Question

A sample of number of NFL injuries including Concussion, ACL tears, MCL tears and PCL tears from 1999 to 2019 are as follows 1 :

1- Develop one hypothesis test on means (Two examples could be: (1) Compare mean of ACL and PCL from 1999 to 2019, or (2) you can compare the concussion injuries from 1999 to 2008 with concussion injuries from 2009 to 2019). You may compare other two means. (alpha=5%)

PCL Year 1999 2000 2001 2002 2003 2004 2005 2006 MCL 95 102 125 123 98 130 85 93 78 81 145 137 65 102 96 105 83 75 78 2007 2008 2009 2010 NFL Injuries Concussion ACL 185 85 211 71 192 45 215 93 255 65 246 60 203 58 199 55 205 62 241 48 295 71 200 66 245 55 261 62 229 61 206 49 275 59 243 56 281 54 214 57 224 47 88 92 125 122 108 99 115 131 134 140 160 126 144 101 2011 2012 2013 2014 2015 2016 2017 2018 2019 84 91 77 79 83 80 132 109 87

Please answer the comparison for concussions. Thanks.

Answer 1

1)

	ACL	PCL
	85	85
	71	93
	45	78
	93	81
	65	65
	60	102
	58	96
	55	105
	62	83
	48	75
	71	78
	66	88
	55	92
	62	101
	61	84
	49	91
	59	77
	56	79
	54	83
	57	80
	47	87
Mean	60.90476	85.85714	AVERAGE()
Sd	11.7979	9.900938	STDEV()
n	21	21

Hypothesis :		α=	0.05
		df	40	n1+n2-2
Ho:	μ1​ = μ2
Ha:	μ1​ not = μ2
t Critical Value :
tc	+/-2.02107539	T.INV.2T(alpha,df)	TWO
Rejection region:
ts	< for -	tc	TWO	To reject
ts	> for +	tc	TWO	To reject
Test :
Sp^2	118.6091326	((n1-1)S1^2+(n2-1)S2^2)/(n1+n2-2)
t	-7.424140255	(X1 bar-X2 bar )/SQRT(Sp^2*(1/n1 + 1/n2))	Equal vriance
P value :
P value	4.84641E-09	T.DIST.2T(ts,df)	TWO
Decision :
P value	<	α	Reject H0

Conclusion:

There is enough evidence to conclude that there is a mean difference between ACL and PCL from 1999 to 2019 at 5% significance level

2)

	Concussion(1999-2008)	Concussion(2009-2019)
	185	295
	211	200
	192	245
	215	261
	255	229
	246	206
	203	275
	199	243
	205	281
	241	214
		224
Mean	215.2	243
sd	23.9944438	31.80565987
n	10	11

Hypothesis :		α=	0.05
		df	19	n1+n2-2
Ho:	μ1​ = μ2
Ha:	μ1​ not = μ2
t Critical Value :
tc	+/-2.093024054	T.INV.2T(alpha,df)	TWO
Rejection region:
ts	< for -	tc	TWO	To reject
ts	> for +	tc	TWO	To reject
Test :
Sp^2	805.1371898	((n1-1)S1^2+(n2-1)S2^2)/(n1+n2-2)
t	-2.242314614	(X1 bar-X2 bar )/SQRT(Sp^2*(1/n1 + 1/n2))	Equal vriance
P value :
P value	0.037067647	T.DIST.2T(ts,df)	TWO
Decision :
P value	<	α	Reject H0

Conclusion:

There is enough evidence to conclude that there is a mean difference between concussion injuries from 1999 to 2008 and concussion injuries from 2009 to 2019 at 5% significance level