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FOOTBALL ASSIGNMENT #5 – LEAST SQUARES REGRESSION EQUATION Hello Everybody. FOOTBALL ASSIGNMENT #5 USE THE LEAST SQUARES REGRESSION EQUATION on Table 7.1 (page 159) You will need from your previous football equation (chapter 6) the SSx, the SSy, the Means of both X and Y and the r that you came up with. Then go to Table 7.3 on page 163 and Find Your Standard Error of Estimate. We’ll just plug the NFL Scoring Mean of 23 into the Equation as our Home Score and interpret the results of what would be our correlating Away Score (please see my example) Football Assignment #5: Least Squares Regression Equation…TAMPA BAY BUCS EXAMPLE Top of Form IN CHAPTER 7, WE USE THE SAME DATA WE USE TO OUR PEARSON “r” and we even use the Pearson “r’ TO PREDICT…………………….What Y would be, given a particular X. I DID ALL 3 OF MY r EXAMPLES, to show you how to get the Y 3 Different Numbers and 3 Different Distributions, ALL WITH TOTALLY DIFFERENT KINDS OF RESULTS I DID THE HOME vs AWAY IN SCORING ORDER HOME VS AWAY IN GAME ORDER AND JUST GAME ORDER 1-8, 9-16 THERE’S ALSO A NEW TERM THAT WILL BECOME EXTREMELY IMPORTANT TO US THE REST OF THE SEMESTER. FIRST, REMEMBER, THIS IS A EQUATION TO PREDICT A SCORE PREDICT SHOULD MAKE YOU THINK OF INFERENTIAL STATISTICS, BECAUSE THAT’S KIND OF WHAT WE ARE DOING. IN THIS CHAPTER WE ARE INTRODUCED TO THE WORD STANDARD ERROR THIS TERM BECOMES OUR NEW TERM THAT REPRESENTS THE OLD STANDARD DEVIATION!!!! Home vs Away……In Scoring Order, LOWEST TO HIGHEST Top of Form 1) OK, as stated before, we need our r, SSx and SSy from the Previous Equation in Chapter 6 r = .91 SSx = 1,124 SSy = 898 2) b = The square root of SSy/SSx Times the r Square Root of (898/1,124) Times .91 (Square root of .80) x .91 or .89 X .91 = .81 b = .81 3) Again from our Sums in Chapter 6, we divide by 8 to get the Means of X and Y X-Bar = 228/8 = 28.5 Y-Bar = 264/8 = 33 4) a= Y-Bar – (b)(X-bar) = 33 – (.81) (28.5) = 33 – 23.085 = 9.915 5) Y = (.81) (X) + 9.915   STANDARD ERROR 1) SSy = 898 , r = .91 2) Sy/x = Square Root of SSy (1-r squared)/n-2 or the Square root of 898 (1-.91 squared)/8-2 or Square root of 898 (1 – .8281)/6 or Square root of (898 x .172)/6 or Square Root of 154.37/6 = square root of 25.73 = 5.07   Now we plug in the NFL Average of 23 as the X (which would represent the Home Score) Y = (.81) (23) + 9.915 = 18.63 + 9.915 = 28.545 Plus/Minus the Standard Error of 5.07 and the Bucs Away Score would be between 23.475 and 33.62 NOTICE MY MEANS ARE 5.5 POINTS APART and IF MY X = 23 and Predicted Y = 28.545 that’s a difference of that 5.5………BECAUSE WE ARE DEALING WITH AN ALMOST PERFECT RELATIONSHIP r = .91 . This was my equation from the Correlating Chapter 6 work to get the pearson “r’ Hello Everybody, Remember we are going to correlate Home Games (X) with Away Games  (Y),  Put each set in order from Lowest To Highest. Home(X)          Away(Y)         XY                X(2)               Y(2)   3                     19                  57                    9               361 24                     23                552                576               529 24                     25                600                576               625 26                     28                728                676               784 31                     31                961                961               961 38                     45              1710              1444             2025 38                     46              1748              1444             2116 44                     47              2068              1936             2209 228                 264              8434              7622             9610 n=8 (E = Sum) OKAY YOU SHOULD RECOGNIZE THE SUM OF SQUARES EQUATIONS FROM THE COMPUTATION FORMULA FOR STANDARD DEVIATION IT’S THE SUM of the SQUARED COLUMN or for when we multiply XY, the SUM OF THAT COLUMN  MINUS THE SUM OF EITHER X x Y or X-squared and Y-squared DIVIDED by n (the Number of our Sample Size. SPxy =  Exy – (Ex)(EY)/n = 8,434 – (228)(264)/8 = 8,434 – 60,192/8 = 8,434 – 7,524 = 910 SSx = EX(2) – (EX)2/n = 7,622 – (228 x 228)/8 = 7,622 – 51,984/8 = 7,622 – 6,498 = 1,124 SSy = EY(2) – (EY)2/n = 9,610 – (264 x 264)/8 = 9,610 – 69,696/8 = 9,610 – 8,712 = 898 r = SPxy                    Sq rt (SSx) (SSy)   r= ____910____ sq rt (1,124) (898)   r = __910__ sq rt 1,009,352.   