Question

Data from the National Football League (NFL) from a recent season reported the number of yards gained by each of the league's wide receivers. The data in this Excel file nfl_receiver_yds_webassign shows the yards gained by each wide receiver in the NFL who gained 50 or more yards during the season. The number of yards gained is right-skewed and therefore a normal model is NOT a good model for the data. However, the square root of the number of yards gained can be approximated very well by a N(21.5, 8.3) model.

Question 1. Use the normal model to determine the probability that a wide receiver gained 270 yards or less during the season. (Use 4 decimal places).

Question 2. A particular wide receiver has a bonus clause in his contract that pays him a bonus of $1 million if his yards gained are in the top 14% of yards gained by wide receivers. How many yards does this receiver have to gain to get the bonus? (Round to nearest yard). yards

NFL receivers with 50 or more receiving yards
Player	Yds
Chad Johnson	1369
Marvin Harrison	1366
Reggie Wayne	1310
Roy Williams	1310
Donald Driver	1295
Lee Evans	1292
Anquan Boldin	1203
Torry Holt	1188
Terrell Owens	1180
Steve Smith	1166
Andre Johnson	1147
Laveranues Coles	1098
Isaac Bruce	1098
Mike Furrey	1086
Javon Walker	1084
T.J. Houshmandzadeh	1081
Joey Galloway	1057
Terry Glenn	1047
Marques Colston	1038
Plaxico Burress	988
Hines Ward	975
Jerricho Cotchery	961
Darrell Jackson	956
Larry Fitzgerald	946
Mark Clayton	939
Braylon Edwards	884
Muhsin Muhammad	863
Eddie Kennison	860
Santonio Holmes	824
Reggie Brown	816
Keyshawn Johnson	815
Santana Moss	790
Bernard Berrian	775
Reche Caldwell	760
Derrick Mason	750
Marty Booker	747
Devery Henderson	745
Bryant Johnson	740
Drew Bennett	737
Antonio Bryant	733
Ronald Curry	727
Deion Branch	725
Donte' Stallworth	725
Wes Welker	687
Arnaz Battle	686
Joe Horn	679
Chris Chambers	677
Eric Parker	659
Travis Taylor	651
Matt Jones	643
Greg Jennings	632
Nate Washington	624
Reggie Williams	616
D.J. Hackett	610
Chris Henry	605
Samie Parker	561
Eric Moulds	557
Randy Moss	553
Ernest Wilford	524
Patrick Crayton	516
Rod Smith	512
Roddy White	506
Cedrick Wilson	504
Joe Jurevicius	495
Kevin Curtis	479
Hank Baskett	464
Bobby Wade	461
Troy Williamson	455
Vincent Jackson	453
Keenan McCardell	437
Michael Jenkins	436
Ashley Lelie	430
Josh Reed	410
Peerless Price	402
Demetrius Williams	396
Terrance Copper	385
Troy Brown	384
Brandon Jones	384
Marcus Robinson	381
Brandon Lloyd	365
Amani Toomer	360
Ruvell Martin	358
Drew Carter	357
Michael Clayton	356
Antwaan Randle El	351
Greg Lewis	348
Justin McCareins	347
Doug Gabriel	344
Ike Hilliard	339
Roscoe Parrish	320
Brandon Marshall	309
Billy McMullen	307
Rashied Davis	303
Alvis Whitted	299
Bobby Engram	290
Mark Bradley	282
Tim Carter	253
Dennis Northcutt	228
Derek Hagan	221
Malcom Floyd	210
Troy Walters	209
Dante Hall	204
David Tyree	197
Nate Burleson	192
David Kircus	187
Corey Bradford	164
Kevin Walter	160
Bethel Johnson	156
Chad Jackson	152
James Thrash	151
Bryan Gilmore	150
Az-Zahir Hakim	147
John Madsen	146
Jabar Gaffney	142
Shaun McDonald	136
Carlyle Holiday	126
Roydell Williams	121
Kelley Washington	115
Tim Dwight	112
Jamal Jones	108
David Givens	104
Maurice Stovall	102
Mike Williams	99
Josh Cribbs	91
Koren Robinson	89
Brandon Stokley	85
Doug Gabriel	84
Aaron Moorehead	82
Tab Perry	81
Sam Hurd	75
Johnnie Morant	70
Jason Avant	68
Justin Gage	68
Paris Warren	63
Brad Smith	61
Keary Colbert	56

Answer 1

1)

µ =    21.5          
σ =    8.3          
              
P(X<√270) = P( X ≤    16.4317 ) = P( (X-µ)/σ ≤ (16.4317-21.5) /8.3)      
=P(Z ≤   -0.61   ) =   0.2707   (answer)

2)

µ=   21.5      
σ =    8.3      
P(X≤x) =   0.86      
          
z value at 0.86=   1.0803   (excel formula =NORMSINV(0.86))  
z=(x-µ)/σ          
so, X=zσ+µ=   1.080   *8.3+21.5  
X =   30.467  

required (answer) = 30.467^2 = 928.21 ≈ 929