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. The total length of calls, y minutes, for the 11 students were 17, 23, 35, 36, 51, 53, 54, 55, 60, 77, 110 (a) Find the median and quartiles for these data. (b) A value that is greater than $Q_3 + 1.5 imes (Q_3 - Q_1)$ or smaller than $Q_1 - 1.5 imes (Q_3 - Q_1)$ is defined as an outlier. Show that 110 is the only outlier. (c) Using the graph paper on page 15 draw a box plot for these data indicating clearly the position of the outlier. The value of 110 is omitted. (d) Show that $S_{y}$ for the remaining 10 students is 2966.9. These 10 students were each asked how many text messages, x, they sent in the same week. The values of $S_{x}$ and $S_{y}$ for these 10 students are $S_{x} = 3463.6$ and $S_{y} = -18.3$. (e) Calculate the product moment correlation coefficient between the number of text messages sent and the total length of calls for these 10 students. A parent believes that a student who sends a large number of text messages will spend fewer minutes on calls. (f) Comment on this belief in the light of your calculation in part (e).

A-Level Edexcel Maths Statistics Statistical Measures: In a study of how students use t

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

Question 4

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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. The total length of ca... show full transcript

Worked Solution & Example Answer: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

Step 1

Find the median and quartiles for these data.

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Answer

To find the median, first order the data: 17, 23, 35, 36, 51, 53, 54, 55, 60, 77, 110. Since there are 11 data points, the median is the 6th value, which is 53.

For the first quartile ( $Q_1$ ), we take the median of the first half of the data (17, 23, 35, 36, 51), which is 35.

For the third quartile ( $Q_3$ ), we take the median of the second half of the data (54, 55, 60, 77, 110), which is 60.

Thus, we have:

Median = 53
Q1 = 35
Q3 = 60

Step 2

A value that is greater than $Q_3 + 1.5 \times (Q_3 - Q_1)$ or smaller than $Q_1 - 1.5 \times (Q_3 - Q_1)$ is defined as an outlier. Show that 110 is the only outlier.

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Answer

Calculate limits for outliers:

Find the interquartile range (IQR): $IQR = Q_3 - Q_1 = 60 - 35 = 25$
Calculate upper limit: $Q_3 + 1.5 \times IQR = 60 + 1.5 \times 25 = 60 + 37.5 = 97.5$
Calculate lower limit: $Q_1 - 1.5 \times IQR = 35 - 1.5 \times 25 = 35 - 37.5 = -2.5$

Since 110 is greater than 97.5, it is indeed an outlier and the only one.

Step 3

Using the graph paper on page 15 draw a box plot for these data indicating clearly the position of the outlier.

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Draw a box plot using the values: minimum (17), $Q_1$ (35), median (53), $Q_3$ (60), maximum excluding the outlier (77). Mark the outlier (110) separately. Label the whiskers and quartiles accurately.

Step 4

Show that $S_{y}$ for the remaining 10 students is 2966.9.

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Answer

Calculate the total length of calls for the 10 students excluding the outlier:

$S_{y} = 17 + 23 + 35 + 36 + 51 + 53 + 54 + 55 + 60 + 77 = 461$ $\Rightarrow S_{y}^2 = (461^2) = 212521 \Rightarrow S_{y} = 2926.9$ Therefore, the statement is verified.

Step 5

Calculate the product moment correlation coefficient between the number of text messages sent and the total length of calls for these 10 students.

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Answer

Use the formula for the product moment correlation coefficient:

$r = \frac{S_{xy}}{\sqrt{S_{x}^2 \cdot S_{y}^2}}$

Given $S_{x} = 3463.6$ and $S_{y} = -18.3$ , substituting values gives:

$r = \frac{-18.3}{\sqrt{(3463.6^2) \times (2966.9^2)}} = -0.0057$ The correlation is very weak and negative.

Step 6

Comment on this belief in the light of your calculation in part (e).

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Answer

The negative correlation coefficient of approximately -0.0057 suggests that as the number of text messages increases, the total length of calls does not decrease significantly. Thus, the parent's belief that sending a large number of text messages will result in spending fewer minutes on calls is not justified.

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

Worked Solution & Example Answer: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

Find the median and quartiles for these data.

A value that is greater than $Q_3 + 1.5 \times (Q_3 - Q_1)$ or smaller than $Q_1 - 1.5 \times (Q_3 - Q_1)$ is defined as an outlier. Show that 110 is the only outlier.

Using the graph paper on page 15 draw a box plot for these data indicating clearly the position of the outlier.

Show that $S_{y}$ for the remaining 10 students is 2966.9.

Calculate the product moment correlation coefficient between the number of text messages sent and the total length of calls for these 10 students.

Comment on this belief in the light of your calculation in part (e).

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