Mine1 8260 7830 8350 8070 8040 Mine2 7950 7890 7990 8140 792

Mine1= [8260 7830 8350 8070 8040]

Mine2= [7950 7890 7990 8140 7920 7240]

Find:

1) Use the level of significance alpha=0.05. Test whether there is a significant difference in heat-prodcuing capacity between the two mines. Assume the variances are equal between heat-producing capacity of two mines.

2) Use level of significance alpha= 0.05. Test the assumption of cariance equality.

Solution

1)

Formulating the null and alternative hypotheses,              
              
Ho:   u1 - u2   =   0  
Ha:   u1 - u2   =/   0  
At level of significance =    0.05          
As we can see, this is a    two   tailed test.      
Calculating the means of each group,              
              
X1 =    8110          
X2 =    7855          
              
Calculating the standard deviations of each group,              
              
s1 =    203.1009601          
s2 =    313.7355574          
              
Thus, the pooled standard deviation is given by              
              
S = sqrt[((n1 - 1)s1^2 + (n2 - 1)(s2^2))/(n1 + n2 - 2)]               
              
As n1 =    5   , n2 =    6  
              
Then              
              
S =    270.215963          
              
Thus, the standard error of the difference is              
              
Sd = S sqrt (1/n1 + 1/n2) =    163.6238912          
              
As ud = the hypothesized difference between means =    0   , then      
              
t = [X1 - X2 - ud]/Sd =    1.558452119          
              
Getting the critical value using table/technology,              
df = n1 + n2 - 2 =    9          
tcrit =    +/-   1.833112933      
              
Getting the p value using technology,              
              
p =    0.076777406          
              
Thus, as we see, comparing t and tcrit (or, comparing p and the significance level) we   FAIL TO REJECT THE NULL HYPOTHESIS.          

Thus, there is no significant evidence that the heat-producing capacity between the two mines are different. [CONCLUSION]

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2.

Here,

s1 = 203.1009601
s2 = 313.7355574

Formulating the null and alternative hypotheses,              
              
Ho:   sigma1^2 / sigma2^2   =   1  
Ha:    sigma1^2 / sigma2^2   =/   1  
              
As we can see, this is a    two   tailed test.      
              
Thus, getting the critical chi^2, as alpha =    0.05   ,      
alpha/2 =    0.025          
df1 = n1 - 1 =    4         
df2 = n2 - 1 =    5          
F (crit) =    0.1067866   and   7.387885751  
              
Getting the test statistic, as              
s1 =    203.1009601          
s2 =    313.7355574          
              
Thus, F = s1^2/s2^2 =    0.419079549          
              
As F is between the two critical values, we FAIL TO REJECT THE NULL HYPOTHESIS.              

Thus, there is no significant evidence that the variances of the two mines are not equal. [CONCLUSION]