in this video we are going to talk about approximating the binomial with the normal distribution so whenever we are working to approximate the binomial using our normal distribution we are going to use this equation and so this equation comes from the fact that mu is equal to NP and sigma is equal to the square Ro TK of npq so we've talked about this before so that just plugs into our Z formula so it's the same as this except we've substituted in our necessary values here or not values but um more applicable variables so when we are working with approximating the binomial with the normal we are taking a discrete function and trying to use a continuous function to understand it so we have something like this we have our continuous function the blue line now purple that I'm erasing because it was bad and we have our discret function so each of these bars represents our discret binomial so we want to using our continuous distribution be able to understand our discret function and so to do that if you see where this point is this point represents five right but with our discret function it's actually this whole bar here this whole section here so we need to do what's called a continuity correction of this 0. five adjustment and so we either add or subtract 0.5 in our continuity correction in order to get this whole bar going on we'll do some examples here in a second but just know that that's what a continuity correction does approximating the binomial with the normal is very accurate when we have n is large and P is not extremely close to zero or one and it's still a fairly decent approximation when n is small n p is reasonably close to 12 so now we're going to jump into a problem we have our mu equal to NP function here and sigma squ equal to npq and so this translates our our binomial distribution which we always use NP and Q into our normal distribution so once we know our mu and sigma squ we can draw out our normal distribution and so that's what this blue line is here and the binomial is our bars so blue line equals normal and bars equals binomial and our binomial is discreet so it's countable it's whole numbers all the time and our normal is continuous so it can take on any tiny little decimal point it wants to and so we want to use our normal distribution to approximate the binomial so let's say we have a problem of find the probability that X is equal to 4 for a binomial distribution with 15 trials and P equal to 0.4 so 15 trials means that we have n equal to 15 we have P given to us as 0.4 which means Q is equal to 0.6 because it's 1 minus p and we want to use the binomial so we're specifically looking at the bars and we want to know where X is equal to four so for us that's going to be this whole shaded region here we want to know what this bar is what the probability of Landing in this section is so to find that we're going to go to our binomial distribution table and we want to know the probability of xal 4 and so this is going to be the probability of X less than or equal to 4us the probability of X less than or equal to 3 we have to do the subtraction because our table is a cumulative table and so that is going to be for p equal to 0.4 and N equal to 15 we want to take 4 minus 3 so 2173 - 095 and that gives us that our probability of xal to 4 is equal to 0.12 68 and so this is using the binomial this is what we've done a couple weeks ago um just straight up binomial calculation so now we want to use a normal approximation of our binomial to find that same value so the probability of x x = 4 so to do that we need to use on our curve we have to find this section and so that's going to be between 3.5 and 4.5 so that's adding in our continuity correction so now we're going to have an upper and lower zv value so we are going to have let's first write down our mu which is n * P which is given to us up on the previous section of the page um S6 right up here have six and then our Sigma squared npq is 3.6 so then we also have our Z function x minus mu over Sigma note that this is Sigma squ over here so we're going to have to remember to take the square root of that and so we want to know to find the probability of x equal to 4 we need to find the probability that X is between 3.5 and 4.5 so that's going to give us this whole bar section in our distribution so we need to convert these to Z values now so we're going to have Z1 is our lower bound this is going to be 3.5 - 6 / the < TK of 3.6 that gives us a Z1 value of NE 1.32 and then we're going to have our upper bound or Z2 equal to 4.5 - 6 / theare < TK of 3.6 and that gives us 0.79 so now I can find these values on my Z table I'm going to add them to my diagram here so 1.32 is going to be somewhere around here and 0.79 is going to be somewhere around here soga 1.32 0.79 and now we can see that this kind of looks like this how cool and so we want to know this shaded region so just like we did with our binomial distribution and like we've done with other normal um problems we need to find the probability of Z less than 0.79 so that finds from here over and then we need to actually subtract out this section so that then we have probability of Z less than 1.32 so then if I go to my table I get Negative 0.79 07 9 is this 2148 value and then negative 1.32 13012 is 934 so this is equal to 0.248 minus 0.934 is equal to 01214 so the probability of x equal to 4 using a normal approximation is 12 .1% and up here we have 12.7% so pretty similar um and so that's how we do our normal approximation of the binomial distribution so you're probably wondering why would we do this um the main purpose here is that the normal distribution is just so well known that being able to apply it to a discrete situation when we have a binomial problem um just sometimes makes the calculations easier makes it a little bit easier to understand um just because the normal is so so welln so that's why we want to know how to use a normal approximation for our binomial distribution and binomial problems