in this video we're gonna go over the standard normal distribution as well as some equations that you need to be familiar with after that we're gonna work on some problems so you could see how to put these formulas to good use so the normal distribution has the shape of a bell curve it looks something like that now the notation for it perhaps you've seen this in your book is the random variable X so for a normal distribution you have two important parameters that you need to know that is the mean and the standard deviation represented by the symbol Sigma that kind of looks like Fayette Emma there it is now the mean is right in the middle of the bell curve and here this would be one standard deviation from the mean and over here this will be about one standard deviation away from the mean on the left side the z-score that corresponds to one standard deviation is simply one when X is less than the mean the z-scores are negative so two standard deviations away from the mean z is equal to two on the left side Z is going to be negative two and let's say over here this is three Stevi a shion's from the mean so Z is going to be three and the same is true for the other side now the formula that you need in order to calculate the z-score is this Z that's a terrible looking Z let's do that again Z is equal to X minus the mean divided by the standard deviation sometimes you may need to calculate X X is equal to the mean plus Z times the standard deviation so these are some formulas that you want to make sure you write down for the problems that I'm going to give you later the probability density function for a normal distribution is this function here f of X is equal to 1 divided by the standard deviation times the square root of 2 pi times e raised to the negative 1/2 times X minus the mean divided by the standard deviation squared now keep in mind e is a number it's approximately 2.718 something this continues now in the regular statistics class you won't need this formula you just need to know that it is the PDF or the probability density function for the normal distribution but you really don't need to use it in order to calculate the answer unless you're using it with calculus or something it does involve integral calculus to calculate the probabilities with that formula I've actually did that in another video on YouTube you could find it it was probably posted about a year ago or something if you want to know how to use calculus to get the answer but for this particular video we're not going to go into that much detail but just in case you see a question on a test at least you know that this formula corresponds to the normal distribution now there's something called the empirical rule that you need to be familiar with so the empirical rule tells us that 68% of the x values lie within one standard deviation of the mean ninety-five percent of the x values lie within two standard deviations of the mean and ninety-nine point seven percent of the x values lie within three standard deviations of the mean so knowing that how would you calculate the area under the curve expressed as a percentage in each of these sections well if this area here is 68% if we divide that by two that means this region must be thirty-four percent as well as the left side as well now what is the area in terms of excuse me in terms of a percent for those two regions to get that answer you need to subtract these two numbers 95 minus 68 that's gonna give us 27 and then if you divide that by 2 you're gonna get 13.5 now it's important to understand that the area of a continuous probability distribution function is 1 so the area of this region is gonna be a decimal value it's 0.135 the same is true for this region here so keep that in mind a total area under the curve is equal to one now what is the probability of finding an x value in those two regions so what we need to do is subtract those two numbers and ninety-nine point seven minus 95 that's going to give us four point seven and then divide that by two so you should get two point three five percent that's the probability of finding or get an x value in that region now the remaining portions by the way this should be like very close to the x-axis my drawing wasn't perfect but to find the remaining part it's going to be a hundred minus nine nine point seven which is 0.3 divide that by two and so there's a point fifteen percent chance of finding and or of getting an x value beyond three standard deviations so those are the values when using the empirical rule to solve probability questions relating to the normal distribution we're going to talk about how to use this chart later in this video but feel free to write down this information because it's going to be useful shortly so here's the first example problem that we're going to work on in this video now for each of these problems feel free to pause the video if you want to try it yourself before seeing the answer and when you want to check your answer just play the video to see if you got it so let's begin so given this information here what are the values of the mean and standard deviation so what you need to know is that the first number corresponds to the mean the second number corresponds to the standard deviation so the mean is 50 in this example and a standard deviation is 10 now Part B and what value of x has a z-score of 1.4 so if Z is equal to 1.4 what is the value of X what do you think we need to do what formulas should we use so perhaps you wrote down this formula X is equal to the mean plus Z times the standard deviation so the mean is 50 the z-score is 1.4 and the standard deviation is 10 now 1.4 times 10 is 14 and 50 plus 14 and that's going to give us 64 so this is the answer for Part B so that's how you can calculate x given the value of Z now what about Part C what is the z-score that corresponds to a value I mean that corresponds to x