Chapter 4, Problem 8PT1

To determine

To choose: The incorrect option regarding the given data.

Answer to Problem 8PT1

Option e is incorrect.

Explanation of Solution

Given information:

Number of IQ scores $=10$

The IQ scores are

  $145,139,126,122,125,130,96,110,118,118$

Formula used:

Inter quartile range

  $\begin{array}{l}IQR=Q_3-Q_1\\ \text{where}\\ Q_3=3\times {\left(\frac{n+1}{4}\right)}^{th}term\\ Q_1={\left(\frac{n+1}{4}\right)}^{th}term\end{array}$

Calculation:

Arranging in ascending order

  $96,110,118,118,122,125,126,130,139,145$

The IQR for the 10 scores can be calculated as follows

  $\begin{array}{l}{Q}_3=3\times {\left(\frac{10+1}{4}\right)}^{th}term\\ Q_3=3\times {2.75}^{th}term={8.25}^{th}term\\ Q_3=8^{th}term+\frac{1}{4}\times \left(9^{th}term-8^{th}term\right)\\ Q_3=130+\frac{1}{4}\times \left(139-130\right)=132.25\end{array}$

The lower quartile is

  $\begin{array}{l}{Q}_1={\left(\frac{10+1}{4}\right)}^{th}term\\ Q_1={2.75}^{th}term\\ Q_1=2^{th}term+\frac{3}{4}\times \left(3^{th}term-2^{th}term\right)\\ Q_1=110+\frac{3}{4}\times \left(118-110\right)=116\end{array}$

Therefore the IQR for 10 scores is

  $132.25-116=16.25$

Finding the IQR after removing 96

The new set of scores is

  $110,118,118,122,125,126,130,139,145$

The IQR for the 9 scores can be calculated as follows

  $\begin{array}{l}{Q}_3=3\times {\left(\frac{9+1}{4}\right)}^{th}term\\ Q_3=3\times {2.5}^{th}term={7.5}^{th}term\\ Q_3=7^{th}term+\frac{1}{2}\times \left(8^{th}term-7^{th}term\right)\\ Q_3=130+\frac{1}{2}\times \left(139-130\right)=134.5\end{array}$

The lower quartile is

  $\begin{array}{l}{Q}_1={\left(\frac{9+1}{4}\right)}^{th}term\\ Q_1={2.5}^{th}term\\ Q_1=2^{th}term+\frac{1}{2}\times \left(3^{th}term-2^{th}term\right)\\ Q_1=118+\frac{1}{2}\times \left(118-118\right)=118\end{array}$

Therefore the IQR for 9 scores is

  $134.5-118=16.5$

Conclusion:

Therefore, the IQR of scores after removing $96\left(=16.5\right)$ is slightly higher than the original one $\left(=16.25\right)$ which invalidates the statement given in sub part e.