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FOOTBALL ASSIGNMENT #5 - LEAST SQUARES REGRESSION EQUATIONHello 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
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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
