Question

3. (3 pts) A certain large shipment comes with a guarantee that it contains no more than 15% defective items. If the proportion of defective items in the shipment is greater than 15%, the shipment may be returned. You draw a random sample of 10 items. Let X be the number of defective items in the sample. If in fact 15% of the items in the shipment are defective (so that the shipment is good, but just barely), what is P(X2 7)? Based on the answer to part (a), if 15% of the items in the shipment are defective, would 7 defectives in a sample of size 10 be an unusually large number? If you found that 7 of the 10 sample items were defective, would this be convincing evidence that the shipment should be returned? Explain a. b. c. If in fact l 5% of the items in the shipment are defective, what is P(X 2)? Based on the answer to part (d), if 15% of the items in the shipment are defective, would 2 defectives in a sample of size 10 be an unusually large number? If you found that 2 of the 10 sample items were defective, would this be convincing evidence that the shipment should be returned? Explain. d. e. f.

Answer 1

(a)

Here X has binomial distribtution with parameters n=10 and p=0.15. The required probability is

$P(X\geq 7)=\sum_{x=7}^{15}\binom{15}{x}(0.15)^{x}(1-0.15)^{15-x}=0.0001$

(b)

Yes because it is an unusual event. The probabiltiy of part a is less than 0.05 so it is unusual.

c)

Yes

d)

15 P(X > 2)-1-P(X < 1) 1 (0.15)"(1-0.15)TO-r-0.4557

e)

no because probability of part d is geater than 0.05.

f)

No