Applied Probability Trust

Transcription

You sample 36 apples from your farm's harvest of over 200,000 apples. The mean weight of the sample is 112 grams with a 40 gram sample standard deviation. What is the probability that the mean weight of all 200,000 apples is within 100 and 124 grams? Let's think about what they're asking. So there's some distribution of all of the weights of all 200,000 apples or there's more than 200,000. We don't even know how many apples, just a huge number. So there's some population distribution of weights. Maybe it looks something like that. It will have a mean weight. It has a mean weight. We don't know what that mean weight is, and it also has a population standard deviation. So this might be one standard deviation above the population mean, that would be one standard deviation below. And we'll say this distance right here is the population standard deviation. Both of these are parameters that we do not know of the entire population. This is the population distribution right over there. Now, we know from our experience with the last few videos that you can repeatedly take samples, or if you kind of visualize, repeatingly take samples of a certain size-- in this video we're going to focus on sample sizes of 36. And you keep taking the means of those sample size, and you plot the frequency with which you get those means, you would eventually get something called the sampling distribution of the sample mean. Let me write this down. The sampling distribution of the sample mean. So that might look something like this. I'll try to draw it a little bit bigger since we're probably going to use this one a little bit more. It is going to be pretty close to a normal distribution. It's going to have some mean, and we specify that-- let me draw it down here. It has mean when we show that this is the mean of the sampling distribution. And we know that the mean of the sampling distribution, that the means of all of your means, or of the actual distribution of means, is actually going to be your original population mean. So this is going to be your population mean over here. And it also has some standard deviation. So maybe this is a standard deviation above the mean, this is a standard deviation below the mean, right over there. And we can specify that by the standard deviation of the sampling distribution of the means. And we know that this can be a given, or I guess approximated, because for fairly large samples this gives a pretty good indicator. There's a couple of correction factors if you get to smaller samples. But this is going to be our population standard deviation divided by the square root-- and we saw this in the last two videos-- divided by the square root of the number of samples we have when we calculate each of those means. And we know in this example that we are taking 36 samples. So this is the square root of 36. This is a sampling distribution of the sample mean-- let me write it over here-- 4n equals 36. For each of our sample buckets or baskets to have 36 items and then we take their mean. And then that is essentially each of those means is a sample from this distribution right over here. The means are the sample from this-- the things that we're using to calculate the samples are samples, or the things that were used to calculating the means are samples from that. Hopefully that isn't too confusing. But this isn't the first time we've seen it. Anyway, this distribution's standard deviation is going to be the standard deviation of this population standard deviation divided by 6. But we still don't know this. We still don't know this parameter up here. Now with that said let's refocus on what they're actually asking us. They want to know the probability that the mean weight of all 200,000 apples. Well the mean weight of all 200,000 apples is that parameter right over there. And they want to know what is a probably that it is between 100 and 124 grams. So they're actually asking us if something is between 100 and 124 grams it is within 12 of our sample mean. Right? That's all they're saying. What is a probability that this thing is within 12 of our sample mean? Because if you're less than 12 or if you're 12 less you're going to get to 100. If you're 12 more you're going to get to 124. So what they're asking us is what is the probability that our population mean, this parameter, this unknown parameter, is within 12 of the mean of our one sample. Now if I told you that I'm within 5 feet of you, then that also means that you're within 5 feet of me. So this is the exact same thing as the probability that the sample mean is within 12 of the actual mean. I really you to-- this should make sense. If I said what's the probability that I'm either 5 behind you or 5 ahead of you, that's the same thing as a probability that you're either 5 behind me or 5 behind you. This is asking what's the probability that we're 12 apart, or what's the probability that I am within 12 feet of you. And this is the probability that you're within 12 feet of me. They're asking the exact same thing. But when you phrase it this way it might dawn on you that you might be able to use the sampling distribution of the sampling mean. There's some unknown mean here, which is the same thing as this value right here. So this thing-- let me make it very clear-- this is also the same thing because this value and this value are the same thing. This is exactly the same thing as asking what is the probability that our one sample mean is within 12 of the actual mean of the sampling distribution. So we're just saying what's the probability that that one sample mean we have is within 12 of this actual sampling mean. At this point your brain should