Homework 1

Due Monday Sept. 9 at 11:59pm

Homework policies and submission instructions

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 \(\{x_i\} \) be a dataset consisting of \( N \) real numbers, \( x_1, \ldots, x_N \) .
    1. Prove 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 1, 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 \( (\sum_{i=1}^N |x_i-d| - \sum_{i=1}^N |x_i-m|) >= 0 \) for \(d\) >= \(m\), then for \(d < m\) .