To start this problem off, we've got to talk about what the empirical rule is. The empirical rule is a rule that governs bell shaped curves or bell shaped distributions, and as long as your data is classified as being bell shaped, the center would be where your average would be found. It would also be where your median and your mode is found. And the empirical rules states that if you were to look at your data and look within one standard deviation above and one standard deviation below your average, you will find 68 of your data. Then if you were to go two standard deviations out in each direction, it would account for 95 of your data. And if you were to go three standard deviations out in each direction. Yeah, It would account for 99.7 of the data. So therefore we can break our bell curve up into various sections. So since on the left side and the right side are symmetric, We can take the 68 and we can break it up into 34 to the left of the mean And 34 to the right of the mean. And if the four sections 12 three Four have to total up to 95 and we've already accounted for 68%. That means there's 27 left to be split between these two sections, which would then make this section .135 And this section .1 35 And if from here To hear has to give us 99.7 and we've already accounted for 95, that means there's 4.7 left over to be split evenly between the other two little sections. So if I take 4.7 and split it up, I'm going to get 2.35. So we could say we've got .0235 in this section And .0235 in this section. So we're going to use that data to answer part B in this problem. The other thing that is mentioned in this problem is asking you to calculate a Z score. So is the score is a way of standardizing every bell shaped set of data so that we can make comparisons. So what happens is the average Equates to a zero Z score. So the average would have a Z score of zero, and if you're one standard deviation to the right, it would be a Z score of one. If you're two standard deviations to the right, we would have a Z score of two. If you're three standard deviations to the right, it will be a Z score of three. If you're one standard deviation to the left, it would be a Z score of negative one, two. Standard deviations to the left of the mean would be a Z score of negative two and three. Standard deviations to the left of the mean would be a Z score of negative three. And we have a formula that can help you transition any data set or data point into its corresponding Z score, and you would do that by saying whatever the raw score is, subtract the mean and divided by the standard deviation. So now that we have a little bit of the um theory, let's answer our question. So, in this particular problem, we are talking about a certain brand of automobile tires and you are told that they're mean life span of these tires would be 35,000 miles. The standard deviation of that particular brand of tire is 2250 miles. And it tells us that the lifespan of tires have this bell shaped distribution. So we're going to draw a bell shaped curve for our data And we are going to place our mean in the center. So we're going to put 35,000 miles in the center and we are going to select three different, Um, randomly selected tires and we find that their lifespans are a 34,000. So 34,000 would be like right in here, We're going to select a tire that had 37,000, well 37,000 is going to be above the 35,000. And we're going to pick one with 30,000. So 30,000 is gonna be back here and we want to calculate the Z scores for each of these lifespans. So to find the Z score for the 37,000, We'll do 37,000 minus the average, which was 35,000 Divided by the standard deviation, which was 2250. And when you do that we get a Z score of approximately 0.89 Now we're going to do the Z score for the 34,000. So we're going to do z equals 34,000 -35000, Divided by 2250. And we will get a Z score of approximately -44. And then we're going to do the Z score for 30,000. So we'll do z equals 30,000 minus 35,000 over the standard deviation. And we're going to get a Z score of approximately -2.22 and we want to determine which of these is unusual. So any time you have a Z score greater than two Or less than -2, it's classified as unusual. If you have a Z score that's greater than three or less than negative three, it's very unusual. So therefore We had a Z score that was less than -2. So this life span of the tire is unusual. So that was part A. Now in part B. We want to do something very similar and we want to use the empirical rule to find the percentile. So again, for part B, I'm going to draw a bell shaped curve. We're going to place that average in the center and we want to Start with the 30,500 miles. So 30,500 miles would be back here. So we're going to calculate its Z score. So we do 30,000 500 -35000. And we're going to divide it by 2250. And in doing so we are going to get a Z score of, let me bring in our calculator. So we get 30,500 -35000, Divided by 2250. And we're getting a Z score of -2. So when you look up at our curve with all the bell values filled in, we could see that From the center up would be 50 of our curve. And then with the Z score of -1, we have a 34 in here and we have a .135 in there. So our goal is to figure out the percentile and the percentile, anytime you're dealing with a percentile, you want to know what percent did worse. So in essence we're trying to find this area in here. So what we'll do is we're going to think about the fact that entire curve is 100% or if we add up the areas it would be one. So I'm gonna start with the entire curve, I'm going to take away the 50%, that is in the Right half of the curve. I'm going to take away the 34%, that's between a Z score of zero and a negative one and I'm going to take away the .135, which is the area between the Z score of -1 and -2. And I'm going to be left with a .025 as the area down here in the left part of the graph. So therefore a tire that lasts 30,500 miles is in the 2.5%ile Or is at the 25%ile. Let's look at the next one. The next one we're gonna draw are bell shaped curve. Again, we're going to place our average in the center And this time we want to look at a tire that lasted 37,250 miles. So we're going to go to 37 250 mi. So again, we want to find its disease score. So we'll do 37,250 -35000 mi, divided by the standard deviation. So I'll bring in my calculator. So we're going to do 37-50 -35000. And we're going to divide that by 2250 and we're getting a Z score of one. So we want to find the percentile. So again, I want you to think about the fact that in the lower half of the bell Would be .50. That's between a Z score of zero and all the way into the left half. And if this is a Z score of one, that means there's 34 in there as well. So when we're talking percentile we're talking about the percent less. So we're talking about the entire curve to the left of that boundary line. So we could say a tire that last 3007 or sorry, 37,250 miles is at the 84th percentile Because when I add up the .34 and the .50, I'm getting a .84. And then the final part of part B is to talk about a tire that lasts exactly 35 1000 miles. So if I think of the bell shaped curve, Here's 35,000 miles. So I want to know what percent is less than that. Well, that would be half of the curve. So I could then say, a tire that lasts 35,000 miles is at the 50th percentile. So just to recap, the empirical rule governs bell shaped curves and these would be the percentages between various data sets. R Z scores would be a way of standardizing various data data sets that have different means and different standard deviations and we can convert all data into corresponding Z scores by using that formula of X minus mu over sigma. It's unusual when you have a Z score less than negative, two or greater than positive too. And it's very unusual when you have Z scores greater than three or less than negative three.