Just Siles are the points that mark off the lowest 10% and highest 10% of a density curve. So this problem is asking us to find the death styles of the standard normal distribution. So first thing we wanna do is we want to look at the standard normal distribution. The standard normal distribution is are bell shaped curve, and our units are the Z scores. So with the standard normal curve, your average is zero, and your standard deviation is one. So all of these numbers on this axis are your Z scores. And what we're trying to find is the bottom 10%. So where would the bottom 10% B So there's 10% in this tale and then where would the top 10% bay? So that means there's 10% in this tale. And by doing that and keeping in mind that the full standard normal curve has an area of one, that means there's 80% in here. So we're trying to find this cut off that separates the bottom 10% and this cut off, which separates the top 10%. So to find the bottom cut off, we're going to use your inverse norm feature on your calculator, and in doing so, you need to talk about the area in the left tail, the average and the standard deviation. So to find the bottom or the left death style, we're going to use inverse norm. The area in the left tail is 0.10 The average of the standard normal curve is zero, and the standard deviation is one. So I'm going to bring in the graphing calculator and we're going to hit second. There's and we're going to do number three inverse norm 0.10 comma zero comma one. And we find that one of the death styles is approximately negative. 1.28 Now, to find the other deaths style, I'm going to change colors for you. So to find this death style right here, we're going to do Z equals in verse, norm. And this time going to the left would be 80% plus 10%. So we would have to use 0.90 for the going into the left tail comma. Zero comma one. So again, I'm going to bring in my graphing calculator and we're going to do second. There's inverse norm 0.90 comma, zero comma one and we find that the other deaths style is positive. 1.28 So for part a, the death styles of the standard normal curve are Z equals plus and minus 1.28 part to be part B is asking us what the death styles are for a distribution about the heights of young women. So again we have our curve, and this time our horizontal axis will be the heights measured in inches. And we know that the mean height is 64.5 and we know that the standard deviation of the heights of women would be 2.5 inches. And we want to find the death styles of that distribution. Well, keep in mind that the Death Siles are still in the same place at Z scores of negative 1.28 and positive 1.28 which will separate the 10% the lowest 10% and the highest 10%. So we have all the information we need to go ahead and find the corresponding raw pieces of data, or are ex values associated with each of those Z values So we have a formula for a Z score that reads Z equals X minus mu, divided by sigma. So for this ex score, we're going to say negative 1.28 equals our unknown X value, minus the average 64.5 over the standard deviation of 2.5. And if I find my cross products meaning I'm going to cross multiply, I will get X minus 64 5 is equal to 2.5 multiplied by negative 1.28 And then if I add 64.5 to both sides, I will get X equals 64.5 Plus what I get when I multiply 2.5 times negative 1.28 So this lower death style is going to be 61.3 inches. To find the upper death style. I'm going to use the same formula, and I'm going tohave Z equals. And in this case, my Z is positive. 1.28 So I'll have one positive 1.28 equals X minus 64.5 over the standard deviation of 2.5. Again, I'll do my cross products so I'll have X minus 64.5 equals 2.5 multiplied by 1.28 And if I add 64.5 to both sides, I will have 64.5 plus 2.5 times 1.28 yielding the second death style to be 67.7 inches. So for Part B, the death Siles of the distribution with a mean equal to 64.5 and a standard deviation of 2.5 will be 61.3 inches and 67.7 inches.