In a study of how students use their mobile telephones, the phone usage of a random sample of 11 students was examined for a particular week - Edexcel - A-Level Maths Statistics - Question 4 - 2009 - Paper 1

First, calculate the interquartile range (IQR):

$IQR = Q3 - Q1 = 60 - 35 = 25$

Define the outlier limits:

• Upper Limit: $Q_3 + 1.5 \times IQR = 60 + 1.5 \times 25 = 60 + 37.5 = 97.5$
• Lower Limit: $Q_1 - 1.5 \times IQR = 35 - 1.5 \times 25 = 35 - 37.5 = -2.5$

The value of 110 exceeds the upper limit of 97.5, thus it is an outlier.

On the box plot:

• The box spans from Q1 (35) to Q3 (60).
• The median (53) is marked inside the box.
• The whiskers extend to the minimum value (17) and the largest non-outlier (77).
• An asterisk (*) or marker represents the outlier (110).

This visual representation highlights the outlier distinctly.

The sum of the values for the remaining 10 students is:

$\sum y = 17 + 23 + 35 + 36 + 51 + 53 + 54 + 55 + 60 + 77 = 461$

Calculate:

$S_{y} = \sum y^2 - \left( \frac{(\sum y)^2}{n} \right)$

Using: $\sum y^2 = 24,219$, we find:

$S_{y} = 24219 - \left( \frac{461^2}{10} \right) = 24219 - 10609.61 = 2966.9$

The correlation coefficient is calculated using:

$r = \frac{n{\sum xy} - (\sum x)(\sum y)}{\sqrt{\left[n\sum x^2 - (\sum x)^2\right]\left[n\sum y^2 - (\sum y)^2\right]}}$

With given values: $S_{x} = 3463.6$, $S_{y} = -18.3$, we find:

Substituting values leads to the final correlation coefficient.

If the correlation coefficient is close to zero, it suggests that there is little to no linear relationship between the number of text messages sent and the total minutes spent on calls. Therefore, the parent's belief that students sending a large number of text messages will spend fewer minutes on calls may not be justified, indicating a weak or potentially nonexistent correlation.