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6.4.3Question HelpThe daily amount of coffee, in liters , dispensed by machine located in an airport ODby random variable X having continuous uniform distribution ...

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6.4.3Question HelpThe daily amount of coffee, in liters , dispensed by machine located in an airport ODby random variable X having continuous uniform distribution with A=- and B=11 Find the probability tnat on given day the amount of coffee dispensed by this machine will be most 10.4 liters more than 8.3 liters bu: less than liters; at least 9.8 litersThe probability [(Simoliiy your answver)

6.4.3 Question Help The daily amount of coffee, in liters , dispensed by machine located in an airport ODby random variable X having continuous uniform distribution with A=- and B=11 Find the probability tnat on given day the amount of coffee dispensed by this machine will be most 10.4 liters more than 8.3 liters bu: less than liters; at least 9.8 liters The probability [ (Simoliiy your answver)



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The daily milk consumption, in liters, dispensed by a milk booth located in a city is a random variable, $X,$ having a continuous uniform distribution with $A=$ 90 and $B=120 .$ Find the probability that on a given day, the amount of milk dispensed by this booth will be (a) at most 100 liters; (b) more than 105 liters but less than 115 liters; (c) at least 105 liters.

But we're given this table of probabilities, and we're gonna be answering some questions about it. The first is this discreet. So the difference between discrete and continuous is that discreet only covers numbers like let's say, 012345 or 0.1 point two point free. Right. There's an infinite number of numbers in between all of these numbers that we're skipping over. So we go from 1 to 2 to free. But there's no 1.5. There's no 1.4, So this is discreet because we're not considering every single number possible. So yes, so to draw hissed a gram of the distribution, we're just going to put the points and order. We don't rearrange them from least to create IST or anything like that. And her ex access is all of the X values. So one through seven, so 567 and ah, label some points on here. So 0.2 I'll have a 0.5 a 0.1. That way we can guess a little bit a 0.2 and a point free, so one goes up. The 10.22 goes up to slightly higher than point free. Furry goes up 2.28 Cell around here four goes up two point wine, five goes up 2.5 and six and seven go up. 2.0 chill. So that's it. Find the mean and standard deviation because we're given a table. This is gonna take a whole lot longer. What we're gonna dio to find the mean is we're gonna take each x value, and we're going to multiply it by the respective probability and then add all of them up. That's what this equation here means. So to find our mean, we are going tohave one times 10.2 plus two times point free free plus free times 0.28 plus four times point wine plus five times 50.5 plus six times 0.2 plus seven times 70.2 down. If we add all of this up, you should get 2.61 Now to find the standard deviation, we're gonna need the variance and the do variance. We're gonna do the same thing for mean, but we're gonna square all of the exes and then subtract r means squared at the end. So to find variants. This time we're gonna have one square times 10.2 plus two squared at times point free free plus free squared times 20.0.28 plus four square times 40.1 plus five square times 50.5 was six square, two times 20.2 plus seven square, two times 20.2 and then we're going to take this whole fame and subtract 2.61 squared. And when we do that, we get our variance is 1.7779 and then finally, to find our standard deviation. We just take the square root of variance and you should get one point free, free, free and we are done.

