All right. So, we're looking at the length of human pregnancies, which is approximately normally distributed with a mean 266 days days. And then the standard deviation of the population 16. And were asked some questions about what's the probability randomly selected pregnancy uh lasts less than This many days. 260 days. Well, so they were basically asking basically we are asking the problem a probability that a random sample two away. Mhm. A selected pregnancy. So this one we're just gonna take X and want that to be less than 260 days. So for this one, what we're gonna need is the Z score, which is calculated by x minus mu over sigma, Which for our purposes here is to 60 -2, 66, all over 16, which is going to give us -375. And then we do a look up table and that would tell us that the probability is equal to 0.34, looks 354. Excuse me. Mhm. And then, just as a little picture, here's your normal distribution This .354. So, here's your mean Of to 66- 60 is somewhere over here. We'll call that to 60. And then it's this area, This number .354 is this area. And then be we have the probability that a random sample with 20 as a mean of 26 year last. So we have the probability that the sampling distribution has a mean less than 260. So this one we're gonna apply the we use the central limit theorem which says that the sample distribution has a mean that is approximately normal or that excuse me. That is that the sampling distribution is approximately normal given a and we as long as the problem the population distribution is normal, which this is we can use this. He was essential limit room which says the Z scores was given this way with the sample mean minus mu over the standard error of the mean. Which is given as sigma over route. And um If it's not normally if the population distribution is not normal then we have to have the sample size of and greater than 30 but it's our population is already normal. So we can just apply this. So we're gonna do Z score. But we want the X. Bar which in this case is to the mean to be to 60, That is to 66 all over the The standard error of the mean, which is in this case is 16 over and then N is 20 in this case. So route N if you do that, you end up with the negative 1677. That's the Z score. Mhm. Okay. And then this is going to be equal to You. Do a little table look up or some sort of calculating device to get 0.047. Yeah. And again that would be a the area over here. Although the the sampling distribution Oh and the sampling distribution of a population that's approximately normal would have a mean, That is the population means so we can do that then for c it's similar. Uh except the sample size is now 50, so we do the same kind of thing. Mm The probability of the X. Bar is less than 2 60. However, the mean is different. So just to Put that all together here, this is where N20 20 and this is where and is 50. So it's a larger sample. And that means we're gonna do the same thing with the Z score to 16 -2 66. of Make that six. All over 16 over route 50. That is now the standard deviation of the mean. And then this is going to give us -2 651. So that's all good. And that's the Z. Value. And then we do the table look up And this is .004, I'll draw a picture of it because I was like pictures, pictures and statistics, and pictures and math. Just go together. Like just so well, Peanut Butter and Jelly. Here we go. So here's here's where the mean was to 66. Popular the uh to 60. So There were the means to 66 right in the middle, Normal distribution is symmetric. And so here's to 60. And then this .04.2.004 is this area right here. My graph is my drawing not to scale, but just to give you a picture of it, where is here, I'll do it in red. It's the same kind of thing here is to 66, but the standard deviation or the standard error of the mean changed um And to 66 is right here to 60. Excuse me, is right here And .04 is right here. So this .47 is right here. So this area, this red area is a bit bigger than this one. And that's because when we increase the sample size, we reduced, we decrease the variability. So the probability of this happening is pretty, pretty rare because with size 50 we're getting a better picture of what the, the mean is and that's actually what the next question is. He says, what can we conclude if this sample of 50 pregnancies gave us the mean gestation of Of 260 days or less? Well, that my friends would be very rare, that means one of two things. one we either picked a very unique, So we either picked a very unique set of 50 people who have a, I mean that's 260 or less, we impact a we randomly selected ah a unique group Who had the mean of 200 of less than 260. Well, when you put sample mean two days, the gestation period, uh there's that or you know what, that could, that could mean that the population is wrong or that the population mean is wrong. So this could mean this could be a telling sign for us to say, you know what? We should reevaluate this, this uh This assumption that the mean is 266 days. So we, this could also tell us that perhaps the The population mean is not to 66. Let's use symbols perhaps. Oh my gosh, having trouble writing today. She really, perhaps new is not 66. You know, I love symbols of And the last party. What the probability that Sample size of 15, we'll have a gestation within 10 days of the mean. So um like the calculations will be the same here. Um and but what we're looking for, basically the probability That the sample mean is within 10 of the population. So the sample mean, the population mean is to 66 within 10 would be like, Right, if here's to 66 Shortly. Here's to 66, you could go 10 this way, which take you 2 to 76 or 10 this way Be right plus 10 or -10. It should be 2:56. And what's the probability of landing within here. And so if we grab this out, just kind of super do that. Here's to 66 and we want Here's one standard deviation two Standard deviations three, Sandy aviation's government standing ovations where it is, which is actually the Z score, we would want this total area right here, So we want the Z score for 2 56 and Z score for 2 76 which this is a beautiful thing about normal distribution. It's symmetric people, so this is pretty, pretty cool. Make that better. Mhm Yeah, so once we get the Z scores with the mean, it's the standard normal. So The mean is zero You get plus 24, two -2.42, which hopefully That shouldn't seem random to that should seem pretty understanding because you're 10 away on the left and 10 away on the right, so that you're going to be the same distance away on either side, we're going to have the same standard deviations away, positive. End of the negative. And then you do your table look up and then basically what you end up doing is the probability of X less than two 0.42 minus. But Princess, usually 1 -1 X are less than To -2.42. And this is going to end up giving us .984 and what we're doing, so we're doing this, we're doing the Z look up and you're doing the same table look up. But just to get you that the concept here, this portion here, X bar minus, Excuse me, the probability that export is less than 2.42 is all this stuff and then probably the X bar is less than point Excuse me. The probability that export is less than negative, 2.42 is this stuff? So we need to subtract this magenta portion away from the whole green thing, which will give us This magical air we want. So that's why we do one that probability. Um So you go. And if you're just to make sure you're getting that, this ends up being .992 and you're gonna subtract that from 1 -1. Oh no, sorry, this ends up, this whole thing ends up being mhm .008. That's that's where you get. So there you go.