r = 910/1,004.67 = .91 .91 is obviously a strong correlation, BECAUSE WE PUT THEM IN ORDER OURSELVES, IT WASN’T RANDOM………….L JUST WANT YOU TO GET THE FEEL OF THE EQUATION. BUT WE’LL SHOW YOU SOME OTHER FORMATS TOO Bottom of Form     Home vs Away in “REAL” order from 1st game to 8th game Top of Form 1) OK, as stated before, we need our r, SSx and SSy from the Previous Equation in Chapter 6 r = -.31 SSx = 1,124 SSy = 898 2) b = The square root of SSy/SSx Times the r Square Root of (898/1,124) Times -.31 (Square root of .80) x -.31 or .89 X -.31 = -.276 b = -.276 3) Again from our Sums in Chapter 6, we divide by 8 to get the Means of X and Y X-Bar = 228/8 = 28.5 Y-Bar = 264/8 = 33 4) a= Y-Bar – (b)(X-bar) = 33 – (-.276) (28.5) = 33 – -7.86 = 40.86 5) Y = (-.276) (X) + 40.86   STANDARD ERROR 1) SSy = 898 , r = -.31 2) Sy/x = Square Root of SSy (1-r squared)/n-2 or the Square root of 898 (1- -.31 squared)/8-2 or Square root of 898 (1 – .0961)/6 or Square root of (898 x .904)/6 or Square Root of 811.7/6 = square root of 135.28 = 11.63   Now we plug in the NFL Average of 23 as the X (which would represent the Home Score) Y = (-.276) (23) + 40.86 = -6.35 + 40.86 = 34.3 Plus/Minus Standard Error of 11.63 and the Bucs Away Score would be between 22.67 and 45.93 OK, IN THIS ONE…..we were dealing with a NEGATIVE Pearson r (-.31) But more importantly, an equation with a very small relationship……..even with the Negative, we see that the Y (which had a higher Mean than X) is still Higher then the X………..and our STANDARD ERROR IS MUCH HIGHER This was my equation from the Correlating Chapter 6 work to get the pearson “r’ Hello Everybody, Remember we are going to correlate Home Games (X) with Away Games  (Y),  But this time, NOT in order from Lowest To Highest……..In order of appearance Home(X)          Away(Y)          XY               X(2)            Y(2) 31                      23                713               961              529 38                      28              1064             1444              784 38                      19                722             1444              361   3                      45                135                   9            2025 24                      25                600               576              625 24                      46              1104               576            2116 26                      31                806               676              961 44                      47              2068             1936            2209 228                  264              7212             7622            9610 n=8 (E = Sum) OKAY YOU SHOULD RECOGNIZE THE SUM OF SQUARES EQUATIONS FROM THE COMPUTATION FORMULA FOR STANDARD DEVIATION IT’S THE SUM of the SQUARED COLUMN or for when we multiply XY, the SUM OF THAT COLUMN  MINUS THE SUM OF EITHER X x Y or X-squared and Y-squared DIVIDED by n (the Number of our Sample Size. SPxy =  Exy – (Ex)(EY)/n = 7,212 – (228)(264)/8 = 7,212 – 60,192/8 = 7,212 – 7524 = -312 SSx = EX(2) – (EX)2/n = 7,622 – (228 x 228)/8 = 7,622 – 51,984/8 = 7,622 – 6,498 = 1,124 SSy = EY(2) – (EY)2/n = 9,610 – (264 x 264)/8 = 9,610 – 69,696/8 = 9,610 – 8,712 = 898 r = SPxy                    Sq rt (SSx) (SSy)   r= ____-312_____ sq rt (1,124) (898)   r = _  -312__ sq rt 1,009,352   r = -312/1,004.67 = -.31   OKAY, WANT YOU TO NOTICE TWO THINGS 1) ONLY THE XY COLUMN (the 3rd One Changed)…….the order of the other changed, but the SUMS ARE WHAT THEY ARE NO MATTER THE ORDER, including the X-squared and Y-squared Columns 2) WE CAN HAVE A NEGATIVE r…….our r’s can be anywhere from -1.00 to 1.00, can not exceed in either direction -1 or 1.  