Problems

1. (10 points) A teacher gives 5 students a multiple choice test, in which each problem is worth 1 point and there is no penalty with negative points. The median and mean scores turn out to be 9 and 10 points, respectively.
   1. What is the minimum of the possible top scores?
   2. What is the maximum of the possible top score?
   3. What is the minimum of the possible standard deviations?
   4. What is the maximum of the possible standard deviations?
2. (10 points) Let $\{x_i\}$ be a dataset consisting of $N$ real numbers, $x_1, \ldots, x_N$ .
   1. Prove from definitions or proved properties in the textboook that the standardized data set $\{\widehat{x_{i}}\}$ that is derived from $\{x_i\}$ has mean = 0 and standard deviation = 1
   2. If the median of data set $\{\widehat{x_{i}}\}$ is -0.5, is the data symmetric, left-skewed or right-skewed?
3. (10 points) Textbook problem 1.11 (data)
4. (10 points) Textbook problem 1.12 (data)
5. (10 points) Textbook problem 1.13 (data)
6. (Extra credit 5 points) Let $\{x_i\}$ be a dataset consisting of $N$ real numbers, $x_1, \ldots, x_N$. Prove the function $g(m) = \sum_{i=1}^N |x_i-m|$ is minimized when $m=\text{median}(\{x_i\})$. Hint: try to prove $(\sum_{i=1}^N |x_i-d| - \sum_{i=1}^N |x_i-median|) >= 0$ for $d$ >= $median$, then for $d < median$, d is any real number .

Note: Whenever the problem asks the student to plot a graph, the graph should be submitted by pasting it in the document. For Ex. 3-5, the program code should be inserted in the document.