Question

An unnoticed mechanical failure has caused one-fourth of a machine shop’s production of 10,000 pistol firing pins to be defective. A random sample of 25 firing pins was drawn from the population.

a.) Explain why this random variable has a binomial distribution? (2 points)

b.) What are the mean and the standard deviation of the random variable? (2 points)

c.) Find P(X > 4). Use the binomcdf function on your calculator. (2 points)

Answer 1

So where are you stuck?

Answer 2

I don't really understand what a binomial distribution is or how it relates to the question. Neither do I know how to calculate the standard deviation and mean from the given information. I am unsure of what C is asking for.

Answer 3

Okay, a binomial distribution is a function for calculating the probability of a certain model.

Answer 4

The model is that you have \(n\) different events. Each event is independent of the other event, and they all have the same probability \(p\) of happening.

Answer 5

So essentially, you have the same event happening \(n\) times, and each time the result is unaffected by the previous result.
This model works for things like flipping coins, rolling dice, etc.

In this case each of the 25 firing pins is a sample event.

Answer 6

Does that make sense?

Answer 7

Yes it does! I am confused as to how this scenario is a binomial distribution though?

Answer 8

Well, the firing pins are randomly selected. It's not like they're all from the same line of defective firing pins.

Thus we can see them as being independent of each other. They all should have 1/4th probability of being defective. Even if the first 20 are defective, we still don't know whether or not the next 5 will be.

Answer 9

Okay gotcha!

Answer 10

Thank you!

Answer 11

How do I calculate the standard deviation?

Answer 12

There is some predetermined formula. I think it was \(\sqrt{np(1-p)}\) or something.

Answer 13

What does it mean by 'random variable", like how do I know what numbers to use? We know there are 10,000 defective firing pistols.

Answer 14

A random variable is some variable that does not have any particular value, it just has a set of values and probabilities for each value.

Answer 15

In this case, our random variable would be the number, out of 25, of defective firing pins.

Answer 16

So it could be 0, 1, 2, ..., 24, 25. Each of these have a certain probability associated with them, and when you add them up you have a probability of 1.

Answer 17

How do I solve the standard deviation? I am confused.