A report shows the mean family income for a small town in th

A report shows the mean family income for a small town in the US is $27,000 per year. An evaluation of 25 families from this town reveals a mean to be $30,000 with a sample standard deviation of $10,000. Does this information disagree with the report? Apply the 0.01 significance level and run a hypothesis test.

Solution

Formulating the null and alternative hypotheses,              
              
Ho:   u   =   27000  
Ha:    u   =/   27000  
              
As we can see, this is a    two   tailed test.      
              
Thus, getting the critical t,              
df = n - 1 =    24          
tcrit =    +/-   2.796939505      
              
Getting the test statistic, as              
              
X = sample mean =    30000          
uo = hypothesized mean =    27000          
n = sample size =    25          
s = standard deviation =    10000          
              
Thus, t = (X - uo) * sqrt(n) / s =    1.5          
              
Also, the p value is, for a two tailed test,              
              
p =    0.146655646          
              
Comparing |t| < 2.7969, (or, p >0.01), we   FAIL TO REJECT THE NULL HYPOTHESIS.          
              
Thus, there is no significant evidence that the mean family income for that small town in US is not $27000 per year. [CONCLUSION]