be reading that gee, if I could figure out how many standard deviations that is, how many standard deviations away that is on this distribution, I can then use a Z-table to actually figure out the probability. And that's exactly what we're going to do. But there's one slight complication here. We don't know the actual standard deviation of the sampling distribution. We just know that it's this thing right over here divided by 6. But we don't know this thing. So what we're going to do is get our best estimate of this thing. So we need a good estimator for the actual population standard deviation. What's our best estimate of that? What's going to be our sample standard deviation? We sampled 36 things, and we had a sample distribution of 40. So we have a sample distribution of-- let me write this way. This is going to be approximately equal to our sample distribution or sample standard deviation, which we got to be 40. So we literally just took-- we found the mean of our 36 apples, mean weight was 112 grams. Then we found the square distance from each of the apples' weights to this. Took the average of those. Well we didn't take the straight up average, we divided by n minus 1. We learned all of this many, many videos ago. And then we took the square root of that. This gave us the sample standard deviation, it is our best estimator for this. So if this is our best estimator for that, our best estimator for this thing right here is going to be equal to our sample standard deviation divided by 6, which is equal to 40 over 6, which is equal to-- let's get our calculator out. So if we have 40 divided by 6, we have 6.6-- I'll just write down 6.67. So this thing right here is 6.67. So our best estimate of the standard deviation of the sample distribution of the sample mean is 6.67. So this distance right here is 6.67. So how many standard deviations is 12 if you look at this distribution right over here? Well we just divide 12 by 6.67. So let me get the calculator back up. So if we have 12-- I'll just divide it by that 6.-- actually this exact number-- 12 divided by-- answer just means the last answer we got-- that gives us 1.8 exactly. The numbers just happened to work out well. So this is completely analogous to saying what is the probability that our one sample mean is within 1.8 standard deviations. Let me write it this way. It was in 1.8 standard deviations of the sample mean-- within 1.8 of these-- of our actual mean of our sampling distribution. So we're literally just asking that. So if you look at this distribution up here, within 1.8 standard deviations, this is one standard deviation, another 0.8 would maybe get us right about there. And we're within 1.8 above and 1.8 below, so this is 1.8 standard deviations above the mean, this is 1.8 standard deviations below the mean. So we're just going to say what's the probability when we just took this one sample of 36 apples that we lie in this space over here? And to figure this out what I'm going to do is I'm going to use our Z-table to figure out essentially this space over here. Just what's the probability of being 12 above it. And then we can just double it because a normal distribution is symmetric. So let's go to our Z-table. So what's the probability of being between the mean and 1.8 standard deviations above the mean? So if you just go straight to your Z-table, 1.80 is this right over here. Now you get this 0.9641 number. But be very careful. This 0.9641 number that gives you-- so if I draw a normal distribution-- let me draw a better normal distribution. If I draw a normal distribution like that, and this is our mean, this 0.9641 number tells us the probability that we are less than 1.8 standard deviations above the mean. So this is 1.8 standard deviations up here, then this is our mean right over here. This is giving us this entire area right over there. So if I want just this area right here, what I need to do is from this value, from this 0.9641, I need to subtract this, the probability that you're essentially directly less than the mean. And that is this. This is the probability that you're less than the mean, or you're less than the mean plus 0 standard deviations. So this value right here is 0.50. This whole area that I just showed you right over there is 0.9641. So this area right over here is going to be 0.9641 minus 0.5, which is going to be equal to 0.4641. So this area right here, just what's in the magenta. Just between there and there is 0.4641. Let me make sure I got that right. 0.4641. And if I want this entire area I just double it. If I want to include this as well, I just have to double that. So let me get my calculator out, let me get the trusty calculator. So we're going to have 0.4641 times 2 is equal to 0.9282. So this whole area right over here is equal to 0.9282. So we did something neat. The probability that the-- remember that well, we're answering right here. The probably that our sample mean just happens to be within 1.8 standard deviations-- remember, that was 1.8 standard deviations of the sample means from the actual mean is 0.9282, or there's a 92.82% chance. But that's also saying that there's a 92.82% chance that the actual mean is within 12 of our measured sample mean. And that is neat. Because for the first time ever-- we started with very little information. We just started with a little sample over here. And we were able to get as much information about that sample as possible. But we can now say that there's a 92.82% chance that the actual mean is within 12 of the mean we measured. That the actual mean is between 100 and 124, or that we're 92.82% confident that the actual mean is in the range between 100 and 124 grams. I think that's pretty neat.