The Number is the Strength, .91 is stronger than .31……………………..A Negative .91 would be stronger than a Positive .31 The SIGN (positive or negative) just tells us the direction. In a Positive Relationship, HIGH SCORES ARE ASSOCIATED WITH OTHER HIGH SCORES and LOW WITH LOW in a NEGATIVE Relationship, High Scores are paired with Low Scores and Vice Versa (the 3 being paired with the 45 on the 4th line, played a huge role in our negative number) David Geber INSTRUCTOR MANAGER  Not Home vs Away…just 1st 8 scores vs 2nd 8 scores….NO RELATIONSHIP Top of Form 1) OK, as stated before, we need our r, SSx and SSy from the Previous Equation in Chapter 6 r = .16 SSx = 546.875 SSy = 1,555.875 2) b = The square root of SSy/SSx Times the r Square Root of (1,555.875/546.875) Times .16 (Square root of 2.845) x .16 or 1.69 X .16 = .27 b = .27 3) Again from our Sums in Chapter 6, we divide by 8 to get the Means of X and Y X-Bar = 247/8 = 30.875 Y-Bar = 245/8 = 30.625 4) a= Y-Bar – (b)(X-bar) = 30.625 – (.27) (30.875) = 30.625 – 8.33 = 22.295 5) Y = (.27) (X) + 22.295   STANDARD ERROR 1) SSy = 1,555.875 , r = .16 2) Sy/x = Square Root of SSy (1-r squared)/n-2 or the Square root of 1,555.875 (1- .16 squared)/8-2 or Square root of 1,555.875 (1 – .0256)/6 or Square root of (1,555.875 x .9744)/6 or Square Root of 1,516/6 = square root of 252.67 = 15.9   Now we plug in the NFL Average of 23 as the X (which would represent the Home Score) Y = (.27) (23) + 22.295 = 6.21 + 22.295 = 28.50 Plus/Minus Standard Error of 15.9 and the Bucs Score For the 2nd 8 would be between 12.6 and 44.4 OK, IN THIS ONE……16 is pretty much NO RELATIONSHIP AT ALL………….So this is a maddening exercise in Futility………….JUST DO THAT FIRST ONE WITH THE STRONG RELATIONSHIP!!! This was my equation from the Correlating Chapter 6 work to get the pearson “r’ Hello Everybody, This time we are going to ignore Home and Away and just do the correlation between the first 8 games and the Last 8 Games First(X)          Last(Y)           XY               X(2)               Y(2) 23                      3                 69               529                   9 31                    46             1426               961            2,116 28                    24               672               784               576 38                    24               912             1444               576 19                    26               234               361               676 38                    31             1178             1444               961 45                    47             2115             2025             2209 25                    44             1110               625             1936 247                245             7716             8173             9059 n=8 (E = Sum) OKAY YOU SHOULD RECOGNIZE THE SUM OF SQUARES EQUATIONS FROM THE COMPUTATION FORMULA FOR STANDARD DEVIATION IT’S THE SUM of the SQUARED COLUMN or for when we multiply XY, the SUM OF THAT COLUMN  MINUS THE SUM OF EITHER X x Y or X-squared and Y-squared DIVIDED by n (the Number of our Sample Size. SPxy =  Exy – (Ex)(EY)/n = 7,716 – (247)(245)/8 = 7,716 – 60,515/8 = 7,716 – 7,564.375 = 151.625 SSx = EX(2) – (EX)2/n = 8,173 – (247 x 247)/8 = 8,173 – 61,009/8 = 8,173 – 7,626.125 = 546.875 SSy = EY(2) – (EY)2/n = 9.059 – (245 x 245)/8 = 9.059 – 60,025/8 = 9,059 – 7.503.125 = 1,555.875 r = SPxy                    Sq rt (SSx) (SSy)   r= ____151.625_____ sq rt (546.875) (1,555.875)   r = __151.625__ sq rt 850,869.141   r = 151.625/922.43 = .16 REMEMBER NEGATIVE .31 is way stronger than POSITIVE .16 Bottom of Form SAN FRANCISCO 49ers (NFC West) for your Football Assignments. This is the Information you will need. Game 1: They scored 20 points, the game was at Home Game 2: Scored 31 points, game was Away Game 3: 36 points, Away Game 4: 20 points, Home Game 5: 17 points, Home Game 6: 24 points, Home Game 7: 33 points, Away Game 8: 27 points, Away Game 9: 17 points, Home Game 10: 13 points, Away Game 11: 23 points, Away Game 12: 24 points, Home Game 13: 15 points, Away Game 14: 33 points, Home Game 15: 20 points, Away Game 16: 23 points, Home
Below are the files with assignment
Chapter 7: REGRESSION PUTTING THE “r” from Chapter 6 to work Question 7.10D I DID THE PROGRESS CHECKS AGAIN, HAVE MY ANSWERS/EXAMPLES HANDY WHILE YOU ARE READING THE CHAPTER IF YOU HAVE A SCATTERPLOT QUESTION, YOU CAN JUST DO IT ON A PIECE OF PAPER, TAKE A PICTURE WITH YOUR PHONE AND ATTACH IT.  Whatever is easiest for you.  Top of Form 7.10 Assume that r-squared equals .50 for the relationship between height and weight for adults, indicate whether the following statements are true of false. (d) Fifty percent of the variability in weights is predictable from heights. Bottom of Form