equals 30 so if X is 30 what is Z so now we're going to use the rearranged formed let me excuse me the rearranged form of that formula so here it is the z-score is X minus the mean divided by the standard deviation so X is 30 the mean is 50 and the standard deviation is 10 so 30 minus 50 is negative 20 20 divided by 10 is 2 so negative 20 divided by 10 is negative 2 so that's the answer for Part C now let's move on to Part D what is the difference between positive and negative Z values let's create a number line and let's put the mean in the middle now when X was 64 this correspond to a z value of positive 1.4 and when X was 30 the Z value that it corresponded to was negative 2 so as you can see negative Z values are below the mean they correspond to X values that are below the mean and positive z values correspond to X values that are above the mean and so that's really the difference between positive and negative Z values the negative Z values are going to be to the left of the mean and the positive z values will be towards the right of me now let's move on to the next problem the average test score in a certain statistics class was 74 with a standard deviation of 8 there are 2,000 students in this class use the empirical rule to answer the following questions so the first thing we're gonna do is draw a picture before we begin because this picture will be very helpful okay maybe I need to draw a better picture let's say that's our bell curve and in the middle somewhere is the mean the mean is 74 now this is going to be one standard deviation away from the mean which is 8 so if we add 74 and 8 this will give us 82 two standard deviations away from the mean will be 90 and three standard deviations will be 98 now we're going to do the same for the other side 74 minus 8 will give us 66 66 minus 8 that's gonna give us 58 and then subtract up by eight again you should get fifty so now I'm going to fill in the values that we had before so we know that this is 34% so the area under the curve for this region is 0.34 so keep that in mind next we said this was thirteen point five and then this area here that's two point three five now the last part is going to be point 15% so now we can answer the questions let's start with the first one what percentage of students scored less than 58 so the probability of selecting a student who scored less than 58 is going to be the sum of these two values because they are to the left of 58 so all we need to do is just add point 15% plus two point three five percent and that's going to give us two point five percent so this means that two point five percent of all of the students scored less than 58 so the area under this curve here is point zero two five as a decimal now what about Part B what is the probability that a student scored between 66 and 82 on exam perhaps that question should then phrase better let's say if you randomly selected a student in this class what is the probability that the student selected would have scored between 66 and 82 in exam what would you say so using a chart we could find the answer so all we need to do is add up the percentages between these two numbers and so it's 34 plus 34 34 plus 34 that's gonna give us 68 percent so that's the answer for Part B now let's move on to Part C how many students scored at most 90 so what's the probability that X is less than or equal to 90 so here's 90 we need everything up to 90 so to get the percentage we need to add everything in this region everything below 90 so that's going to be point 15 plus 2 point 3 5 plus 13.5 plus 34 plus 34 and then Plus this one as well 13.5 so the percentage that you should get is ninety seven point five percent so this means that ninety seven point five percent of students scored at most ninety now we're not done yet but there's another way in which you can get this answer as well you could say the probability that X is less than or equal to 90 is equal to one minus the probability that X is greater than ninety and the calculation for this will be easier now one as a percentage is a hundred percent and probability that X is greater than ninety it's going to be the sum of those values as a percentage so it's two point three five plus point fifteen percent so if you type that in you should get the same answer which is ninety seven point five percent now we're not looking for a percentage for our answer because Part C it asks us how many students scored at most ninety so we're looking for the number of students and there are two thousand students in this class so we need to determine what is ninety seven point five percent of two thousand ninety seven point five percent as a decimal is 0.975 to get that number just take the percentage divided by one hundred two thousand times 0.975 that's 1950 so this is the answer for Part C so 1950 students scored at most ninety now let's move on to Part D what percentage of students scored at least sixty six go ahead and try that so let's determine the probability that X is at least 66 which means it can be 66 or more so it's greater than or equal to 66 so therefore we need everything to the right of 66 so we're gonna add 34 percent plus another 34 percent plus 13.5 percent and then two point three five percent plus point fifteen percent so the answer is going to be 84 percent that's it for Part II I mean not me Part D Part II how many students scored more than 98 on the tests so what is the probability that X is greater than 98 so the only thing above 98 is point 15% now once again we're looking for the number of students so we got to find out what point 15% of 2000s so first let's convert this into a decimal as you said before to change a percentage into a decimal divided by a hundred so point 15 divided by 100 is 0.0 0.5 so what we need to do is multiply 2000 by that number 2000 times point zero zero one five that's going to give us three so about three