The local medical society claims that doctors earn more ton

The local medical society claims that doctors earn more ton average than lawyers. A sample of 50 lawyers yields a mean of $80,000 and a standard deviation of $10,000. A sample of 50 doctors yields a mean of $83,000 and a standard deviation if $3000. Test this claim at a = 0.02

Solution

Formulating the null and alternative hypotheses,              
              
Ho:   u1 - u2   <=   0  
Ha:   u1 - u2   >   0  
At level of significance =    0.02          
As we can see, this is a    1   tailed test.      
Calculating the means of each group,              
              
X1 =    83000          
X2 =    80000          
              
Calculating the standard deviations of each group,              
              
s1 =    3000          
s2 =    10000          
              
Thus, the standard error of their difference is, by using sD = sqrt(s1^2/n1 + s2^2/n2):              
              
n1 = sample size of group 1 =    50          
n2 = sample size of group 2 =    50          
Thus, df = n1 + n2 - 2 =    98          
Also, sD =    1476.482306          
              
Thus, the t statistic will be              
              
t = [X1 - X2 - uD]/sD =    2.031856384          
              
where uD = hypothesized difference =    0          
              
Now, the critical value for t is              
              
tcrit =    2.081444921          
              
Thus, comparing t and tcrit, we decide to   WE FAIL TO REJECT THE NULL HYPOTHESIS.          
              
Also, using p values,              
              
p =    0.022437372          
              
Comparing this to the significance level,    WE FAIL TO REJECT THE NULL HYPOTHESIS.          
              
              
Thus, we have no significant evidence that doctors earn more than that lawyers. [CONCLUSION]