Suppose that the IRS assigns auditing rates per state by randomly selecting 50 a

Question

Suppose that the IRS assigns auditing rates per state by randomly selecting 50 auditing percentages from a normal distribution with a mean equal to 1.75% and a standard deviation of 0.25%. (a) What is the probability that a particular state would have more than 2% of its tax returns audited? (Round your answer to four decimal places.) (b) What is the expected value of x, the number of states that will have more than 2% of their income tax returns audited? (Round your answer to three decimal places.) (c) Is it likely that as many as 11 of the 50 states will have more than 2% of their income tax returns audited?

Explanation / Answer

A)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    2      
u = mean =    1.75      
          
s = standard deviation =    0.25      
          
Thus,          
          
z = (x - u) / s =    1      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   1   ) =    0.158655254 [ANSWER]

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b)

E(X) = n p = 50*0.158655254 = 7.9327627 [ANSWER]

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c)

Here, n = 50, p = 0.158655254.

We first get the z score for the critical value:          
          
x = critical value =    11      
u = mean = np =    7.9327627      
          
s = standard deviation = sqrt(np(1-p)) =    2.583445029      
          
Thus, the corresponding z score is          
          
z = (x-u)/s =    1.187266331      

As |z| < 2, YES, IT IS LIKELY. [ANSWER]

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