Problems
- (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.
- What is the minimum of the possible top scores?
- What is the maximum of the possible top score?
- What is the minimum of the possible standard deviations?
- What is the maximum of the possible standard deviations?
- (10 points) Let \(\{x_i\} \) be a dataset consisting of \( N \) real numbers, \( x_1, \ldots, x_N \) .
- 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
- If the median of data set \(\{\widehat{x_{i}}\}\) is -0.5, is the data symmetric, left-skewed or right-skewed?
- (10 points) Textbook problem 1.11 (data)
- (10 points) Textbook problem 1.12 (data)
- (10 points) Textbook problem 1.13 (data)
- (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.