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A certain large shipment comes with a guarantee that it
Chapter 4, Problem 19E(choose chapter or problem)
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.
a. If in fact 15% of the items in the shipment are defective (so that the shipment is good, but just barely), what is \(P(X \geq 7)\)?
b. 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?
c. If you found that 7 of the 10 sample items were defective, would this be convincing evidence that the shipment should be returned? Explain.
d. If in fact 15% of the items in the shipment are defective, what is \(P(X \geq 2)\)?
e. 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?
f. If you found that 2 of the 10 sample items were defective, would this be convincing evidence that the shipment should be returned? Explain.
Equation Transcription:
Text Transcription:
P(X geq 7)
P(X geq 2)
Questions & Answers
QUESTION:
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.
a. If in fact 15% of the items in the shipment are defective (so that the shipment is good, but just barely), what is \(P(X \geq 7)\)?
b. 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?
c. If you found that 7 of the 10 sample items were defective, would this be convincing evidence that the shipment should be returned? Explain.
d. If in fact 15% of the items in the shipment are defective, what is \(P(X \geq 2)\)?
e. 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?
f. If you found that 2 of the 10 sample items were defective, would this be convincing evidence that the shipment should be returned? Explain.
Equation Transcription:
Text Transcription:
P(X geq 7)
P(X geq 2)
ANSWER:Solution 19E
Step1 of 7:
Let us consider a random variable X it presents number of defective items in the sample.
Then X follows binomial distribution with parameters “n and p” that is X B(n, p),
The probability mass function of binomial distribution is given by
, x = 0,1,2,...,n.
Where,
n = sample size
= 10
x = random variable
p = probability of success
= 0.15
q = 1 - p (probability of failure)
= 1 - 0.15
= 0.85
Here our goal is:
a).We need to find , when 15% of the items in the shipment are defective.
b).We need to check whether 7 defectives in a sample of size 10 be an unusually large number.
c).We need to check whether this would be convincing evidence that the shipment should be returned or not if yes explain.
d).We need to find , when 15% of the items in the shipment are defective.
e).We need to check whether 2 defectives in a sample of size 10 be an unusually large number.
f).We need to check whether this would be convincing evidence that the shipment should be returned or not if yes explain.
Step2 of 7:
a).
We have n = 10 and p = 0.15.
Here can be obtained from Excel by using the function “=binomdist(X,n,p,false)”
X |
|
7 |
0.0001259 |
8 |