according to the irs study it take a mean 190 minutes for ta

according to the irs study it take a mean 190 minutes for taxpayer to prepare, copy and electronically transfer a 1040tax form. The standard deviation of this distribution is 65 minutes. A consumer watch dog agency selects a random study sample of 75 taxpayers for further study. What is the shape, mean, and standard deviation of the sampling distribution of the sample mean for samples of size 75? What is the probability that the sample mean is less than 3 hours?

Solution

a)

By central limit theorem,

1. It is bell shaped.

2. It has the same mean, u(X) = 190 minutes.

3. The standard deviation is given by

sigma(X) = sigma/sqrt(n) = 65/sqrt(75) = 7.505553499

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b)

Note that 3 hours = 180 minutes.

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    180      
u = mean =    190      
          
s = standard deviation =    7.505553499      
          
Thus,          
          
z = (x - u) / s =    -1.332346775      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   -1.332346775   ) =    0.091373132 [ANSWER]