The US Tennis Association is considering switching the type

The US Tennis Association is considering switching the type of tennis ball used for it

tournament events. For a brand of balls to be “acceptable” to the USTA, when

dropped from a certain height, the balls must bounce an average of at least 4 feet. The

USTA is reasonable happy with the current balls they use, but would be willing to

switch to a new brand if they were confident that the new brand of balls were more

consistent (had a lower variance) compared to the current balls they use. The standard

deviation of the heights the current brand of balls bounced when dropped used is .48

feet. In a sample of 91 balls from the new brand being tested, the sample standard

deviation is .38 feet. At the 5% level of significance, is there enough evidence that the

new balls are more consistent (have a lower variance) than the current brand? Please show work.

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.              

There is signficant evidence that the new brand of balls have a lower variance than the current brand. [CONCLUSION]