2 By statistics faculty with rank of assistant professor fin

2. By statistics, faculty with rank of assistant professor finishing their second year of employ- ment at a higher education institution in Ontario earn an average of $65,500 per year with a standard deviation of $3500. In an attempt to verify this salary level, a random sample of 64 assistant professor with two years of experience was selected from a personnel database for all higher education institutions in Ontario.

[5] a. Describe the sampling distribution of the sample mean, X , of the average salary of these 64 assistant professors.
[5] b. Within what limit would you expect the sample mean to fall with probability 0.95 [5] c. Obtain the probability that X is greater than 66,000.

Solution

a)

By central limit theorem, it will have the same mean, u(X) = 65500.

It will have a standard deviation given by

sigma(X) = sigma/sqrt(n) = 3500/sqrt(64) = 437.5.

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b)

First, we get the z score from the given left tailed area. As          
          
Left tailed area =    0.95      
          
Then, using table or technology,          
          
z =    1.644853627      
          
As x = u + z * s,          
          
where          
          
u = mean =    65500      
z = the critical z score =    1.644853627      
s = standard deviation =    437.5      
          
Then          
          
x = critical value =    66219.62346   [ANSWER]

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c)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    66000      
u = mean =    65500      
          
s = standard deviation =    437.5      
          
Thus,          
          
z = (x - u) / s =    1.142857143      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   1.142857143   ) =    0.126548954 [ANSWER]