# Probability of a defective sample.

• WendysRules
In summary: You can do this with a calculator or using the dbinom function in R or similar functions in other statistics packages. In summary, the probability of having at least 3 defective units is approximately .8784.
WendysRules

## Homework Statement

In a manufacturing plant, a sample of a 100 items is taken from an assembly line. For each item in the sample, the probability of being defective is .06.

What is the probability that there are 3 or more defective units in the sample?

## Homework Equations

z = (x - mean)/standard deviation
mean = n*p
standard deviation = sqrt(n*p*(1-p)

## The Attempt at a Solution

Well, mean is just $$100*.06 = 6$$. The standard deviation is just $$\sqrt{100*.06*.94} = 2.375$$ so the Z = (3-6)/(2.375) = -3/2.375.

So, the prob(x>3) is just $$1- \frac{1}{\sqrt{2\pi}} \int^{-3/2.375}_{-\infty} e^{-.5x^2} dx = .8967$$ Which isn't an answer choice I'm given. So, obviously, I did something wrong! Any help is appreciated.

Answer choices given: .7040, .8784, .9306, .8498, .8843.

WendysRules said:

## Homework Statement

In a manufacturing plant, a sample of a 100 items is taken from an assembly line. For each item in the sample, the probability of being defective is .06.

What is the probability that there are 3 or more defective units in the sample?

## Homework Equations

z = (x - mean)/standard deviation
mean = n*p
standard deviation = sqrt(n*p*(1-p)

## The Attempt at a Solution

Well, mean is just $$100*.06 = 6$$. The standard deviation is just $$\sqrt{100*.06*.94} = 2.375$$ so the Z = (3-6)/(2.375) = -3/2.375.

So, the prob(x>3) is just $$1- \frac{1}{\sqrt{2\pi}} \int^{-3/2.375}_{-\infty} e^{-.5x^2} dx = .8967$$ Which isn't an answer choice I'm given. So, obviously, I did something wrong! Any help is appreciated.

Answer choices given: .7040, .8784, .9306, .8498, .8843.

One of your choices is fairly close to the true value, and is also not far from the normal approximation, when you do it properly. Remember the "1/2" correction: If the true random variable ##N## is binomial and the approximate random variable ##X## is normal (with the same mean and variance as ##N##), then ##P(N \leq k) \doteq P(X \leq k + 1/2)## for integer values of ##k##.

There are several ways to see this, but one way is to note that for a discrete random variable ##N## on ##\{ 0,1,2,\ldots, n \}## we have ##P(N \leq k ) = P(N \leq k + 0.5)## exactly (because ##N## takes integer values). Use the normal approximation on ##k + .5## instead of ##k##. For moderate values of ##n## and if ##np## is not really small or really large, the 1/2-correction often gives you one or two more decimal places of accuracy.

However, in this problem you can just as easily get an exact answer by performing the relatively easy calculation ##P(N \leq 2)## for the exact binomial.

scottdave

## 1. What is the probability of a defective sample?

The probability of a defective sample refers to the likelihood that a randomly selected item from a sample will be defective. It is typically expressed as a decimal or percentage between 0 and 1, with 0 representing no chance of a defective sample and 1 representing a 100% chance of a defective sample.

## 2. How is the probability of a defective sample calculated?

The probability of a defective sample is calculated by dividing the number of defective items in a sample by the total number of items in the sample. This can be expressed as a fraction, decimal, or percentage.

## 3. Can the probability of a defective sample change over time?

Yes, the probability of a defective sample can change over time. It may change due to changes in the manufacturing process, changes in the quality of materials used, or other external factors. It is important to regularly monitor and reassess the probability of a defective sample to ensure product quality.

## 4. How can the probability of a defective sample be reduced?

The probability of a defective sample can be reduced by implementing quality control measures, such as regularly inspecting and testing samples, identifying and addressing any issues in the manufacturing process, and using high-quality materials. Additionally, having a smaller sample size can also decrease the probability of a defective sample.

## 5. Is the probability of a defective sample the same as the probability of a defective product?

No, the probability of a defective sample and the probability of a defective product are not the same. The probability of a defective product refers to the likelihood that a single item randomly selected from the entire product line will be defective. This can be different from the probability of a defective sample, which only considers a small portion of the product line.

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