Question

data contained in the chart below (PGA Tour website, November 1, 2012) was used to develop an estimated regression equation to predict the average number of yards per drive given the ball speed and the launch angle.

Player Club Head Speed Ball Speed Launch Angle Total Distance

Bubba Watson 124.69 184.98 8.79 309.2

Dustin Johnson 121.62 180.57 11.30 301.8

J.B. Holmes 121.33 180.00 12.97 300.6

Rory McIlroy 120.21 178.07 11.25 298.6

Charlie Beljan 122.77 180.71 9.79 296.9

Robert Garrigus 121.39 180.21 12.31 296.5

John Daly 119.07 176.70 13.13 295.9

Gary Woodland 123.80 181.59 9.54 295.6

Adam Scott 119.30 176.81 9.29 295.3

Scott Piercy 117.59 174.42 11.64 295.0

Kyle Stanley 119.51 177.02 12.59 294.5

Keegan Bradley 116.83 172.35 12.61 294.3

Martin Laird 117.74 174.88 10.41 293.5

Ryan Palmer 116.12 172.43 13.31 293.2

Scott Stallings 118.85 175.48 10.74 293.1

Jason Day 119.16 176.87 11.88 292.7

Josh Teater 115.89 171.96 11.98 292.6

Harrison Frazar 114.39 169.80 12.32 292.1

Jhonattan Vegas 122.47 180.38 9.99 291.7

Louis Oosthuizen 116.79 173.32 10.59 291.3

Troy Matteson 116.55 172.48 13.09 291.1

Edward Loar 117.69 172.53 11.53 291.0

Sean O'Hair 116.52 172.87 11.84 290.7

Seung-Yul Noh 118.35 175.08 9.89 290.0

Graham DeLaet 120.04 176.71 11.86 289.7

Troy Kelly 115.64 171.38 9.97 289.7

Angel Cabrera 119.08 176.09 9.95 289.4

Lee Westwood 117.20 173.38 9.68 289.2

Jason Kokrak 118.14 175.31 10.72 288.9

Aaron Baddeley 119.05 176.75 9.04 288.7

Bill Haas 115.55 171.27 10.57 288.7

Martin Flores 118.93 175.38 9.73 288.6

Nick Watney 118.69 175.82 9.20 288.6

Phil Mickelson 116.81 173.18 10.35 288.5

Carl Pettersson 113.85 169.03 11.11 288.2

Harris English 117.43 174.16 10.03 288.2

Boo Weekley 114.87 170.52 9.41 287.9

Bo Van Pelt 114.09 169.38 11.91 287.8

Bobby Gates 119.63 175.51 9.03 287.8

Tiger Woods 120.94 177.69 10.67 287.8

Charles Howell III 118.16 173.88 8.45 287.5

Charley Hoffman 115.76 171.17 11.31 287.5

Vijay Singh 113.75 168.32 11.71 287.4

Rickie Fowler 115.64 171.41 11.72 287.3

Davis Love III 115.21 171.09 11.25 287.1

Erik Compton 114.56 170.04 12.06 287.1

Tommy Gainey 112.94 167.58 12.25 287.0

Daniel Chopra 116.88 172.20 11.74 286.8

John Rollins 113.54 168.34 11.53 286.7

Jeff Overton 112.04 166.07 11.65 286.6

Mark Anderson 117.84 174.76 9.58 286.4

Chris Couch 119.61 174.46 9.99 286.1

Jason Dufner 112.60 167.05 11.02 286.1

John Senden 115.90 171.66 9.80 286.0

Webb Simpson 113.34 166.74 10.47 285.8

Ernie Els 113.20 168.09 12.67 285.7

Kevin Chappell 117.22 172.05 10.88 285.7

Andres Romero 115.42 171.26 10.57 285.5

Charl Schwartzel 116.89 172.76 9.38 285.5

Robert Karlsson 116.86 173.67 10.85 285.5

Bud Cauley 112.39 166.45 13.32 285.4

Jonathan Byrd 115.14 170.67 10.47 285.4

Bill Lunde 111.97 166.16 10.97 285.3

Sergio Garcia 119.13 175.35 8.28 285.3

Greg Owen 118.85 174.38 9.11 285.2

Garth Mulroy 114.82 169.78 11.89 285.1

Steve Stricker 112.22 166.29 10.89 285.1

Hunter Mahan 112.05 166.15 11.12 284.8

Matt Jones 116.29 172.23 10.15 284.8

Rory Sabbatini 112.20 166.59 11.16 284.8

Scott Brown 112.53 165.95 11.90 284.8

Brian Harman 111.08 164.87 11.35 284.7

Jimmy Walker 117.43 174.04 9.93 284.7

Ryo Ishikawa 113.55 168.13 11.43 284.7

Billy Horschel 111.19 164.38 12.11 284.3

Chris Kirk 112.79 167.15 11.78 284.0

Camilo Villegas 113.03 167.86 11.31 283.9

