Question

In: Statistics and Probability

A manufacturing lot contains 40 items. It is known that 6 items are defective. A quality...

A manufacturing lot contains 40 items. It is known that 6 items are defective. A quality assurance engineer selects a random sample of 10 items and checks each to see if it is defective.

i. What is the mean and standard deviation of the number of defective items that she will sample. [4]

ii. What is the probability that she observes two or fewer defective items.

A road surface is being inspected for potholes. The number of potholes per kilometre is distributed as a Poisson random variable with rate parameter λ = 6.

i. What is the probability of not observing any potholes in a kilometre of road? [4]

ii. Suppose that an engineer inspects separate kilometre stretches of road until he observes one that contains potholes. What is the expected number of kilometre stretches of road that he will need to inspect before he observes one with potholes? [3]

iii. Suppose that 10 separate kilometre stretches of road are sampled at random. What is the expected value and standard deviation of the total number of potholes observed on the sampled roads?

Solutions

Expert Solution

Let X is a random variable shows the number of defective out of 10. Here X has hypergeometric distribution with following parameters

Population size: N = 40

Number of defective in population: M = 6

Sample size; n=10

(i)

(ii)

The probability that she observes two or fewer defective items is

-----------------------

Let X is a random number of potholes per kilometre. The pdf of X is

(i)

The probability of not observing any potholes in a kilometre of road is

(ii)

The probability of observing one potholes in a kilometre of road is

Using geometric distribution with parameter p = 0.0149 the expected number of kilometre stretches of road that he will need to inspect before he observes one with potholes is

1 / p = 1 / 0.0149 = 67.11

Answer: 67.11 or 67

(iii)

'Number of potholes in 10 separate kilometre stretches will be Poisson distribution with parameter λ = 6*10 = 60.

The expected value of the total number of potholes observed on the sampled roads is

The standard deviation of the total number of potholes observed on the sampled roads is

orchestra answered 1 year ago

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