Problem 1. (10 POINTS) DATA SUMMARIES AND LINEAR TRANSFORMATIONS. In this problem, you will explore how data summaries change under liner transformation of the data. Consider a sample x1,...,xn. Let I, I, 82, and IQR, denote the sample mean, median, standard deviation, and the interquartile range of the sample. Let Yi = a + xi. Express , , sy, and IQR, in terms of , , sr, and IQR. Problem 2. (10 POINTS) OPTIMIZATION INTERPRETATION OF I AND . The sample mean and median have interesting optimization interpretations. Show that n = arg min (x; a), i=1 n I = arg min xa. - i=1 (1) (2) (3) Remark: Notation a* = arg mina f(a) means that a* is a value of a that minimizes the function f(a). Problem 3. (10 POINTS) INTERPRETATION OF QQ PLOTS. Let x1,...,x be a sample. We know that if the normal-quantile plot, i.e. a collection of points {(z (k))}, falls roughly on the line y = x, then the sample has approximately the standard normal distribution. What can you say about the distribution of the sample if points {(__, (^))}, instead if y=x, fall on the line y = ax + b? Problem 4. (10 POINTS) READABILITY OF QQ PLOTS In this problem, you will investigate the "quality" of normal-quantile plots. Daniel and Wood (1980) "Fitting equations to data" present normal-quantile plots for samples generated from the standard normal distribution. Study of these plots is helpful in acquiring a feel for how much deviation from the straight line is acceptable. They show that small sample sizes often produce normal-quantile plots that deviate substantially from linearity. For large sample sizes the plots are much better. Let us investigate this issue. (a) (3 points) Draw a random sample of size n = 15 from N(0, 1) and plot both the normal-quantile plot and the histogram. Do the points on the QQ plot appear to fall on a straight line? Is the histogram symmetric, unimodal, and bell-shaped? Do this several times and summarize you observations. (b) (3 points) Repeat (a) for samples of sizes n = 50, n = 100, and n 1000. 1 = (c) (4 points) After experimenting with normal samples in (a) and (b), what would be your estimate for the "critical" sample size n*, such that for samples of size larger than n*, the normal-quantile plots are stable enough to be easily interpreted (i.e. do not deviate substantially from linearity)? Write a script that implements this task. Write you results and conclusions as comments in the script.