1 The tIS Tennis Association is considering switching the ty

1 The tIS Tennis Association is considering switching the type of tennis vali nsed for it toamament evenis. Fot a ornnd of ha\'ls tu be zccep.able\" ta the USTA, when dropped frum a certaun hcight, the halls awst baunoe aa nverage ol at ieast 4 lier. The L SI A is rensinatie Lappy with the current hallsoy use, but wuuld he willing to USIA is rensunable swilclh to a new hrand if they were unfiden thai the vea hrand of oalls were more comsistent (hnd wer varinncei cpere deviation of the heighis tbe current braud of hal\'s bouneed wheu dcopped used is 4 foct. In a sample of 91 balls from the uew brund beang tested, the sample stsidard ei to ihe carrent hal\'s they use. The standar deviaton is.38 Leet At the 5% level ol\'significanoc, is there enough evidence uiat the . new halls are more consisternt (have a lower variance) thats the cureal beand

Solution

Formulating the null and alternative hypotheses,              
              
Ho:   sigma   >=   0.48  
Ha:    sigma   <   0.48  
              
As we can see, this is a    left   tailed test.      
              
Thus, getting the critical chi^2, as alpha =    0.05   ,      
alpha =    0.05          
df = N - 1 =    90          
chi^2 (crit) =    69.12603043        
              
Getting the test statistic, as              
s = sample standard deviation =    0.38          
sigmao = hypothesized standard deviation =    0.48          
n = sample size =    91          
              
              
Thus, chi^2 = (N - 1)(s/sigmao)^2 =    56.40625          
              
Comparing chi^2 < chi^2(crit), we REJECT THE NULL HYPOTHESIS.  

Thus, there is significant evidence at 0.05 level that the standard deviation of the new brand is less than the current brand. [CONCLUSION]