 Well hello there, we are now going to discuss applications of normal distributions. So what's the difference between a standard normal distribution and a normal distribution? Well, if you take a non-standard normal distribution with mean u, standard deviation sigma, and when you apply the z-score formula to every single data value, the z-score formula is x minus mu over sigma, then everything is converted to z-scores and that's where standard normal distribution comes into play. And a standard normal distribution has always has a mean of zero and a standard deviation of one. So in other words, the standard normal distribution comes from applying the z-score formulas to data values that are normally distributed. So standard normal means mean of zero, standard deviation of one. So when finding areas with a non-standard normal distribution, you have a couple of options. First, you convert the value of interest, the data value of interest to a z-score and use standard normal probability tables to find that specific probability associated with that z-score or data value. That's a lot of work kind of. Method two, do not convert the value to a z-score. Instead, use technology, in this case Google Sheets, with the given mean and standard deviation of the normal distribution. So instead of a mean of zero and a standard deviation of one, they will give us a specific mean and a specific standard deviation in the question. So we just need to take note of those two numbers. So a roller coaster, who loves roller coasters, yeehaw. A roller coaster has a requirement that people must be at least 55 inches tall. Given that people have normally distributed heights with a mean of 57.1 inches and a standard deviation of 2.1 inches, find the percentage of people who satisfy that height requirement. So I have my picture here and using a data value, we're going to find area under the curve. So your mean always goes in the middle. In this context of this question, the mean is 57.1. And I want to find the percentage of people who satisfy the height requirement. Well who satisfies the height requirement? Anyone that is at least 55 inches tall. So let me plot 55, obviously it's going to be lower than 57.1 so it's to the left of it, and the probability that a person is more than 55 inches tall or at least 55 inches tall is going to be represented by the area to the right of 55. So we are finding the probability that some person's height, a data value, it's not as you score in this case so I just used the arbitrary letter X is greater than or greater than or equal to. It doesn't matter which one you pick as long as you say 55 afterwards. So in Google Sheets you need to make a note of what is the mean, what is the standard deviation, what is my lower bound, and what is my upper bound of my shaded region. So the mean is actually 57.1 and sigma the standard deviation is 2.1. My lower bound is 55 and my upper bound technically it's infinity because the x-axis goes on forever but you just use 999999 this is to ensure we get the most accurate answer possible. So the Google Sheets document we type in four numbers. Where do we go in the Google Sheets document? Well we go to the compute tab and the normal region. We will now type in what is mu so enter whatever mu is 57.1 enter type in sigma your standard deviation 2.1 enter, lower bound was 55 and upper bound was 6.9 and the probability is .8413 so you get .84 the area of that region is .8413 but they want to know percentage. What is .8413 as a percentage? Well the decimal to the right two spots and you actually get 84.13%. No answer locked in so the picture can help you make sure you pick the appropriate lower and upper bound very important. In this example we have IQ scores that are normally distributed with a mean of 100 and a standard deviation of 15. What is the probability someone has a score between 78 and 110? So I'm calculating the probability that someone has a score that some data value is between 78 and 110 and my picture my bell curve I will put the mean right in the middle that's 100 you need to mark 78 somewhere along this x-axis on the bottom and then you need to mark 110 somewhere. The important thing is you're trying to find the area the probability between 78 and 110 so we shade that region. So all we have to do now is resort to our Google Sheets document. In the Google Sheets document in just a moment we will input mu that's the mean of 100. We will input sigma that's the standard deviation of 15. We will input our lower bound so 78 and then we will input our upper bound which is 110. So we will use this in the Google Sheets document now. In the Google Sheets document we go to our compute tab we go to the normal region and we start just typing away our mu is going to be 100 our sigma is going to be 15 our lower bound is going to be 78 and upper bound is 110 you'll notice the probability pop up and it's 0.6763 when you round to four decimal places. You have a 2 in the fourth decimal place since there's a 7 in the fifth decimal place that's 5 or more you bumped the 2 up to a 3.6763 so we will make a note of this. Probability is 0.6763 and look at that there's our answer. So now let's find values from known areas or probabilities. When designing aircraft cabins how often do we do that in our daily lives or I should say how often do we almost hit our heads on airplanes in our daily lives when we're standing up in the aisle way when we're flying. What ceiling height will allow 95% of men to stand without bumping their heads? Men's heights are normally distributed with a mean of 69.5 and the standard deviation of 2.4 and they use men instead of women simply because men have a tendency on average to be taller. So we have a pretty picture here. We know that the mean is 69.5 so what ceiling height will allow 95% of men to stand without bumping their heads? Well remember the x-axis down here represents heights. Where are the people that are going to be bumping their heads? Well they're going to be the ones over here in the right tail so those are going to be the people, the tall people that are always bumping their heads, poor folks, sorry. My goal is to find a data value that will separate the 95% that don't bump their heads with the 5% or .05 that do bump their heads. So I'm looking for this magic height. At this magic height 95% will be able to stand without bumping their heads and then you have 5% that unfortunately will still bump their heads. So the area to the left of my data value is .95. So I'm trying to find that data value. So consulting Google Sheets when you're finding data values from area you obviously still write what is mu, 69.5, what is sigma, your standard deviation, 2.4 and then area to the left which is .95 because you want 95% to not hit their head on the ceiling. So Google Sheets is going to tell us all you have to do is type in mu, 69.5 and type in sigma, 2.4 and the only other thing you care about, you don't care about lower bound, upper bound, we're not calculating a probability is your left tail area. Your left tail area is going to be .95. And then your data value will appear just below 73.45. I like the round data values to 2 decimal places. 73.45 inches of course. 73.45 inches. So this airline company has contracted you out and they're like, hey, we only want 5% of our customers to hit their head on the ceiling or 5% of men to hit their head on the ceiling. What height should we make our cabins? And then you spit back, hey, 73.45 inches. That's the magic number you need. They pay you lots of money, end of story. Statistics is great, isn't it? Let's do another example. IQ scores. Again? Yeah. IQ scores are normally distributed with a mean of 100, standard deviation of 15. What score separates 65% of the results? Then my intention here, after I put my mean of 100 in the middle, is I'm looking for the data value that has 65% of the results below it. That's what my intention was here. So I'm looking for the data value whose area to the left is .65. So Google Sheets. What is your mu, 100 you say? What is your sigma? We're making our list. And what is area to the left? You need area to the left to find the data value. Well, it's .65. You will type these three magic numbers into Google Sheets and it will do all of the hard work for you. 15, area to the left or left-held area, .65, 105.78, 105.78, two decimal places for data values, 105.78. That is the IQ score that separates 65% of the results. How about that? That's pretty cool. So thank you Google Sheets for a good time and that's all I have for now. Thank you for watching.