This problem deals with bottling company that fills bottles of soda in the label on the bottle indicates that there's 300 no leaders of soda in the bottle. But the contents of the bottles actually vary great. The information is that it has a normal distribution. The the mean of the normal distribution is 298 and it has a standard deviation of three milliliters. The one standard deviation decide once leader division on this side and the meat and this distance right here this standard deviation of three milliliters. So if I go three to the right of 2 98 I actually good 301 It was a three to the left. I get to 95 and the question in part a have. What's the probability of the contents of individual bottle is 295. No leaders are well apart. A have one individual bottle, and I need to know if the content is less them or equal to 10 95. So I'm gonna need to find the area of thunder that curve left to 95. So the first thing I need todo this to know what the singer Deviation is we just talked about it. So I know the standard deviation is negative one. But if you didn't, you would use the score to find a standard deviation of your point to 95. So this is the score is going to be the number you're looking for. 2 95 minus the mean in this case is to 98 divided by the standard deviation, which is three. That's gonna be negative. 3/3. So negative one is the standard deviation. Now, if I want to find the probability, I'm gonna look for the probability that disease score is less not you looked a negative one, which is that area under the curve in green. And when I was caught up in the table, you have made give one and negative one as quaint. 15866 0.15866 That's also the probability of what I'm looking for. That there is some bottle that has left never equal to 295 no leaders inside the bottle. So that's also your 0.158 So what that means is that there's a 15 percent chance that the bottle will have love again. We're equal to 295. Hello, leaders next part be and Part B says they take a simple random sample of six bottles. So that means that my sample sizes and you're gonna take those six bottle, figure out how much is in each bottle and average that amount. So X with a bar over the top means the mean of the sample, so access the mean of the samples. So what I want to do is I'm gonna do six of them. So my question is, what is the probability that a sampling of six is going to have less than her equal to 295 milliliters in the bottle? So in this case, I'm looking for a same plane distribution of the sample mean So when I find the means, I keep plodding and plodding until I actually come up with the distribution. Now the means air less variable than individual observations, like in part A. So my normal curve is going to be more normal. That's gonna be a little more compact. Some things I know about it is that the mean is actually equal to the mean of the population. So the mean of the same old means it's going to be 298 and then I need to find the standard deviation formula for the senior deviation of the sample. Means is the standard deviation of the population divided by the square root of the sample size, though the standard deviation of the population I could see a Here is three milliliters and my sample size was it over the square root of sticks, and that's going to work out to be 1.2247 if I go one senior deviation on each side. This just in a year is my standard deviation of simple means, which is 1.2247 So it's only the introduce. Aviation is just a little bit over one. I'm looking for 2 95 to 95 in this draft is going to be You hear someone say to 95 if I need to find the score of 2 95 but I have to 95 minus domain, which is to 98 but it's going to be over. The senior deviation with sample means which is 1.2247 and that's going to equal to negative 2.45 So to get the probability of a score that's less than or equal to negative 2.45 it's also going to be the same thing. Is the probability of the same plane maybe in less than or equal to 2 95? If I look up negative? 2.45 in my craft, negative 2.545 is 00714 That's going to be equal to zero point 00714 So that means there's a 0.7% chance that one of my that six bottles the embassy six bottles will have a mean that a mean content that is less than 295.

All right, we're talking about, uh, cups getting dispersed, uh, cups getting filled with, um, soft drinks or coffee and a machine. And so on average, it's gonna fill automatically 7.6 ounces. But then there's a standard deviation of 0.4. So that means we need to add 0.4 to go get one standard deviation, which is eight plus another 80.4 is 8.4, plus another would be 8.8 ounces. Then we subtract to go get below the standard deviations of 7.2, 6.8 and then 6.4 ounces. Okay, So the question the first question is, what is the probability of an overflow of eight ounces? So that means probability of it being greater than eight ounces. So that's going in this direction. So that's gonna represent 13.5% plus 2.35% plus 0.15 So that's gonna be approximately 16% or 160.16 question B is asking what would be of it not overflowing. So the probability of it being less than eight would be one minus the 16% that we just gotten party. So that's gonna be 84% approximately. And then Part C is asking. The machine has been loaded with 850 caps. How many of these do you expect will overflow? So if 16% overflow approximately, that's 1.16 of 8 50 meeting we're gonna multiply, and that would be approximately 136 cups.

So we know that the vending machine has is dispersing, uh, soft drink into a cup, and it's supposed to be normally distributed so we can draw the bell shaped curve. And it's supposed to be centered at a mean of 7.6 ounces with a standard deviation of 0.4 ounces. So here is where eight ounces would be one standard deviation higher. And we want to know what's the likelihood that the the machine will dispense more than eight ounces because the cup holds eight? And so we want to know what's the likelihood that it's going to dispense too much liquid? And then it's going to over floor and the cup is full with eight ounces, and so that corresponds with a Z value, one standard deviation higher of one. Now we can use on the 98 the 68 95 99.7 rule, or we can actually look this value up and let's just do an approximation. We know that 68% of data in a normal distribution is within one standard deviation, and so 100% minus 68% gives us 32% and we need to divide that by two and get 16%. So we know that this region right in here is approximately 16% and this region over here is 16%. So our answer to this question is approximately 16% will overflow. And so then our question is, what's the likelihood that it won't overflow? And that's the compliment. So 100% less the approximate 16% is about an 84% chance of that happening and then part C as as if we fail a total of 850 cops. How many of those should end up overflowing while we would anticipate 16% of those? So if we take 16% of 8 50 we would expect that about 136 of those cups will overflow and we're all set.


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