The average salary for a Queens College full professor is 85

The average salary for a Queens College full professor is $85,900. If the average salaries are normally distributed with a standard deviation of $11.000. Find these probabilities: The professor makes more than $91.000 If a sample of 25 professors were selected, what is the probability that the sample mean is more than $91000?

Solution

A)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    91000      
u = mean =    85900      
          
s = standard deviation =    11000      
          
Thus,          
          
z = (x - u) / s =    0.463636364      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   0.463636364   ) =    0.321454148 [ANSWER]

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b)

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    91000      
u = mean =    85900      
n = sample size =    25      
s = standard deviation =    11000      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    2.318181818      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   2.318181818   ) =    0.010219721 [ANSWER]