with 6 gasoline pumps and a traffic value of 4,000 cars per hour? Which of the estimated coefficients is/are significantly different from zero? It was noticed that the correlation coefficient between SIZE and TRAFFIC was Will this present a problem? SOLUTIONS BEGIN ON NEXT PAGE 4

5 6. The T statistics for TT, UU, and WW are non-significant. (The corresponding P values all exceed 0.05.) (b) Yes. The R 2 statistic exceeds 99%, so there is no doubt that this is a good regression. (c) The R 2 value jumps to 97.37% using just variables VV and XX (labeled as D F in the schematic display). R 2 jumps to 99.02% using variables SS, VV, XX (labeled as A D F). Either of these looks reasonable, but the C p statistic certainly favors the three-predictor model, using SS, VV, and XX. The stepwise procedure works its way through three models that seems to fit well. These are, in the order discovered: Independent Variables TT VV XX R % SS TT VV XX 99.04% SS VV XX 99.02% The preference ought to be for the last, namely the regression of Y on SS, VV, and XX. It is somewhat interesting that this is the second three-predictor model that the program finds. 7. (b) The value is , read directly from the output. Y was used as the dependent variable. (c) It s 1.050E+04, meaning 10,500. There are 59 data points. (e) It s (f) Since STUDENT S T = COEFFICIENT STD ERROR, we must have (g) Only for MARSHM, for which the value of STUDENT S T is only That is, we accept : β MARSHM = 0. H 0 5

6 (h) Since SS(REGRESSION) + SS(RESIDUAL) = SS(TOTAL), we have or 8.990E+06 + SS(RESIDUAL) = 1.086E+07 8,990,000 + SS(RESIDUAL) = 10,860,000 which leads to SS(RESIDUAL) = 10,860,000-8,990,000 = 1,870,000. The value was actually printed as 1.873E+06. (i) This is a test of the hypothesis H 0 : β CHIPS = 0, β RAISINS = 0, β MARSHM = 0, β REPACK = 0, β FLAKE = 0 which says that all five coefficients are equal to zero. With a calculated value of 50.89, the null hypothesis is soundly rejected. The official 5% cutoff point, using (5, 53) degrees of freedom, is 2.39; the 1% cutoff point is units of REPACK (j) This estimated slope will have units which are. The actual units of FLAKE SD(REPACK) formula is b REPACK on FLAKE = r REPACK, FLAKE and the numeric SD(FLAKE) value is This is PROFIT i = β 0 + β 1 SIZE i + β 2 TRAFFIC i + ε i where the noise terms ε 1, ε 2,, ε 140 are independent with means 0 and equal standard deviations σ. (Other notational schemes would be permitted here, of course.) (b) Find this as (6) (4) = , meaning $369, (c) The constant is standard errors away from zero, so it is certainly significant. (Most people would not even bother about asking for the significance of the constant.) The coefficient of SIZE is standard errors away from zero, so it 918. would be judged not statistically significant The coefficient of TRAFFIC is 815. be judged statistically significant. standard errors away from zero; it would No. 6