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Problem 64 Hard Difficulty

A machine fills containers with a mean weight per container of 16.0 oz. If no more than $5 \%$ of the containers are to weigh less than 15.8 oz, what must the standard deviation of the weights equal? (Assume normality.)

Answer

$0.120 z$

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Video Transcript

okay for this problem. We are looking at some machine filling data. Um, so and it's assumed that normally distributed. So we always want to draw a diagram to think through what's going on here. So are mean. Way is number is 16 announces your plan here is we're gonna fill the pieces that we know and work backwards to find the pieces that we don't know. So be 16 ounces, but we don't know is this standard deviation. So we want to know the standard deviation. So a little bit different than the previous problems we've been doing. We want to standard deviation. So we must have one of the piece of information here and there will suffer. A little piece of information is no more than 5% can weigh less than 15.8 ounces. They're 16. It's a 5% pretty small number here, so we'll say, uh, no more than 5%. So I'm gonna put a 50.5 here, so no more than 5% way, um, 15.8 ounces, so I don't normal graph 15.8 is here. There's not much variation going on there. So our plan here is gonna t to take this information and we're gonna work backwards and get a Z score that goes with this spot right here. And then we can use our Z score and are no mean to solve the equation and get, um, standard deviation. So let's go over here and there a calculator on. And, um when do in verse in normals the second distribution go to the universe normal because we have a piece of information where we know an area or proportion there that goes with the 5%. So when it changed this to 5% of my area, I'm gonna leave my meanest in a deviation here alone, cause that's, uh, without the context we have here. So for anything with the area of 5% that corresponds to a Z score of negative 1.644 So any kind of Z that goes with 0.5 equals to negative 1.644 So now we're gonna apply our, um, our understanding of Z scores to work backwards and find are unknown. So we know any of these four equals to any value minus the mean of the data set by by the standard deviation. So for this problem, we you know that we're gonna use this negative 1.644 because that's disease for that corresponds to the X. We do know. So the 15.8 is the value we know that has a 5% less than 5% there. Uh, in this case, we do know the mean, which is 16 ounces. Could could you know, that's negative there, which makes sense. And we don't know the standard deviation. It's a little more tricky algebra here, but if we want to get the standard deviation, what I'm gonna do is I'm gonna multiply both sides by the standard deviation, have some good algebra habits there to get it off the bottom. So we got negative. 1.644 equals two, uh, 15.8 minus 16. So that's just point to and, um, and the negative 1.644 His mobile standard deviation there from the standard deviation after divide herbal sides by make it a 1.644 showing on my work here. So the standard deviation equals to whatever. The result is a two divided by one 0.644 So let's go. Something tricky. Oh, yeah. Since I shouldn't be negative. So I think I see my negative mistake there. So this is negative. Two negative as well. 15.8 minus 16 is negative. Two is it's positive value, which it should be. So we're gonna take 0.2. And since my answers appear gonna pull back up from a calculator point to most of the negative in there. Second answer. But so that means our standard deviation is just 0.12 one, which seems like it makes pretty good sense. So since point to early over here and 16 is that so 160.1 to a little bit like 1/2 gets us a step to that 5%. So our standard deviation is negative. One negative 0.1 to 15 ounces with these containers