At the time it takes to wait for a kidney transplant For people in the age group of 35 to 49 is approximated by a normal distribution. So we're going to draw that normal curve. And when you looked at the figure, you found that the average wait time was 1,674 days. With a standard deviation of 212.5 days. So part a is asking you to find the wait time that represents the 80th%ile. So when you're talking, in terms of percentiles, if you score at the 80th%,ile, that means You did better than 80 of the people that took the test. So that means 80 scored lower. So when we're talking the 80th%,ile, we're talking about 80 have a lower weight time in terms of the kidney weight. And if there's 80 on the left side of this boundary line, that means there's 20 that are on the upper side. So the first thing we want to do to figure out the boundary here between the lower 80 in the upper 20 is we're going to find a Z score and the fastest way is to find that Z score will be inverse norm in a graphing calculator. And when you use that function from the calculator, you do need to provide the area in the left tail, followed by the mean of the standard normal curve and the standard deviation. So for our problem, the area in that left tail is 80 of the curve. The mean for a Z score, the Z scores, the standard normal scores and the mean of Z scores is always zero and the standard deviation is always one. So I'm gonna bring in my graphing calculator and I'm going to access the inverse norm by hitting two and the vares button, which gets me into the distributions menu, and it's number three in this menu. So the area in the left tail, followed by the means of z scores, followed by the standard deviation. We end up with a Z score of approximately .84. So we're saying that this boundary is a Z score of about .84 And keep in mind the mean is a Z score of zero. So we now need to find, how many days does that represent. So you have a formula that says Z equals x minus mu divided by sigma. So we're going to take that formula, we're going to use our algebra skills and we're going to transform it so I can perform cross products here and get x minus mu equals z times sigma. I can then add mu to both sides and say that X is the same as new, plus Z times sigma. So I want to apply that formula. So I'm going to say mu plus Z times sigma. And the average wait time to get a kidney was 1,674. The Z score I just found was .84 And the standard deviation provided was 212.5. And when I calculate that out I will get 1 852 0.5 days, Part B in part B. We want to know what waiting time represents. The first quartile. So again, I'm going to draw that bell shaped curve And in terms of court, als, The first quartile represents the 25th%ile. So again, that's saying 25 is lower than that boundary line. We still have our average at 1,674 days. And again we are going to find the Z score attributed with this boundary line by using our inverse norm. The area in our left tail is .25. The mean of the Z scores or standard normal curve is always zero and the standard deviations one. So we'll do in verse norm 0.25 comma zero comma one. And we're going to get a Z score of about negative 10.67 So we want to find the raw score for that boundary line. So again, we're going to use the formula we derived. So we're going to take our average which is 1,674 days plus the Z score associated with this boundary. And multiply that by the standard deviation of 212 5 days and we will get 1531.625 days. So let's just recap What waiting time represents the 80th%ile. So that's saying about 80 of the time you're waiting, or you're you're waiting less than 1,852 5 days and 25 of the time You're waiting less than 1,531 625 days.