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 :

2- Develop one hypothesis test on variances (As an example: compare variance of Concussion and MCL from 1999 to 2019). You may compare other two variances. (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

Answer 1

Sol:

In order to test the differences between variances in two groups we must perform a F test.

We will use the same assumptions as the previous part.

Assumption that the data is randomly sampled from the population and that the data variables follow a normal distribution.

When testing the difference between two variances so we must also assume that the two populations are independent from one another.

In this case we will test whether the variance of concussions is significantly differenet from the variance of ACL injuries from 1999 to 2019.

The hypothesis are the following: (This is considered a two-tailed F-test)

H: 02

H. : # 02

F* SA s?

$S_{1}^{2}:$ variance of concussions from 1999 to 2019 =

$S_{2}^{2}:$ variance of ACL injuries from 1999 to 2019

F* 967.29 139.19 6.94

Now we need to find the critical values.

Numerator degrees of freedom = 21 -1

Numerator degrees of freedom df = 20

Denominator degrees of freedom = 21 - 1

Denominator degrees of freedom = 20

Significance level = 0.05

Critical Value = F(1-$\alpha$/2, N1 - 1, N2 - 1)

= F(97.5,20,20)

Critical Value = 0.406

F($\alpha$/2, N1 - 1, N2 - 1) = F(0.025,20,20)

= 2.464

Rejection region: Reject H₀ if F^* < 0.406 or F^* > 2.464

Hence, since F* > 2.464 we reject H_o and conclude that the variance of concussions is significantly different from the variance of ACL injuries during 1999 to 2019.

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