Question

The quality-control inspector of a production plant will reject a batch of syringes if two or more defective syringes are found in a random sample of nine syringes taken from the batch. Suppose the batch contains 1% defective syringes. (a) Make a histogram showing the probabilities of r = 0, 1, 2, 3, , 8 and 9 defective syringes in a random sample of nine syringes. (b) Find μ. What is the expected number of defective syringes the inspector will find? (c) What is the probability that the batch will be accepted? (d) Find σ. Step 1 (a) Make a histogram showing the probabilities of r = 0, 1, 2, 3, , 8 and 9 defective syringes in a random sample of nine syringes. Recall that the binomial distribution with parameters n, p, and r gives the probability distribution of the number of r successes in a sequence of n trials, each of which yields success with probability p. Here we can let "success" be defined as "finding a defective syringe." The batch contains 1% defective syringes, so there is a 1% chance that any given syringe will be found to be defective. Therefore, p = 0.01 . A random sample of nine syringes are checked for quality, so n = 9

Step 2

We are interested in the probability of r defective syringes when r = 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Use this table to create a table of r values and the corresponding

P(r)

values when n = 9 and p = 0.01. (Round your answers to three decimal places.)

r	P(r)
0	0.914
1	?
2	?
3	0.000
4	?
5	0.000
6	0.000
7	0.000
8	0.000
9	0.000

Please help find 1 ,2 and 4 in the map

Answer 1

a) The batch contains 1% defective syringes, so there is a 1% chance that any given syringe will be found to be defective. That is, the probability that a randomly selected syringe is defective is 0.01

Let R be the number of defective syringes found in a random sample of nine syringes taken from the batch. We can say that R has a Binomial distribution with parameters, number of trials (number of syringes in the sample) n=9 and the success probability (the probability that a randomly selected syringe is defective) p=0.01.

the probability that R=r  defective syringes found in a random sample of nine syringes is

Now we can find the probabilities when r=0,1,2,...,9 as below

Now we can fill in the table as below

r	P(r)
0	0.914
1	0.083
2	0.003
3	0.000
4	0.000
5	0.000
6	0.000
7	0.000
8	0.000
9	0.000

Make the following histogram showing the probabilities of r = 0, 1, 2, 3, , 8 and 9 defective syringes in a random sample of nine syringes.

(b) Find μ. What is the expected number of defective syringes the inspector will find?

The expected value of R is

ans: the expected number of defective syringes the inspector will find is 0.09

Note: Since R has a Binomial distribution, we can calculate the expected value of R using the formula for Binomial expectation

(c) What is the probability that the batch will be accepted?

The probability that the batch will be accepted is same as the probability that the number of defective syringes in a random sample of 9 is less than 2

ans: the probability that the batch will be accepted is 0.997

(d) Find σ.

The expected value of is

the variance of R is

The value of standard deviation is

ans:

Note: Since R has a Binomial distribution, we can calculate the standard deviation of R using the formula for Binomial distribution