in this video we're gonna talk about the normal distribution or the bell-shaped curve so let's begin by drawing a picture of that curve and it's gonna look something like that now right in the middle we have the population mean represented by the Greek letter mu and to the right of that one stand deviation away we have the Greek letter Sigma and this is one standard deviation to the left of the mean and then this is two standard deviations away and then three on both sides all the way to the right we have infinity and all the way to the left negative infinity now the area under the curve represents the probability of an event happening between two points so the probability of an event happening between the mean and the first and deviation you can calculate it if you know the 6895 99.7 rule so within the first and deviation of either side from the mean there's a sixty-eight point two six eight percent chance that an event will happen in this region and then within two standard deviations the percentage is ninety five point four five percent and then within three standard deviations of the mean the percentage is going to be 99.73% now I hope that you're taking notes because you're gonna need to know these numbers it's gonna help you solve some problems that are coming up soon now if we take this number sixty eight point two six eight and divided by two that's going to give us thirty-four point one three four percent so that's the percentage of finding an event between the mean and the first standard deviation now the left side is going to be the same it's gonna be thirty four point one three four percent when you didn't live a normal distribution curve or a bell-shaped curve you can have symmetry in this problem and keep in mind the area of this region is point three four one three four it's the percentage as a decimal so the area is going to be the same as the probability of an event happening within a certain region now what percentage goes in this region if you take the difference between these two numbers so if you subtract a ninety-five point four five by sixty eight point two six eight that will give you twenty seven point one eighty to take that number divided by two and you should get a thirteen point five nine one and this number is going to go on both sides so that's the percentage of an event happening between the first and a second standard deviations of either side now to find the next number follow the same process so we're going to subtract these two numbers ninety-nine point seven three minus ninety five point four five that's four point Julie then divided by two so the percentage in this region is going to be two point one four percent and the same is true in that region as well now to find the percentage that goes in those two regions keep in mind a total percentage from negative infinity to infinity is a hundred percent so take a hundred and subtracted by ninety nine point seven three you should get a point two seven and when you divide that by two the answer in this region is gonna be point one three five percent and the same is true for this area as well so you may want to take a picture or at least write this down because you can use this to solve problems associated with a normal distribution curve now it's important to keep in mind that mu represents the population mean and Sigma is the standard deviation now as we said before the probability of an event happening from negative infinity to infinity is a hundred percent that's the total probability which means that the total area of this curve is one so if you add up all of these numbers you should get a hundred percent and the left side everything up to here has to be fifty percent which is point five as a decimal so now let's work on some practice problems number one the test scores of a physics class with eight hundred students are distributed normally with a mean of 75 and a standard deviation of seven part a what percentage of the class has a test score between 68 and 82 so feel free to pause the video and work on this problem now what I'm gonna do first is I'm gonna draw a picture that's similar to the picture that I had before but instead of using symbols such as mu and Sigma I'm going to use numbers now we said that the mean is 75 and the standard deviation is 7 so the first thing I would recommend doing is put in the mean right in the middle then 1 standard deviation away from the mean we're gonna put the numbers as well so we're gonna add 7 to 75 75 plus 7 is 82 and 75 minus 7 is 68 now let's move another standard deviation away from the next number from 82 and 68 so 82 plus 7 is 89 68 minus 7 is 61 now let's do it one more time 89 plus 7 is 96 61 minus 7 is 54 and we could stop here now let's put the appropriate percentages in each region so we know that within the first and deviation is going to be 34 point one three four percent and then between the first and the second stand deviation it's going to be thirteen point five nine one percent between the second and a third it's going to be two point one four percent and then for this region point one three five percent so now we have everything that we need in order to get the answer so let's calculate the probability or the percentage of the class that has a test score between 68 and 82 so X is going to be a random continuous variable that can vary beyond 75 or less than 75 now all we need to do to get the answer is basically find the area under the curve between these two numbers so we got to find the area of the shaded region so all we need to do is simply add up the percentages between 68 and 82 so it's gonna be thirty four point one three four plus another thirty four point one three four so that's gonna be sixty eight point two six eight percent so that's the percentage of the class that has a test score between 68 and 82 now there's a formula that you could use to get the same answer in fact a more accurate answer using calculus now if you're watching this video and if you haven't taken a calculus class don't worry about it because the