Problems

1. (10 points) A teacher gives 5 students a multiple choice test, in which each problem is worth 1 point. The median and mean scores turn out to be 9 and 10 points, respectively.
   1. What is the minimum possible top score?
   2. What is the maximum possible top score?
   3. What is the minimum possible standard deviation?
   4. What is the maximum possible standard deviation?
2. (10 points) Let \( \{x\} \) be a dataset consisting of \( N \) real numbers, \( x_1, \ldots, x_N \). Read the derivation in section 1.3.1 of the textbook that shows that the function \( f(\mu) = \sum_i (x_i-\mu)^2 \) is minimized when \( \mu=\text{mean}(\{x\}) \).
   1. Assume that \( N=4 \) and that \( x_1 \leq x_2 \leq x_3 \leq x_4 \). Show that the function \( g(\mu) = \sum_i |x_i-\mu| \) is minimized when \( \mu=\text{median}(\{x\}) \).
   2. In at most two sentences, explain why the median is less sensitive to outliers than the mean using what you know about \( f(\mu) \) and \( g(\mu) \).
3. (10 points) Textbook problem 1.11 (data)
4. (10 points) Textbook problem 1.12 (data)
5. (10 points) Textbook problem 1.13 (data)