students scored more than 98 on the exam and that's it for number two so that's how you could use the empirical rule to answer probability problems related to a normal distribution situation now sometimes you have z-scores that are not whole numbers of the stand deviation and you can't use the empirical rule whenever you have z-scores like one two three or negative two negative one you could use the empirical rule to calculate the area under the curve does calculating the probability in question but let's say if your z-score is one point five six you can't use the empirical rule to calculate the area under the curve and if you don't want to use integral calculus you need to use the standard z tables sometimes referred to as the normal distribution tables now I'm going to show you how to use it real quick but let me talk about the graph so here is our standard normal distribution let's say this is the mean and let's say this is the z-score of one point five six so when you get the area under the curve from the Z table which for this problem it's going to be point nine four zero six two you need to understand that this gives you the area to the left of that line let me put this line in red so it gives you the left side the area of the shaded region in blue let's say if your z-score is negative let's say the Z is negative point four three using a table you'll find that the area to the left is point three three three six zero and so the graph if you wish to shade the region will look something like this here's the mean a Z is negative so it's gonna be to the left of the mean so it's gonna be to the left of the dotted line so that shaded region corresponds to an area of 0.333 six which means if you want to convert that to a percentage the probability that X lives in this region somewhere will be thirty three point three six percent you just need to multiply that by a hundred so here we have a positive z-score table so the first thing we're gonna do is look at the first column and find the value that corresponds to one point five now looking at the first row we need to find a value that corresponds to point zero six because one point five plus point zero six gives us the z-score that we want one point five six now we need to find the number that corresponds to the row one point five and the column point zero six and as you can see that number is 0.94 zero six two so that represents the area under the curve given a Z value of one point five six so the area of the blue region shaded in left I mean to the left the area under the curve that's gonna be nine point nine four zero six two so that's how you could find it using the positive z-score table now the next z-score value that we're looking for is negative point four three so in the first column we see the value negative point four and in the first row we need to look for the column that says point zero three so the value that corresponds to a Z value of negative two point four three is point three three three six zero that is going to be the area under the curve given a Z value of negative point four three so that's how you could find the area under the curve if you know the z-score you just got to look it up in the table and you'll get the answer number three normally distributed IQ scores have a mean of 100 and a standard deviation of 15 use the standard z table to answer the following questions what is the probability of randomly selecting someone with an IQ score that is less than 80 so let's fight them what we know the mean is 100 and the standard deviation is 15 now let's draw a picture so here is a rough sketch of our bell curve the mean is 100 and we want to find the probability of selecting someone with a score of less than 80 so what we need is the area to the left of 80 so the first thing we need to do is calculate the z-score and then we could use the Z table to get the area under the curve so we know that Z is going to be X minus the mean divided by the standard deviation so X is 80 based on this problem we're trying to find a probability that X is less than 80 so we're gonna say X is 80 the mean is 100 and the standard deviation is 15 so 80 minus 100 is 20 I mean negative 20 negative 20 divided by 15 is negative one point three three now it's negative one point three repeating but we're gonna stop it at negative one point 33 so what you need to do at this point is use the negative Z table and find the area the value of the area that corresponds to the Z value so using that table you should get point zero nine 176 now that's the answer as a decimal if we multiply this by a hundred we'll get the answer as a percentage which is nine point one seven six percent so that is the probability of selecting someone with an IQ score less than 80 now let's move on to Part B what is the probability of randomly selecting someone with an IQ score that is greater than 136 go ahead and try that so first let's begin by calculating the z-score it's going to be X minus the mean over the standard deviation so that's 136 minus a hundred divided by 15 so 136 minus 100 is 36 and 36 divided by 15 is 224 so using the positive z-score table convert the Z value of 2.4 to the area value so find the area under the curve that corresponds to a z-score of 2.4 go ahead and take a minute to do that so the value that you should get is 0.99 1/8 but that's not our answer yeah one of I track but there's a little more work that we need to do so let's draw a picture so here is the mean of 100 and here is our x value of 136 which corresponds to a z-score of 2.4 now we want to find the probability that X is greater than 136 which is this region the z-score table the positive z-score table it gives us the area to the left so that is the area highlighted in red so that area is 0.99 1/8 we want to find the area highlighted in blue keep in mind the total area is 1 so the probability that X is greater than 