Question

Binomial Events Past records over a long period of time show that 10.0% of the screws manufactured by a particular machine are defective. Given that a random sample of 10 screws are selected, complete the following tasks. a). Calculate the mean. b). Calculate the variance and standard deviation. c). Calculate the probability of 0 to 10 screws being defective [i.e. P(X=0), P(X=1), P(X=2)... P(X=n)]. d). Construct a histogram of the probability of faulty screws versus the number of faulty screws. e). Find the probability that 10% of the screws will be defective. f). Find the probability that at least 2 screws will be defective.

Answer 1

Result:

n=10, p=0.10

a).

mean = np = 1

b).

Variance = np(1 - p) = 0.9

Standard deviation = 0.9487

c).

P(X=x) = (nCx) px (1-p)n-x

Binomial Probabilities Table
	X	P(X)
	0	0.3487
	1	0.3874
	2	0.1937
	3	0.0574
	4	0.0112
	5	0.0015
	6	0.0001
	7	0.0000
	8	0.0000
	9	0.0000
	10	0.0000

d).

Histogram 0.4500 0.3874 0.4000 0.3487 0.3500 0.3000 0.2500 P(X) 0.1937 0.2000 0.1500 0.1000 0.0574 0.0500 0.0112 -0.00150.00010.00000.00000.00000.0000 0.0000 0 1 2 3 4 5 6 7 8 9 10 Number of Successes

e).

10% of the screws of 10 = 1

Therefore P( x=1)

=0.3874

f).

P( atleast 2 screws defect) = P( x ≥2) = 1- P( x <2)

=1-(P( x=0)+P( x=1))

=1-(0.3487+0.3874)

=0.2639