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Current time:0:00Total duration:11:47

Video transcript

this is the same problem that we had in the last video but instead of trying to figure out whether the data supply sufficient evidence to conclude that the engines meet the actual emission requirement and all of the hypothesis testing I thought I would also use the same data that we had in the last video to actually come up with a 95 percent confidence interval so you can ignore the question right here you can ignore all of this I'm just using that same data to come up with a 95 percent confidence interval for the actual mean emission for this new engine design so we want to find a 95 percent 95 percent confidence confidence interval and as you could imagine because we only have 10 samples right here we're going to want to use a t-distribution and right down here I have a t-table and we want a 95% confidence interval so we want to think about the the range of T values that 95 or the range that 95 percent of T values will fall under so let's think about it this way so let me draw a T distribution let me draw a T distribution right over here so the T distribution looks very similar to a normal distribution but it has fatter tails it has fatter tails this end and this end will be fatter than in a normal distribution and then in it we want to find an interval so this is if we this is a normalized t-distribution the mean is going to be 0 and we want to find an interval of T values between some negative value here and some positive value here that contains 95 that contains 95 percent of the probability so this right here has to be 95 percent and to figure what these critical T values are at this end and this end we can just use a T table and we're going to use the two-sided version of this because we're symmetric around the center so you look at the two-sided we want a 95% confidence interval so we're going to look right over here 95% confidence interval we have 10 data points which means we have 9 degrees of freedom so 9 degrees of freedom for our 10 data points we just took 10 minus so if we look over here for that type so for a t-distribution with 9 degrees of freedom you're going to have 95% of the probability is going to be contained within a t-value of so the T value is going to be between negative so this value right here is 2.262 and this value right here is negative 2.262 that's what this right here tells us that if you are if you contain all the values that are less than 2.262 away from the center of your T distribution you will contain 95% of the probability so that is our T distribution right there let me make it very clear this is our T distribution T distribution so if you randomly pick a T value from this T distribution it has a 95% chance of being within this far from the mean or maybe we should write it this way if I pick a random T value if I take a random T statistic there's a 94 let me write it this way there's a 95% chance there is a 95% chance that a random T statistic is going to be less than 2.262 2.262 and greater than and greater than negative negative 2.262 95% chance now when we took this sample we have also we can also derive a random T statistic from this we have our sample mean and our sample standard deviation our sample mean here our sample mean here is 17 point one seven figured that out on the last video just add these up divided by 10 and our sample standard deviation here is two point nine eight so the T statistic that we can derive from this information right over here so let me write it over here the T statistic that we can derive from this and you can view this T statistic as being a random sample from a T distribution a T distribution with 9 degrees of freedom so the T the T statistic that we can derive from that is going to be our mean seventeen point one seven minus the true the true mean of our population or actually we say the true mean of our sampling distribution which is also going to be the same as the true mean of our population that's our population mean over there / / S which is two point nine eight over the square root the square root of our number of samples we've seen this multiple times this right here is the T statistic so by taking this sample you can say that we've randomly sampled at ISTA tist ik for this nine degree of freedom T distribution so there's a 95% chance that this thing right over here this thing right over here is going to be between is going to be less than two point two six two and greater than negative two point two six two so the 95% 95% probability 95% probability still is still applies to this right here now we just have to do some math calculate these things let me get my calculator out and get the calculator out and so let me just calculate this denominator right over here so we have two point nine eight divided by the square root the square root of ten so that's point nine four two three Oh point nine four two three so what I'm going to do is I'm going to multiply both sides of this equation by this expression right over here so if I do that so let me just do that right over so if I multiply this entire this is really two equations one or two inequalities I should say that this quantity is greater than this quantity and that this quantity is greater than that quantity but we can operate on all of them at the same time this entire inequality so what we want to do is multiply this entire inequality by this value right over here and we just calculated that that value let me write it over here the two point nine eight I'll write it all right over here two point nine eight over the square root of 10 is equal to 0.9 point nine four two so if I multiply if I multiply this entire inequality by 0.9 for two I get on this left hand side over here I have negative two point two six two times times point nine four two and it's a positive number that we're multiplying the whole inequality by so the inequality signs are still going to be in the same direction is less than well we're multiplying this we're multiplying this whole expression by the same expression in the denominator so it'll cancel out so we're just going to be less than seventeen point one seven minus our population mean which is going to be less than two point two six two times once again once again 0.94 to let me scroll over to to the right a little bit nine four to just be clear I just I'm just multiplying both all three sides of this inequality by this number right over here in the middle this cancels out so if I multiply I'll just write out here point nine four two point nine four two point nine four to this and this is the same number so that's why those cancel out and now let's hit the calculator to figure out what these numbers are so point if we have the point nine four two times two point two six two so we're going to say times two point two six two is two point one three so this is going to be so this number right over here on the right hand side this number on the right hand side is two point one three two point one three this number on the left is just the negative of that so it's negative two point one three negative two point one three and then we still have our inequalities is going to be less than seventeen point one seven minus the mean which is less than two point one three now what I want to do is I actually want to solve for this mean and I don't like that negative sign of the mean I'd rather have the swapped around I'd rather have the mean minus seventeen point one seven so what I'm going to do is multiply this entire inequality by negative one if you do that if you multiply the entire thing times negative 1 this quantity right here this negative 2 point 1 3 will become a positive 2 point 1 3 but since we are multiplying an inequality by a negative number you have to swap the inequality sign so this less than will become a greater than this negative this negative mu will become a positive mu this positive 17.1 7 will become a negative 17 point 1 7 where we have to swap this inequality sign as well and this positive 2 point 1 3 will become a negative two point one three and we're almost like we just want to solve for M you have this inequality expressed in terms of MU so what we can do is now just add 17 point one seven to all three sides of this inequality and we are left with two point 1 3 plus 17 point 1 7 is greater than mu minus 17 point 1 7 plus 17 point 1 7 is just going to be mu which is greater than so this is greater than mu which is greater than negative two point one 3 plus 17 point one seven or a more natural way to write it since we actually have a bunch of greater than signs that this is actually the largest number and that and this are sorry this is actually the smallest number and this over here is actually the largest number is doctory flip you can just rewrite this inequality the other way so now we can write we can write actually let's just figure out what these values are so we have two point 1 3 plus so we have two point 1 3 plus 17 point 1 7 so that is that is the high end of our range so that is 19 point three so this value right over here so this is 19 let me do it in that same color this value right here is 19 point three is going to be greater than mu which is going to be greater than and this is negative two point one three plus 17 point one seven or we could have 17 point one seven minus two point one three which gives us fifteen point zero 4 15 point oh four 15 0.04 and remember the whole thing all of this we started with there was a 95% chance that a random t-statistic will fall in this interval will fall in this interval we had a random T statistic and all we did is a bunch of math so the 90 there's a 95% chance that any of these steps are true and so there's a 95% chance that this is true there's a 95% chance that the true population mean which is the same thing as the mean of the sampling distribution of the sample mean there's a 95% chance or we're confident that there's a 95% chance that it will fall in this interval and we're done