formula that I'm going to show you you can easily use it with the appropriate calculator and if you don't have a calculator I can show you how to use it with an online calculator at Wolfram Alpha so here's the form that you need the probability a fun in an event between a and B is going to be the definite integral from A to B e e is the inverse of the natural log function of n raised to the negative X minus the population mean squared divided by 2 times the square of Sigma or the stand of EI ssin and this is all divided by Sigma times the square root of 2 pi and then we have a DX symbol here now all you need to do plug in the population mean which you have that 75 you also need to plug in the standard deviation which is 7 and then plug in a and B in this case a is 68 B is 82 and then just type that into the calculator and that should give you the answer so let's go ahead and do that so the percentage of students with a test score between 68 and 82 is going to be the definite integral from 68 to 82 and then it's gonna be e raised to the negative X minus 75 squared divided by two times seven squared divided by seven times the square root of two pi D X so make sure to use these parentheses that I'm showing you when you're plugging into your graphing calculator I'm going to show you how to use Wolfram Alpha towards the end of this video so you can look for it then so let's go ahead and plug this in it's gonna take me a while to type this in so bear with me for one moment and it's gonna take a while for the calculator to get the answer to so what I got is point six eight two six eight nine four nine so if you multiply that by a hundred that's gonna be 68 from two six eight nine four nine percent so as you can see these two answers are not exactly the same but they're close enough so this is an approximation and this answer is more accurate to the exact answer so technically you should be like sixty eight point two 69 instead of 216 but if you rounded sixty eight point three then you should be fine now let's move on to our next problem or the next part of this problem that is Part B approximately how many students have a test score between 61 and 89 so we're not looking for a percentage rather than number of students and keep in mind we have a total of 800 students in this class so first let's find the percentage as we did in Part A so the percentage of students with the test score between 61 and 89 is going to be so we start in that 61 and we wish to end at 89 so we need to find the area under the curve for that entire region so basically we need to add those four numbers so we're gonna add thirteen point five nine one plus thirty four point one three four plus thirty four point one three four again and then another 13 point five nine one and you should get approximately ninety five point four five percent because these two numbers they're within two standard deviations of the mean so that's always gonna be about ninety five point four or five percent whenever you have that situation now let's confirm this answer with a calculator so we're gonna evaluate the definite integral from 61 to 89 and then it's gonna be e raised to the negative x minus 75 squared divided by 2 times 7 squared divided by 7 times the square root of 2 pi so notice that this portion doesn't change because the mean and the standard deviation is the same for Part A and B the only thing that's changing is the limits of integration a and B that's all I need to change in this problem so go ahead and plug that in if you don't know how I'll show you later at the end of this video this calculator takes about five seconds to get the answer so I got point nine five four four nine nine seven three six so that's approximately ninety five point four four nine nine seven percent if you multiply this by 100 so this answer is more accurate but nevertheless this is a good estimate of this answer it rounds pretty well now this is not the final answer for Part B so what this means is that ninety five point four five percent of the class has a test score between 61 and 89 now there's eight hundred students so what's the ninety-five point four or five percent of eight hundred so what I'm going to do is take this number because it's a more accurate value and multiply that by eight hundred my calculator also has one one here so if you take that number and multiply by eight hundred it's gonna give you seven sixty three point six approximately now because we didn't with people we want to round to the nearest whole number so we're gonna say approximately seven hundred sixty four students have a test score between 61 and eighty-nine so this is the answer to Part B now let's move on to Part C what is the probability that a student chosen at random has a test score between 54 and 75 so go ahead and work on that problem so here's 54 and we're gonna stop at 75 so basically we need to add these three numbers so the probability of selecting a student that has a test score between 54 and 75 is going to be so we're gonna add two point one four plus thirteen point five nine one plus 34 point one three four and so you should get 49 point eight six five percent so that's enough rock Meishan pretty much a good approximation now let's get the exact answer or a more accurate answer using calculus so this is gonna be the definite integral from 54 to 75 e negative X minus 75 there's not to forget to square that divided by 2 times 7 squared that's terrible-looking - I gotta fix that divided by 7 times the square root of 2pi DX and so I got point four nine eight six five zero one zero two which is about forty nine point eight six five percent if you take this number and multiply by 100 so this is a very good approximation and so that's the answer for Part C Part D the last part of the problem approximately how many students have a test score and greater than or equal to 96 so using a graph what's the answer so all I need is this region which we can clearly see that it's point one three five percent now we're not looking for the percentage but the number