According to the US Government Accountability Office women h

According to the U.S. Government Accountability Office, women hold 40% of the management positions in the United States. Suppose we take a random sample of 20 people in management positions. Find the

A) all women

B) atleast 16 women

C) no more than 9 women

D) between 10 and 15 women (including 10 and 15)

E) Find the mean of this distirbution

F) Find the standard diviation

Solution

a)

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    20      
p = the probability of a success =    0.4      
x = the number of successes =    20      
          
Thus, the probability is          
          
P (    20   ) =    1.09951*10^-8 [ANSWER]

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b)

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    20      
p = the probability of a success =    0.4      
x = our critical value of successes =    16      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   15   ) =    0.999682969
          
Thus, the probability of at least   16   successes is  
          
P(at least   16   ) =    0.000317031 [ANSWER]

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c)

Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    20      
p = the probability of a success =    0.4      
x = the maximum number of successes =    9      
          
Then the cumulative probability is          
          
P(at most   9   ) =    0.755337203 [ANSWER]

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d)

u = mean = np =    8      

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e)
          
s = standard deviation = sqrt(np(1-p)) =    2.19089023