Approximately, the pth quantile of a set S of n numerical values is the value .5:p such that the proportion of values in S that are less than or equal to 3p equals p. This concept may be familiar from the idea of the 95';h percentile. There are complications arising from the fact that there may not be a value SI) for which the proportion exactly equals p. The "quantile" function in R defaults to one approach to addressing this. The default is acceptable for these exercises. The rst quartile of .S' is the value 5'03 5. The third quartile of S is the value so_75. The median of S is the value 305. The interquartile range of .S' is the value 3035 3025. Given a data set with known values for the median and the interquartile range, one can calculate values of ,u and 0' such that the Normal distribution Normal(,u, 0'2) has the same median and interquartile range. This gives a way to select a Normal distribution approximating the data set. The results of this exercise will be used in the next exercise identify Normal distributions that are reasonable approximations to simulated data sets. Question 3.3 By analogy, for a random variable with cumulative density function F, let :vcp satisfy Fm) = F'- The rst quartile of the random variable is the value x025. The third quartile of the random variable is the value 2:035. The median of the random variable is the value x05. The interquartile range of the random variable is the value x0_75 1:025. Please calculate the values of x0_75 for the Normal distributions with mean 0 and sd in 1,2,3, . . .10 and plot the points consisting of the value of the sd and the corresponding 91:075. This should give an indication of a simple function relating sd and x0_75.[5 points) Question 3.b u2 2\" H31 2 _du. Denote by wp the value such that p Denote by Z], the value such that 39: L00 (x 102 L\": e 262 dx. 231:ch Thus 2p 1 u2 WP 1 -(x-u)2 f eru=f e 21:2 dx -09 21: -00 211'0'2 Note that the change of variable u- transforms the integrand of the integral on the right hand side of the equality to equal the integrand of the integral on the left hand side. Please use this change of variable applied carefully to the limits of Integration to give a formula for 23, in terms of wp, 11, and o'. [5 points) Question 3.c Please give a formula for wags in terms of p, a, and 2025. Please give a formula for w0_75 in terms of p, o', and 2015. Please give a formula for Wo.5 in terms of 11. Please give a formula for w0_75 w0_25 in terms of o', and 205,5. Please use the results of 3.b to address these questions. [5 points) Question 3.d Suppose a data set has median equal to m and interquartile range equal to q. What values of p and a result in a random variable Normalm, 02) with the same median and interquartile range? Please use 2035 in your solution. The symmetry of the standard Normal density function implies that 2025 = zo_75. [5 points)