Roberto Castro 112.82 166.77 10.73 283.8

John Merrick 113.11 167.87 9.87 283.7

Will Claxton 111.70 165.53 10.92 283.7

D.J. Trahan 114.75 169.13 11.31 283.6

Kevin Kisner 110.97 164.10 11.60 283.6

Kevin Stadler 113.62 168.43 11.20 283.6

Kevin Streelman 115.47 169.53 10.94 283.6

Danny Lee 113.89 168.05 11.01 283.5

Mathew Goggin 112.82 167.40 11.42 283.4

Justin Rose 114.63 167.47 11.99 283.3

Daniel Summerhays 113.02 167.09 10.28 283.2

Fredrik Jacobson 113.47 166.26 9.56 283.2

Pat Perez 115.40 169.33 10.84 283.2

Roland Thatcher 112.64 166.92 12.20 283.2

Stephen Gangluff 114.55 169.97 10.35 283.2

Ben Crane 109.99 163.20 12.29 283.0

Cameron Tringale 114.76 169.63 10.78 283.0

Geoff Ogilvy 114.96 170.40 10.88 283.0

J.J. Killeen 116.26 171.08 8.23 282.9

Brendan Steele 112.26 166.28 12.02 282.8

Miguel Angel Carballo 113.62 168.60 10.16 282.7

Sang-Moon Bae 114.12 168.73 9.33 282.7

Brandt Jobe 115.70 171.65 10.69 282.6

Marc Leishman 115.52 171.26 11.24 282.2

Kyle Reifers 111.59 165.44 11.70 281.8

Tim Herron 112.73 167.17 10.60 281.6

Stewart Cink 115.95 171.13 11.20 281.4

James Driscoll 117.58 173.27 8.75 281.3

Steve Wheatcroft 109.67 162.76 11.61 281.3

Jonas Blixt 111.92 164.16 11.49 281.2

Padraig Harrington 114.92 170.53 10.47 281.2

Tommy Biershenk 109.44 162.21 12.65 281.2

Tom Gillis 113.96 167.07 9.04 281.1

Chad Campbell 112.89 167.23 9.77 281.0

John Huh 111.11 164.87 11.88 281.0

Ian Poulter 110.97 164.73 9.85 280.9

Rod Pampling 112.31 166.09 9.42 280.9

Brandt Snedeker 110.75 164.04 12.61 280.8

Jeff Maggert 106.09 157.48 13.05 280.8

Johnson Wagner 110.58 164.09 10.31 280.8

Matt Every 113.42 167.54 10.98 280.8

Derek Lamely 116.06 172.10 10.75 280.6

Blake Adams 113.65 168.14 10.04 280.5

D.A. Points 110.48 163.95 11.40 280.5

Billy Mayfair 110.08 163.34 13.37 280.4

Matt Bettencourt 113.05 167.89 11.39 280.3

K.J. Choi 109.62 162.04 10.75 280.2

Brendon de Jonge 115.65 170.32 8.57 280.1

Graeme McDowell 112.40 165.25 10.04 280.0

Hunter Haas 111.96 165.31 8.51 280.0

Matt Kuchar 108.13 160.51 11.36 280.0

Ryan Moore 111.51 165.08 10.22 280.0

Robert Allenby 111.62 165.58 11.79 279.9

Marco Dawson 111.22 164.84 10.66 279.8

Ricky Barnes 116.05 170.84 8.44 279.7

Trevor Immelman 118.20 172.33 6.72 279.7

Spencer Levin 110.50 164.05 9.77 279.4

Vaughn Taylor 108.42 161.00 13.30 279.4

George McNeill 115.66 171.40 9.74 279.2

J.J. Henry 114.03 168.60 11.15 279.1

David Hearn 109.74 162.69 14.06 279.0

Y.E. Yang 112.73 166.05 9.38 278.8

Richard H. Lee 109.79 162.97 10.73 278.7

Ted Potter, Jr. 109.70 161.75 9.77 278.7

William McGirt 112.89 167.43 9.80 278.7

Cameron Beckman 109.79 162.39 10.88 278.3

Chris Stroud 111.16 164.80 10.85 278.1

Arjun Atwal 109.72 162.87 11.23 277.8

Kris Blanks 112.19 164.63 10.34 277.7

Alexandre Rocha 109.32 162.34 11.78 277.3

Chez Reavie 109.86 163.06 11.78 277.3

Kevin Na 112.97 165.33 10.04 277.3

Russell Knox 109.50 162.41 10.82 277.2

Kyle Thompson 107.68 159.89 11.65 277.1

Charlie Wi 111.44 164.82 11.15 276.9

Scott Dunlap 110.76 162.39 10.19 276.4

Zach Johnson 107.57 159.21 12.29 276.4

Stuart Appleby 112.43 165.42 9.24 276.3

Brendon Todd 110.66 163.73 10.69 276.2

Sung Kang 109.76 162.70 11.81 276.1

Michael Thompson 111.63 165.53 11.93 276.0

Jim Furyk 109.60 162.12 9.56 275.9

Dicky Pride 109.35 162.21 11.13 275.7

Nathan Green 115.02 169.92 7.63 275.6