of students which we'll get to later so how can we get the same answer using calculus so we know we need to calculate the definite integral from A to B now we could see that a is 96 but we only have one number here so what's B what should we put for B now if you're thinking of a hundred you shouldn't use that because granted most head scores are graded based on 100% like the highest value is 100 but sometimes teachers may have extra credit so some students could have a a score of 105 or 110 if you put a hundred here you won't get 0.135 percent let's call the function f of X instead of writing the whole thing I'm gonna plug it in so give me one minute to do that and so if you integrate it from 96 to 100 you're going to get point zero zero 1 1 7 2 3 8 so if we multiply that by 100 that correlates to point 1 1 7 percent approximately which is not the same as this answer so what you need to keep in mind is that this goes to infinity and the left side goes to negative infinity so you have to integrate it from 96 to infinity now this is going to be a problem because for the most part we can't plug in infinity into a calculator so we can approximate the answer by picking a number that's I need to get rid of this we need to pick a number that is very high that can represent affinity so a hundred wasn't good enough so let's pick a much bigger number like a thousand now the answer will be exact but it's going to be a very very good approximation so if you go ahead and plug this in you should get an answer that's a very very close to this answer so go ahead and type that into your calculator now my calculator gave me this answer point zero zero one three four nine eight nine zero three so multiply that by a hundred and that's about point one three four nine eight nine and I'll stop there percent so this rounds two point one three five percent so that's a good approximation now let's determine the number of students that have a test score that's equal to or greater than ninety six so we're going to take this number and multiply it by the number of students in class eight hundred so point zero zero one three four nine eight nine zero three times eight hundred you should get one point zero seven nine nine and so for this we're gonna round it to the nearest whole number so approximately one student scored a ninety six or more on his physics exam so that's how you can use the normal distribution curve to answer problems such as this one now go ahead and open up your internet browser I'm going to use the Google search engine and then when you're ready type in online calculator indefinite integral and then you can also type in Wolfram Alpha so I'm gonna go to this site Wolfram Alpha widgets definite integral calculator and you should see a page that looks like this so type it in exactly the way I'm going to show to you so type in e and then shift six to get the arrow negative parenthesis if you don't put the negative sign you will get it wrong X minus 75 squared at shift six and then we also need to open up another parenthesis in front of the negative sign so nice open up two parenthesis to keep that in mind and then divided by open a new friend disease too Schiff eight that's the multiplication symbol times seven squared close parentheses and close it again this entire thing has to be on the exponent of E that's why we have to open up an additional set of parenthesis and then divided by open a new set of parenthesis seven square root s QRT open parenthesis again 2 pi 2 P I and then close the parenthesis twice now we're going to integrate it from 68 to 82 and then click Submit so you can see the answer here point six eight two six eight nine so that's about sixty eight point two six and nine percent if you multiply it by a hundred now the good thing about this is you don't need to type in the integral expression because that part is not going to change the only thing he needs to change are the limits of integration so for Part B we need to find a probability from 61 to 89 so let's change these numbers and then just click Submit and so we have point nine five four five one ninety five point four five percent if you need more digits click the more digits link here and then you can get a more accurate answer which we had point nine five four four nine nine seven and if you want more numbers beyond that you can write to your heart's pleasure now let's move on to Part C so it was from 54 to 75 and so we did get this answer point four nine eight six five and see a more accurate answer point four nine eight six five zero one and we rounded to 0-2 at least my calculator did that but we can leave it as point four nine eight six five four forty nine point eight six five percent now for the last one now I mentioned not to integrate it from 96 to 100 because it won't give you point one three five percent you need to go to infinity so as you can see point zero zero one one seven so this is not an accurate answer but as you increase this value let's say to 105 it's going to approach a more accurate answer notice point zero zero one three four so it's getting close to point zero and one three five so let's try 110 now notice that it's converging it's getting closer to point zero zero one three five so we'll need to pick on the large number that represents infinity so we can try a thousand so we get point zero zero one three four nine nine and you can select more digits if you want now let's pick a number that's bigger than a thousand to see if it's going to stay at this value if it's going to converge to it if the value changes then we don't have the right answer so if we go to ten thousand it might take longer to calculate notice that this number didn't change I guess it takes a longer time to go up to 10,000 but you can still run it two point zero zero one three five so if we go to a hundred thousand it won't change that means your final answer is if you multiply this by 100 point one three five percent because 100,000 is a good approximation of infinity and so that's it for this video thanks for watching