Question

A sC60:G100ecretarial service found that 90% of their invoices are paid promptly (within 10 days). A random sample of 12 invoices is checked. Use this situation to answer the parts below.
					(a)	Recognizing that this is a binomial situation, give the meaning S and F in this context. That is, define what you will classify as a "success" and what you will classify as a "failure" when one invoice is selected and checked to see if it was paid promptly.
	S =	invoices paid within 10 days
	F =	invoices paid after 10 days
(b)	Next, give the values of n, p, and q.
	n =	p =	q =
(c)	Construct the complete binomial probability distribution for this situation in a table out to the right.
(d)	Using your table, find the probability that exactly eight of twelve randomly selected invoices were paid promptly.
(e)	Find the probability that at least eight of the twelve were paid promptly.
(f)	Find the probability that fewer than six were paid promptly.
(g)	Find the mean and standard deviation of this binomial probability distribution.
(h)	By writing a sentence, interpret the meaning of the mean value found in (g) as tied to the context of invoices being paid promptly.
(i)	Is it unusual to have all twelve invoices be paid promptly? Briefly explain your answer giving supporting numerical evidence. (Hint: 2 Sigma Rule)

Answer 1

Part a)

Success S =  invoices are paid promptly (within 10 days)

Failure F =  invoices are not paid promptly.

Part b)

n is number of trials or random sample = 12

p is probability of success ( % of people whose invoices are paid promptly ) = 0.9

q is probability if failure = 1 - p = 1 - 0.9 = 0.1

Part c)

part d)

probability that exactly eight of twelve randomly selected invoices were paid promptly.

P( x = 8 ) = 0.0213

Part e) probability that at least eight of the twelve were paid promptly.

P( x >= 8) = P(8)+P(9)+P(10)+P(11)+P(12)

=0.0213+0.0852+0.2301+0.3766+0.2824

P( x >= 8)=0.9956

Part f)  probability that fewer than six were paid promptly.

P( x < 6 ) = P( x <=5) = P(0) + P(1)+P( 2)+P(3)+P(4)+P(5)

= 0+0+0+0+0+0

P( x < 6 ) = 0

Part g) mean and standard deviation of this binomial probability distribution.  

Mean = n*p = 12*0.9

Mean = 10.8

Standard deviation = =

Standard deviation = 1.0392

Part h) Expected number of people whose invoices are paid promptly are 10.8

Part i)

We have to find 2 sigma limits

Lower limit = Mean -(2*standard deviation) = 10.8 - (2*1.0392 )  

Lower limit = 8.72 ~ 9

Upper limit = Mean +(2*standard deviation) = 10.8 + (2*1.0392 )

Upper limit = 12.9 ~ 13

So the values outside the interval ( 9,13) are considered as unusual .

12 fall between the interval (9,13) , therefore it is not unusual.