Assume that a population is normally distributed with a mean

Assume that a population is normally distributed with a mean of 100 and a standard deviation of 15. Would it be unusual for the mean of a sample of 3 to be 115 or more? Why or why not?

Solution

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    115      
u = mean =    100      
n = sample size =    3      
s = standard deviation =    15      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    1.732050808      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   1.732050808   ) =    0.041632258 [ANSWER]

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If this is unusual depends on the standard. Some classes us P < 0.05 as unusual. In that case, it is unusual.