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The article "Computer Assisted Net Weight Control" (Qual.Prog. $1983 : 22-25$ ) suggests a normal distribution with mean 137.2 oz and standard deviation $1.6 \mathrm{oz},$ for the actual contents of jars of a certain type. The stated contents was 135 oz.(a) What is the probability that a single jar contains more than the stated contents?(b) Among ten randomly selected jars, what is the probability that at least contain more than the stated contents?(c) Assuming that the mean remains at $137.2,$ to what value would the standard deviation have to be changed so that 95$\%$ of all jars contain more than the stated contents?

Intro Stats / AP Statistics

Chapter 3

Continuous Random Variables and Probability Distributions

Section 9

Supplementary Exercises

Continuous Random Variables

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All right. So the question that is given to us about a normal distribution, So given that he was normal distribution to solve this problem so mean is given as ah 1 37 I want to go there on the Sigma, but is the standard division This is given as 1.6. So now, if I use the definition of standard normal distribution there is given by X minus mu upon sigma. So the first part that they have told us actually to find before that I'm going to substitute the value of mule or yours. And one sigma values 1.6. So first part, which is a bar they have asked us to find the probability that the single jar contains more than the state of contents. So the state of contents are given us 1 35 So which means I can take X greater than 1 35 This is what we're supposed to find. So to start with, I'm going to first find the value of said at X equal 1 35 so we can substitute X over here as 1 35 minus 1 37 going toe upon 1.6 to simplify, you're gonna get minus 1.37 or here after you simplify and get the value of said we can therefore now right, the probability off this particular Tom Tom's upset as Zell is greater than minus one wind, three said So since from the table we need to look this value we can write the greater than sign as one minus probability off send a lesson equal to minus 1.37 Now this condition that we have the scandal represented as one minus five off minus 1.37 So you want to take this valley from the table off the Sander normal distribution table and it won't get the blow off 1.3 and column of 0.7 I think the value and then value comes as a 0.8 So if I performed the subtraction, the answer that I'm getting here for the probability is 0.9162 So this is the first part that we've done now moving on to the second part for the second part there said that we need to find that the jars can't be more than scary contents are at least take So which means that over here the vast refused my normal distribution because n over here is given asked them their moral dangers on DDE. Therefore, I applied by normal distribution this can Britain s Stanko mother probably TV and just found you know why 9162 So the definition of by a normal distribution to find the probability and begin as n c x b days twigs and this is cuties to end minus X where we can see that Q was given by one minus b So in this particular problem, you know P is 0.96 To soak you is going to be one minus B which will be 0.8 three So no morning on the mean part probably that they told us to find is probably off They have said X should be granted an equal So say it because they say that there should be a two least eight jars So this can Britain as probability off X is equal to eight Bless probability off exit nine and next is probably off X That is equal then. So if I substituted these values or here in the formula off by a normal distribution we will have. In a way, he understands this is 10. See it be a zero wind 916 to this will be raised to eight. And he was zero pines. It'll end three years. That'll be 10 minus eight, which is to plus, if I write for X is equal to nine consistency. Nine. You know why? 9162 days to nine. Andi, don't mind. You ate 38 These 21 and the next is going to be only 10. See 10 and 0.9162 days to 10 and this time becomes a stew zero, which is actually one. So I did not write that bar the calculator. If you put all of these terms and you simplify it, an individual answers. I'm writing her. I'm getting 0.1568 Plus, the second term is giving me 0.381 plus stardom is 0.416 10. If you add all of these values on so getting here 0.9547 So we've added these probabilities because at a point of time. Only one Republican occurred. We cannot have. All the three are occurring at the same time. So it reflects or and or means audition. So we've been with the B partner going onto the sea part again for the sea parted. Told us he was a normal distribution. So normal distribution. We know Muir's 1 37.2 and ah signifies 1.6. But we write Sigma squared, which is the villians or here. So now they're told that, uh, you need to actually, So this is the first problem. They have said that the mean remains 1 37.2 But you to find the standard deviation value such that nine different person off all jars will contain more than the slated contents. So in this case, we actually have to solve this as the worst problem where they have said that 95 percenters already the number of jars which contain, you know more than the skated condense. So I can therefore write down that probability off zed. All right. Oh, is good. And then some small value off there which was given as 95% so far right that in decimal that'll be 0.95 So not from the normal table that we generally look. What we need to do is we need to find a nearby value 2.9, find the table and take that particular rule and call him and find out the value of this small. They so from the table. If I find this well, I'm getting the value off small that as minus 1.645 So no, Since I've got the value of that, we also know human sigma so I can use the formula and I can find out the value off the new Sigma. Add 0.95 probability. So I can say that they're interesting. My eyes therefore equal to X minus mu. So in this case, simplifying this part funder we could get So this is Ah, we have X is 1 35 which will be equal to the new partners 1 37.2 Bless zealot, which is minus 1.645 into Sigma. So if I simplify the subtraction and taking the terms on the other side, I get 1.645 Sigma is actually called Toto going to. So therefore, the value of signifies going to divide by 1.645 which actually gives me while I'm buying 34 or was there so this way we have done with the starving off older?

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