Problems

1. (5 points) Textbook problem 1.4
2. (5 points) Show that \( \text{std} (\{kx\}) = |k|\text{std} (\{x\}) \) by substituting into the definition. You'll need to use the properties of the mean.
3. (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 \( g(\mu) \).
4. (10 points) Textbook problem 1.11 (data, description)
5. (10 points) Textbook problem 1.13 (data, description)
6. (10 points) Textbook problem 1.15