 Hey there, instead of using z-scores to find the area under a curve We're now going to be given the area under the curve and we have to find the z-score that goes with that corresponding area So we're going to draw a bell shape curve and identify the region under the curve that Corresponds to the given probability once again We'll really focus on drawing a picture for every single question and then using the cumulative area to the left So using the area to the left of that unknown data value. We're trying to find We'll use our table to identify the data value that gives that probability the best so We'll find the closest probability in the body of the normal probability table, and then I'll show you how to use technology So the probability table is just for you to understand how things used to be done and how some people still do stuff Using the same bone density test as our previous video find the bone density score that separates the bottom 2.5% and find the score that separates the top 2.5 percent in my picture my mean is zero because I'm dealing with the standard normal distribution and I'm looking for the data value That separates the bottom 2.5 percent So that means I have a region in the lower tail of the graph on the left side of the graph with the area of point 0.025 and then I have another data value another z score that separates an upper region of point 025 So those regions represent the probabilities of the data values That are the top 2.5 percent and the bottom 2.5 percent All right, so the key ingredients that we need here I use subscript z sub 1 z sub 2 just because I'm looking for two z scores So let's talk about the first z score to find the first z score to find that first data value I need to find its area to the left So the area to the left of my first z score z sub 1 is point 025 So you look at the body of your table and Look for point 025. So let's look for point 025 in the body of our table. I am looking for 0.025 right here at the very bottom. I have point 025 And you see you got negative one point nine the second decimal place is six negative one point nine six So my first data value is negative one point nine six What about the second data value You have to know the area to the left to find any data value along the bottom along the x-axis You must know the area to the left of the unknown data value. You're trying to find well. We have a problem I only know the area to the right What is the area to the left? Well, if you know the entire area under the curve is one the area to the left will be one minus Point 025 one minus the area to the right So point nine seven five So you look for that in your table now using common sense here my mean of my standard normal distribution is zero and my Z sub two is to the right of zero So it should be a positive number All right, so we're gonna look at our table of positive values positive z scores, and I'm looking for point nine seven five So look in the body of the table till you find point nine seven five Found it, and that's one point nine six That's one point nine six So you found your two data values So we can use area under a curve to find data values It's important. You know that you must know the area to the left of the data value you are finding So using these tables is a lot of work Assume the thermometer readings were normally distributed with a mean of zero degrees Celsius and a standard deviation of one degree Celsius find a temperature reading corresponding to piece of 85 that means the 85th percentile in case you forgot that notation It's the temperature which separates the bottom eighty-five percent from the top fifteen percent So think about your picture. I have zero Right in the middle, that's my mean and I'm looking for the data value that separates the bottom eighty-five percent from the top fifteen percent So I'm looking for the z score or the data value that has an area to the left of point eighty five and an area to the right of point fifteen Remember to find a data value. You always have to know area to the left So I'm going to introduce you to Google Sheets here because using the tables tiring in Google Sheets your mu will be equal to zero Sigma will be equal to one when you're trying to find data values from probabilities or from areas The only other thing you have to type into your sheet is the area to the left The area to the left of the data value I'm trying to find is point eighty five use your picture to your advantage So let's show you How to use Google Sheets now? So in Google Sheets will go to our compute tab and we're focused on the normal region So mu is zero and sigma is one You don't care about anything else except the left tail area when you're trying to find data values You use left tail area. Well, it's point eighty five So the corresponding data value will do two decimal places for data values is one point zero four One point zero four after you round. That's one point zero four So my data value is one point zero four How about that so the area to the left is key to the left to the left we use the area to the left So using Google Sheets is so much fun Let's do some more practice So assume that thermometer readings are normally distributed with the mean of zero and a standard deviation of one degrees Celsius Assume two point eight percent of thermometers are rejected because how they have readings that are too high And there's another two point eight percent of the thermometers rejected because their readings are too low So these are thermometers that just are terrible at reading the temperature. We need to throw them out quality control stepping in here Find the cutoff values which separate the rejected thermometers from the others So standard normal means my mean in the middle is zero So I'm finding the data value that cuts off a bottom region of Point oh two eight. That's two point eight percent And I'm trying to find the other data value that cuts off the upper region with the area of point oh two eight Alright, so I'm looking for two values. Well, let's label them Z1 and Z2. Let's use Google Sheets So Google Sheets Mu is zero Sigma is one we're trying to find Z1 first that smaller cutoff value and My area to the left as shown in my curve remember always draw that picture This point oh two eight. This is going to help us find. What is Z1? Now what about Z2 well, we know that mu is equal to zero still This doesn't change when you're dealing with standard normal data Sigma is one in the area to the left. We don't know We have an issue Or remember how to find area to the left. It's one minus the area to the right one minus point oh two eight Subtract that right hand area from one So one minus point oh two eight will actually give you in this case Point nine seven two And that's going to tell me what Z2 is that upper cutoff value All right, so what's going to happen here is let's type all this information into Google Sheets We have two separate things we'll be typing in So Google Sheets to the rescue mu and Sigma are already zero and one so my left tail area for my Lower cutoff value is going to be All right, so left tail area is point oh two eight you get negative one point nine one and Then the left tail area for my larger cutoff value will be point nine seven two the area to the left is And seven two what it's positive one point nine one. How could this be? It's the same number just a negative version in the positive version Why is that? Well, first let's write down our answer then we can talk about that, right? So Z sub one is negative one point nine one That's your lower cutoff value and Z sub two is positive one point nine one That's your upper cutoff value And the reason why They are the same number just one's positive one's negative is because Our two tails on either side of the curve Both had area of point oh two eight and Then our mean of a standard normal distribution is zero so because of that symmetry That's why you have negative one point nine one positive one point nine one zero in the middle Same area in both tails the values will be opposite of each other All right, so that is how to find data values using area under a curve. Thanks for watching