Suppose college faculty members with the rank of professor at two-year instituti

Question

Suppose college faculty members with the rank of professor at two-year institutions earn an average of $52,500 per year with a standard deviation of $4,000. In an attempt to verify this salary level, a random sample of 60 professors was selected from a personnel database for all two-year institutions in the United States.

a What are the mean and standard deviation of the sampling distribution for n = 60?

b What’s the shape of the sampling distribution for n = 60?

c Calculate the probability the sample mean x-bar is greater than $55,000.

d If you drew a random sample with a mean of $55,000, would you consider this sample unusual? What conclusions might you draw?

Explanation / Answer

a)

By central limit theorem,

Same mean: u(X) = u = 52500 [ANSWER]

Reduced standard deviation:

sigma(X) = sigma/sqrt(n) = 4000/sqrt(60) = 516.3977795 [ANSWER]

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b)

By central limit theorem, it is bell shaped. [ANSWER]

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c)

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    55000      
u = mean =    52500      
n = sample size =    60      
s = standard deviation =    4000      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    4.841229183      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   4.841229183   ) =    6.45192*10^-7 [ANSWER]

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D)

YES, BECAUSE THIS IS A VERY SMALL PROBABILITY.

I might conclude that the true mean os actually greater than 52500.

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