The Professional Golfers Association PGA maintains data on p

The Professional Golfers Association (PGA) maintains data on performance and earnings for members of the PGA Tour. For the 2012 season Bubba Watson led all players in total driving distance, with an average of 309.2 yards per drive. Some of the factors thought to influence driving distance are club head speed, ball speed, and launch angle. For the 2012 season Bubba Watson had an average club head speed of 124.69 miles per hour, an average ball speed of 184.98 miles per hour, and an average launch angle of 8.79 degrees.
The file named PGADrivingDist contains data on total driving distance and the factors related to driving distance for 190 members of the PGA Tour (PGA Tour website, November 1, 2012). Descriptions for the variables in the data set follow.
Club Head Speed: Speed at which the club impacts the ball (mph).
Ball Speed: Peak speed of the golf ball at launch (mph).
Launch Angle: Vertical launch angle of the ball immediately after leaving the club (degrees).
Total Distance: The average number of yards per drive.
a. Develop an estimated regression equation that can be used to predict the average number of yards per drive given the club head speed.
b. Develop an estimated regression equation that can be used to predict the average number of yards per drive given the ball speed.
c. A recommendation has been made to develop an estimated regression equation that uses both club head speed and ball speed to predict the average number of yards per drive. Do you agree with this? Explain.
d. Develop an estimated regression equation that can be used to predict the average number of yards per drive given the ball speed and the launch angle.
e. Suppose a new member of the PGA Tour for 2013 has a ball speed of 170 miles per hour and a launch angle of 11 degrees. Use the estimated regression equation in part (d) to predict the average number of yards per drive for this player.

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
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
Sean O\'Hair	116.52	172.87	11.84	290.7
Seung-Yul Noh	118.35	175.08	9.89	290
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
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
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
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
Cameron Tringale	114.76	169.63	10.78	283
Geoff Ogilvy	114.96	170.40	10.88	283
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
John Huh	111.11	164.87	11.88	281
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
Hunter Haas	111.96	165.31	8.51	280
Matt Kuchar	108.13	160.51	11.36	280
Ryan Moore	111.51	165.08	10.22	280
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
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
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
Mark Wilson	107.55	159.14	12.75	273
Brian Gay	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

The regression equation is

Y = 124.716 + 1.394 X

Where Y = Average number of yards per drive

X = Total head speed

Adjusted R² = 0.706

The estimated regression equation is

Y = 117.139 + 0.988X

Where Y = Average number of yards per drive

X = Ball speed

Adjusted R² = 0.746

The recommendation is good. The total distance of a shot does not only depends on the ball speed, but also on the club head speed also. both of these two factors simultaneously effects the distance covered by the ball. If we estiamte a regression equation for total distance with both of these factors, we can see that the variability in the total distance variable is explained better than the previous two cases where there was only one expalanatory variable in both the cases.

The regression equation is,

Y = 113.851 - 2.23X₁ + 2.515 X₂

Where Y = average number of yards per drive

X₁ = club head speed

X₂ = ball speed

We can see that adjusted R² is 0.77 which is better than the previous two cases.

The estimated regression equation is

Y = 81.596 + 1.093X₁ + 1.646X₂

where Y = averge number of yards per drive

X₁ = ball speed

X₂ = Launch angle

Adjusted R² = 0.826