Chapter 4, Problem 47E

(a)

To determine

To find: the median, inter quartile range, mean and the standard deviation on the basis of given data.

Answer to Problem 47E

Median is 239, IQR is 9, Mean is 237.6 and Standard deviation is 5.7

Explanation of Solution

Given:

  

  

Calculation:

Median is the mid number of an ordered data set. Here data display mathematics achievement scores for eight graders in all 50 states that even number of values in data set so median is average of 25^th and 26^th values. 25^th values is 239 and 26^th value is also 239 so the median is 239.

To find IQR let us fist find $Q_1$ and $Q_3$ . Lower quartile is middle value of lower half.

Lower half of data has 25 values so the lower quartile is

  $\begin{array}{c}{Q}_1=13\text{ th value}\\ \text{=233}\end{array}$

Upper quartile is middle value of upper half. Upper half of data has 25 values so the upper quartile is

  $\begin{array}{c}{Q}_3=13\text{ th value of upper half}\\ \text{=242}\end{array}$

Therefore IQR is

  $\begin{array}{c}\text{IQR= }{Q}_3-Q_1\\ =242-233\\ =9\end{array}$

  $\begin{array}{c}\text{Mean = }\overline{x}\\ =\frac{11882}{50}\\ =237.6\end{array}$

So standard deviation would be

  $\begin{array}{c}\text{SD}\left(s\right)=\sqrt{\frac{{\left(x-\overline{x}\right)}^2}{n-1}}\\ =\sqrt{\frac{1587.7}{49}}\\ =\sqrt{32.40}\\ =5.7\end{array}$

Hence for state averages: median is 239, IQR is 9, Mean is 237.6 and Standard deviation is 5.7.

(b)

To determine

To Explain: the summary statistics would report for these data.

Explanation of Solution

Given:

  

  

Graph:

  

By seeing the histogram, it is noticed that it appears that the distribution is skewed to left. Whenever the distribution is skewed that time it would be better to use median as the measure of centre of the distribution and IQR as the measure of spread of the distribution.

(c)

To determine

To Explain: the brief summary of the performance of eighth graders nationwide.

Explanation of Solution

Given:

  

  

By seeing the distribution, it is observed that it is skewed to the left or negatively skewed and is looks to unimodal. Almost 20 states have scores between 240 and 244. One of the states has the less than 225 scores and 2 states have scores between the 225 and 229. Almost 4 states have scores greater than the mode of distribution.