An unnoticed mechanical failure has caused onefourth of a ma

   An unnoticed mechanical failure has caused one-fourth of a machine shop’s production of 10000 pistol firing pins to be defective. A random sample of 25 firing pins was drawn from the population.

a.)   Explain why this random variable has a binomial distribution? (2 points)
b.)   What are the mean and the standard deviation of the random variable? (2 points)
c.)   Find P(X > 4). Use the binomcdf function on your calculator. (2 points)
d.)   Is it appropriate to use the normal approximation to the binomial for this problem? Support your answer with numbers and a test. (3 points)
e.)   Using the normal approximation with the continuity correction, calculate the probability that you’ll observe more than 4 defective firing pins in the random sample of 25. Indicate the X-values and the z-scores involved. (3 points)
f.)   How does your answer compare with the one you have in part c.)? (2 points)
g.)   How do you account for the difference in parts c.) and e.)? (2 points )
h.)   Sketch the probability distribution and the area you found using the normal approximation with the continuity correction. (2 points)

Solution

a.

Because the number trials are fixed (25), the samples are independent, and the probability of success is constant (1/4 = 0.25).

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b.

As n = 25, p = 0.25,

mean = n p = 6.25

standard deviation = sqrt [n p (1 - p)] = 2.165063509

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c.

Note that P(more than x) = 1 - P(at most x).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    25      
p = the probability of a success =    0.25      
x = our critical value of successes =    4      
          
Then the cumulative probability of P(at most x) from a table/technology is          
          
P(at most   4   ) =    0.213740923
          
Thus, the probability of at least   5   successes is  
          
P(more than   4   ) =    0.786259077

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