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Problem 59 Hard Difficulty

A brewery's filling machine is adjusted to fill quart bottles with a mean of 32.0 oz of ale and a variance of $0.003 .$ Periodically, a bottle is checked and the amount of ale is noted.
a. Assuming the amount of fill is normally distributed, what is the probability that the next randomly checked bottle contains more than 32.02 oz?
b. Let's say you buy 100 quart bottles of this ale for a party; how many bottles would you expect to find containing more than 32.02 oz of ale?

Answer

a .3581; b. about 35-36 bottles

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Video Transcript

okay for this problem. We are given some information about ah, a breweries filling machine of court bottles. So we want Teoh drawer are normal distribution. That said, that's Norman. Normally distributed with the mean of 32 ounces is what it's supposed to fill at. Well sketched that out there and we said, We have a variance of 0.0. Uh, there there are three So most problems have been giving us the standard deviation. So with the mean and the standard deviation of Newsom's Ys fours and then some either technology or a table Teoh figure out the the probability this was giving us a variance. But we just got to go back to our understanding off the variances. Ah, a value stocks about variability and the standard deviation soda step water was gonna take the square root of the variance to get the standard deviation. So the standard deviation which we use this symbol, this is just the square root of 0.3 So let's just get that out of the way. And then the next two steps will be binding our boundaries, looking at our direction and then working backwards. We're working using the technology to get the probabilities. So it's first things first. Let's take this number 0.3 square root of it and thats gonna tell us that corresponds to ah, standard deviation is 0.5 It was dinner TV show. You can see that's ounces. So 0.54055 ounces. So that means that we're going above 0.55 each of those steps up and then below. I'm not gonna sketch that all out, but maybe we'll get somebody of the boundaries. So let's go over here and look at A We want to find its probability of 32.2 ounces more than so second stuff, we're gonna write what we want using probability notation. So we want to the probability that, given this information, um, it's normally the field is normally distributed. So what? The probably next bottle contains more than 32 point zero to write that out some more than 32.2 So I'm gonna sketch that just like to look at the direction. So 32 eyes here 0.5 the whole standard deviation above the 32.2 He's just a little bit above the the meaning there, but not even a full standard deviation above, um, it says more than 32.2. So actually going to sketch that to the rate I d like to show my directions. And I bet your teacher I want you to think your show your direction. Um, your table A table three user, You can use a table to get that. Since we have a calculator up here, we're gonna use our calculator. And were you second distribution? We know it's a normal distribution. I almost think their frequencies here. So in this case, we're going to look. A boundary of 32.2 is our left boundary upper boundary is going to the right towards infinity typing up ninth. That many nines is gonna give us plenty of digits to evaluate that. And, ah, we got to use the information we're given for this problem. We were told that 32 ounces is the mean and we were told the variants, but we figured out the standard deviation is 0.558 typical distance from the mean. So let's look at this and it says the 35% chance of rain over filler bottles there. So, uh, right that answer down 350.358 Dan, it's above that threshold. Okay, um and then we'll look at part B and see what it asks us. So for power bi, they see by 100 court bottles of the sale for party. How many bottles would you expect to be containing more than 32.2 ounces of Ailes? So we want to know for part B is rephrase that. Our own words. We want to know if we have 100 bottles. How money you're gonna be overfilled defense we call overfilled. So our plan here is just to take that simple Crip probability, Um, and just multiply it, um, the three d printer to So let's just take our number here, draw A little euro se will take him. We've been we figured out from party that about 36% more precisely 30.3581 overfilled If we consider that being overfilled, you were the most play that by 100. And we know that this is going with the decimal over to places. So 34 5.81 bottles. Okay, so let's say, um, everybody, would you expect? We'll say about 35 Maybe could see 36. But since there's not a full bottle, say, about 35 bottles. So that's what we do probability to make predictions. All right, so that's our problem about the feeling of bottles.