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EI

# Boxes are labeled as containing 500 g of cereal. The machine filling the boxes produces weights that are normally distributed with standard deviation 12 g.(a) If the target weight is 500 g, what is the probability that the machine produces a box with less than 480 g of cereal? (b) Suppose a law states that no more than $5 \%$ of a manufacturer's cereal boxes can contain less than the stated weight of 500 g. At what target weight should the manufacturer set its filling machine?

## A. $\approx 4.78 \%$B. Target weight must be 520 $\mathrm{g}$

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Applications of Integration

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##### Kristen K.

University of Michigan - Ann Arbor

##### Michael J.

Idaho State University

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### Video Transcript

Okay. And this problem we're going to answer some questions about the normal distribution. So the first part asked, What's the probability producing less than 480 g. So, let's take a look. We want if X is a normal distribution With mean 500 and standard deviation 12, then we can rewrite this problem as the probability that X is less than 480, because that will be the probability of Proceeding a box with less than 480 g of cereal. So in order to get this probability, we use the pdf of the normal distribution and we integrate like this from negative infinity 24 80 of one over sigma square to pie, X minus mu squared over two sigma squared dx. In this problem we have that New equals 500 and sigma equals 12. So we can rewrite this as the integral. I'm gonna plug in 12. 4 sigma and then plug in 504 mu Yeah. Now you would have to use a calculator to actually find the value because uh the function e to the negative X square doesn't have an easy anti derivative. Yeah. So when you use a calculator, you'll get .0478, which is 4.7, Yeah. So there's a 4.78% chance that this machine will produce a box of less than 480. Okay, For the next part, what should we set the target way to be If we want to make sure that 5% of boxes Uh don't have less than 500. So the interesting thing is that we can use the previous part to answer this question. This four seven, It's Pretty close to 5%. And the idea is if we were to shift the average over then the probability is going to be the same. So let's call why X plus 20 then the probability that why is less than Yeah, 500 is the exact same thing as the probability that X Is less than 480. You plug in X-plus 20 into here, and you'll get that. Why is less than 500? Okay, So this percent is 5%. But what would that mean? That the mean of why? It is? Yeah, so the mean of why is the mean of X plus 20? So it turns out that why is normal? Yeah, With the mean 520 and standard deviation equal to 12? And you can see this by uh huh, substituting Yeah, Y equals 20 plus x into that integral. What if you had a um if you want to replace this function with a function of why then you would have Why -520? The standard deviation doesn't change this limit would go to 500 and you get the exact same percent. So the target weight should be 520

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