136 is going to be 1 minus the probability X is less than 136 or you can say less than or equal to so this answer that we have here and that corresponds to the probability that X is either less than 136 or less than or equal to because it's going to be the same so the answer is 1 minus 0.99 1/8 and that's going to be point zero zero eight two so there is an eight point two or rather there is a point a two percent chance that a person selected at random will have an IQ score that's more than 136 so this is the answer to Part B Part C what is the probability of randomly selecting someone with an IQ score that is between 95 and 110 so what do you think we need to do for that problem well let's begin by drawing a picture just to get a good visual of what we need to do so we need to calculate the area of this region highlighted in blue between 95 and 110 in order to do that in order to calculate the probability that X is between 95 and 110 we need to take the difference of the probability that X is less than 110 and the probability that X is less than 95 the probability that X is less than 110 looks like this here's the shaded region that corresponds to it and we needs to subtract that region by what we have here so if we take the region shade in red subtracted by this region it will give us the area under the curve between 95 and 110 so that's what we need to do in this problem but we need to calculate the z-scores that correspond to those two numbers first so Z is X minus the mean over the standard deviation so let's start with an x value of 110 and so this is going to be 110 minus 100 which is 10 and 10 divided by 15 this is point six repeating but we're going to round that to 0.67 now when X is 95 this is going to be 95 over or minus 100 which is negative 5 and negative 5 divided by 15 that's negative point three repeating but we're going to use negative 0.33 so the area that corresponds to a z-score of positive point six seven using the positive z-score table that is 0.74 eight five seven and the area that corresponds to a z-score of negative point 33 using the negative z-score table that's positive point three seven zero seven zero so this value corresponds to the probability that X is less than 110 and this value corresponds to the probability that X is less than 95 so let's go ahead and subtract these two values 27 for 857 minus 0.3 707 that's going to be 0.37 787 so that is the area under the curve between the x-values 95 and 110 so we could say there is a thirty seven point seven eight seven percent chance that a randomly selected person will have an IQ between 95 and 110 now what about Part D what IQ score corresponds to the 90th percentile go ahead and try that problem let's begin by drawing a picture we have drawn many pictures in this video so far so here is the mean the mean is at the 50th percentile this is zero this is a hundred so the 90th percentile would be somewhere in that area approximately and there is an x value that corresponds to the 90th percentile we need to determine what that x value is now what we know is that the area to the left of that x value is based on the percentile if the percentile is 90 the area to the left has to correspond to 0.9 zero and so using a z table we can kind of work backwards in Reverse here we can take that area and find a z-score that corresponds to it so find the z-score that corresponds to an area of 0.9 zero we need to use the positive z table because the area is greater than 0.5 now there's two values of interests a z-score of positive 1.28 has an area of 0.89 973 and a z-score of 1.29 has an area to the left of 0.9 0 1 4 7 now this value is a lot closer to 0.9 and then is this value it's about five times as close if you take the difference between this number and point nine you're going to see that it's a lot smaller then the difference between this number and point nine so I'm going to choose a z-score of one point two eight so now that I have the z-score that corresponds to the 90th percentile I can calculate the x value using this formula X is going to be the mean plus C times the standard deviation so the mean is 100 Z is 1 point 28 and the standard deviation is 15 so go ahead and type that in the answer that you should have is 119 point two so this is the IQ score that corresponds to the 90th percentile Part II the middle 30% of IQs fall between what two values so what do you think we need to do to get that answer Part II is very similar to Part D let me show you why so let's put the mean here so this is going to be a hundred and we're gonna call this value X 2 and this 1 X 1 those are going to be the two IQ values that we're looking for now granted is not drawn to scale so this is going to be the middle 30% which means that 15% will be on the right side and 15% will be on the left side so we need to correspond I mean wow we need to find the x values and that correspond to these percentages now understand that this is the 50th percentile so 15 percent to the right of that will bring us to the 65th percentile and 50 minus 15 on the left will give us the 35th percentile so we need to find the x value that corresponds to the 35th percentile and the x value that corresponds to the 65th percentile so that's how it's similar to Part D so let's start with the 65th percentile that means the area to the left that is everything from here if you shade the we find the area under the curve all of this will have an area of 0.65 so now using the positive Z table what z-score corresponds to an area of 0.65 what would you say now there's two values of interests when Z is 0.38 the area is 0.64 8:03 it's pretty close to 0.65 and when Z is 0.39 the area is going to be 0.65 173 so these two values are almost equidistant from 0.65 therefore if we average them we'll get a number close to 0.65 so I'm going to take the average of those z-scores so I'm going to choose a z-score value