Rocco Mediate 107.57 159.75 11.74 275.6

Gary Christian 109.31 162.23 10.54 275.3

Ben Curtis 108.23 160.08 12.00 275.2

Ken Duke 110.93 164.09 8.66 275.2

Tim Clark 104.37 154.88 12.37 275.2

Bob Estes 111.90 165.98 10.02 275.1

Michael Bradley 114.15 169.53 10.24 275.1

David Toms 104.29 154.98 13.99 274.9

Heath Slocum 106.18 157.45 11.72 274.9

Stephen Ames 109.85 161.89 10.38 274.9

Luke Donald 110.69 163.94 11.45 274.7

Patrick Sheehan 107.82 159.98 11.12 274.7

Greg Chalmers 109.83 162.54 9.29 274.6

Jason Bohn 109.76 162.83 10.33 274.4

Brian Davis 105.83 156.96 12.09 274.2

David Mathis 107.65 159.39 10.82 274.1

Justin Leonard 107.33 159.25 11.02 273.5

Bryce Molder 109.28 161.82 10.70 273.0

Mark Wilson 107.55 159.14 12.75 273.0

Brian Gary 104.59 155.17 13.09 272.8

John Mallinger 107.99 160.27 11.66 272.4

Billy Hurley III 109.15 158.94 10.26 270.9

Ryuji Imada 109.89 162.56 10.37 270.9

Jerry Kelly 105.40 155.50 12.64 270.4

Chris DiMarco 107.13 157.41 11.35 270.1

Colt Knost 106.76 157.74 10.76 268.8

Gavin Coles 104.75 154.43 10.55 268.5

Tom Pernice Jr. 107.03 158.41 11.54 268.2

Nick O'Hern 104.66 155.27 11.54 265.7

1Does the estimated regression equation provide a good fit to the data? Explain.

2 In part (b) of a different question , an estimated regression equation was developed using only ball speed to predict the average number of yards per drive. Compare the fit obtained using just ball speed to the fit obtained using ball speed and the launch angle.

Answer 1

Sol:

Install analysis tool apck in regression

go to data >data analysis>regression

you will get

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.909863463
R Square	0.827851522
Adjusted R Square	0.826010362
Standard Error	2.848721026
Observations	190
ANOVA
	df	SS	MS	F	Significance F
Regression	2	7297.779	3648.89	449.6358	3.60536E-72
Residual	187	1517.545	8.115211
Total	189	8815.324
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	81.59638703	6.952834	11.7357	3.46E-24	67.88031532	95.31246
Ball Speed	1.092659468	0.036437	29.98778	2.31E-73	1.020779409	1.16454
Launch Angle	1.646453663	0.176476	9.329625	3.1E-17	1.298314256	1.994593

regression eq is

total distance=81.596+1.093*ballspeed+1.646*launchangle

R sq=0.8278

82.78% variation is total distance is explained by model.Good model.

F(2,187)=449.64

p=0.0000

p<0.05

Model is significant .we can use model for predicting average distance

2 In part (b) of a different question , an estimated regression equation was developed using only ball speed to predict the average number of yards per drive. Compare the fit obtained using just ball speed to the fit obtained using ball speed and the launch angle.

Now y=

total distance

x=launch angle

you will get

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.002092
R Square	4.38E-06
Adjusted R Square	-0.00531
Standard Error	6.847614
Observations	190
ANOVA
	df	SS	MS	F	Significance F
Regression	1	0.038593	0.038593	0.000823	0.977143
Residual	188	8815.285	46.88981
Total	189	8815.324
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	282.6682	4.421338	63.93272	1.7E-129	273.9464	291.39
Launch Angle	0.011575	0.403454	0.028689	0.977143	-0.7843	0.807454

total distance=282.668+0.012*launch angle

R sq=

R Square

4.38E-06

F(1,188)=0.0008

p=0.9777

p>0.05

Model is not significant.

R sq is less with only launch angle as predictor.

Not a good model.