In a sample of n16 selected from an underlying normal popula

In a sample of n=16 selected from an underlying normal population with S=10.

What is the value of X^2 statistic if you are testing the null hypothesis Ho that =7.5.

What is your statistical decision if

a.) H1 7.5

b.) H1 > 7.5

Solution

NOTE: Here, the level of significance is not given, so we assume it is 0.05.

a)

Formulating the null and alternative hypotheses,              
              
Ho:   sigma   =   7.5  
Ha:    sigma   =/   7.5  
              
As we can see, this is a    two   tailed test.      
              
Thus, getting the critical chi^2, as alpha =    0.05   ,      
alpha/2 =    0.025          
df = N - 1 =    15          
chi^2 (crit) =    6.262137795   and   27.48839286  
              
Getting the test statistic, as              
s = sample standard deviation =    10          
sigmao = hypothesized standard deviation =    7.5          
n = sample size =    16          
              
              
Thus, chi^2 = (N - 1)(s/sigmao)^2 =    26.66666667 [ANSWER, CHI^2 STATISTIC]          
              
As chi^2 is between the two critical values, we FAIL TO REJECT THE NULL HYPOTHESIS.   [

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b)

Formulating the null and alternative hypotheses,              
              
Ho:   sigma   <=   7.5  
Ha:    sigma   >   7.5  
              
As we can see, this is a    right   tailed test.      
              
Thus, getting the critical chi^2, as alpha =    0.05   ,      
alpha =    0.05          
df = N - 1 =    15          
chi^2 (crit) =    24.99579014        
              
Getting the test statistic, as              
s = sample standard deviation =    10          
sigmao = hypothesized standard deviation =    7.5          
n = sample size =    16          
              
              
Thus, chi^2 = (N - 1)(s/sigmao)^2 =    26.66666667          
              
As chi^2 > chi^2(crit), then we REJECT THE NULL HYPOTHESIS.