of 0.38 5 because point 65 is in the middle of these two numbers so now let's get rid of this so the z-score that corresponds to X 2 is 0.38 5 now due to the symmetry of these percentiles the z-score that corresponds to x1 is negative 0.385 and you can check that out if you look up an area value of 0.35 which corresponds to the 35th percentile you'll get a z-score of negative point 385 using the negative z-score table so now that we have the z-scores what we need to do is calculate the X values so let's start with x2 this is going to be z2 and this is z1 so x2 is going to be the mean times I mean plus z2 times the standard deviation so that's a hundred plus positive point three eight five times 15 and so x2 is 105 point 8 the exact answer that I got in my calculator is 105.7 75 now for x1 we're going to use the same formula but using Z 1 instead of Z 2 so it's a hundred plus negative point three eight five time is 15 so x1 is going to be ninety four point two to five but I'm going to round out to ninety four point two so these are the two x values that correspond to the middle of 30% of all the IQs based on the standard normal distribution so that's it for number three now let's move on to number four company XYZ manufactures an average of four hundred thousand tires annually on average two percent of the tires were manufactured with a defect a random sample of 500 tires were selected for quality control Part A calculate the mean and standard deviation of the defective tires in a sample so take a minute to work on this problem the first thing that we want to take into account is that two percent of the tires have a defect so the probability of getting a tire with a defect is point zero two the probability of not getting the tire with a defect that's gonna be Q that's one minus point zero 2 which is 0.98 now we're choosing a sample size of 500 tires so n is 500 so we're choosing a small sample out of the 400,000 tires that are manufactured annually now to calculate the mean in this situation we could use this formula it's going to be n times P so it's 500 times point zero two so basically we're looking forward to percent of 500 point zero two times 500 is 10 so about 10 of the 500 tires on average will have a defect so that is the mean now let's calculate the standard deviation the formula that we're going to use is this one it's the square root of n times P times Q so n is 500 P is point zero two and then Q is 0.98 so this is going to be three point one three so that is the standard deviation in this problem so that's it for Part A now Part B what is the probability that less than eight tires will be defective in the sample first let's rewrite the mean and a standard deviation so I can create more available space so we're looking for the probability that X is less than 8 so we need to use the Z table but first we need to calculate the z-score so Z is going to be X minus the mean divided by the standard deviation X is 8 the mean is 10 and a standard deviation is 3 point 1 3 so 8 minus 10 is negative 2 divide that by 3 point 13 this will give us a z-score of approximately if we round it it's a negative point 6 4 so now using the negative Z table we need to get the area under the curve to the left so negative 0.6 4 corresponds to an area value of 0.26 1 is 0 9 so we could say that the probability that less than 8 tires will be defective in the sample is 26.1% approximately that's a rounded answer so that's it for part 8 I mean that part a about Part B now let's move on to Part C what is the probability that more than 15 tires will have some sort of defect so let's calculate the probability that X will be greater than 15 so once again let's calculate the z-score first so this is going to be 15 minus 10 divided by 3 point 1 3 so that's 5 divided by 3 point 1 3 and so that's 1 point 5 9 7 but we're going to round it to 1 point 6 so that's the z-score now using the positive Z table 1 point 6 corresponds to an area of 0.9 for 5 to 0 from the left now if we draw the picture we want the area on the right so here's the mean let's say this is 15 the area shaded in blue corresponds to this value but we want the area shaded in red so it's going to be 1 minus this answer so it's 1 minus 0.9 4 5 2 and so that's point zero 5 4 8 so that corresponds to five point four eight as percentage so there's a five point four eight percent chance that more than fifteen tires will have some sort of a defect now let's move on to Part D what is the probability that the number of defective tires will be between seven and fourteen so what is the probability that X is between seven and fourteen so this is going to be the probability that X is less than 14 minus the probability that X is less than seven so let's calculate the Z scores for both numbers let's start with 14 so it's 14 minus the mean of 10 divided by three point one three so that's four divided by three point one three and that's going to give us a z-score if we round it to one point two eight now using the positive Z table one point two eight corresponds to a value of 0.8 nine nine seven three now let's calculate the z-score when X is 7 so 7 minus 10 is negative three negative three divided by three point 13 that's gonna be we gotta round this to a negative point nine six using the negative Z table negative 0.96 is corresponds to a value of 0.1 6 8 5 3 so the area under the curve from the left to x equals 14 is 0.8 9 9 7 3 and the probability that X is less than 7 is going to be 0.16 8 5 3 so we got to take the difference of those two values and that's going to give us this answer point seven three one two so the probability that the number of defective tires will be between seven and fourteen is 73 point 12 percent and that is it so that's it for this video if you find it to be helpful don't forget to subscribe to this channel if you want more videos like this thanks again for watching