Question

A company operates an assembly line. The company believes that there is a 3% probability that any given item produced on the line will contain a defect. If 100 items are pulled at random from the line, what is the probability there will be exactly 4 defective items? ____

If 100 items are pulled at random from the line, what is the probability there will be 9 or more defective items? _______

If X = the number of defectives in a sample of 100 items from the line find the mean and standard deviation of X.________

Mean of X =

Standard Deviation=

Suppose that the company did a quality control survey and did in fact pull 100 items from the line and 9 were found to be defective. Using your work above, would you question whether the "true" defect rate was actually 3%? (Select all that apply.)

No, the standard deviation of the number of defects is large relative to the mean.

Yes, the probability of obtaining 9 or more defects is quite small under the assumption of a 3% defect rate.

Yes, 9 defects is several standard deviations higher than the expected number of defects.

Answer #1

n =100 p=0.03

Here n>= 100, np = 100*0.03 = 3 <10 we can use poisson probabilities.

P(x = k) =

for a poisson distribution mean = variance =

a) the probability there will be exactly 4 defective items

P(X=4) = = 0.168

b) the probability there will be 9 or more defective items =

= 1- = 1- 0.996 = 0.004

c)for a poisson distribution mean = variance =

d) given 9% is defective,

Yes, the probability of obtaining 9 or more defects is quite small under the assumption of a 3% defect rate.

standard deviation = 3^{0.5} = 1.732

mean = 3

standard deviation = 1.732

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