S1 January 2011 Q3

3. Over a long period of time a small company recorded the amount it received in sales per month. The results are summarised below.

	Amount received in sales (£1000s)
Two lowest values	3, 4
Lower quartile	7
Median	12
Upper quartile	14
Two highest values	20, 25

An outlier is an observation that falls
either 1.5 \(\times\) interquartile range above the upper quartile
or 1.5 \(\times\) interquartile range below the lower quartile.

(a) On the graph paper below, draw a box plot to represent these data, indicating clearly any outliers.

Blank graph paper with a horizontal axis Sales (£1000s) from 0 to 30

(5)

(b) State the skewness of the distribution of the amount of sales received. Justify your answer. (2)

(c) The company claims that for 75% of the months, the amount received per month is greater than £10 000. Comment on this claim, giving a reason for your answer. (2)

Box plot on a scale Sales in £’000 from 0 to 25: box from 7 to 14 with median 12, lower whisker to 3, upper whisker drawn either to 20 or to 24.5, with the outlier at 25 marked — Mark scheme (a)

Scheme	Marks
Outliers \(14 + 1.5 \times (14 - 7) = 24.5\)	M1
\(7 - 1.5 \times (14 - 7) = -3.5\)	A1
Outlier 25 either upper limit acceptable on diagram	M1 A1ft B1
	(5)

Mark scheme (b)

Scheme	Marks
Since \(Q_3 - Q_2 \lt Q_2 - Q_1\). Allow written explanation negatively skew	B1 dB1
	(2)

Notes

1st B1 for \(Q_3 - Q_2 \lt Q_2 - Q_1\) statement or an equivalent statement in words
Use of \(Q_3 - Q_2 \lt Q_2 - Q_1\) does not require differences to be seen.
2nd dB1 for “negative skew” dependent on suitable reason given above. “correlation” is B0
“positive skew” with a supporting argument based on whiskers can score B1B1
e.g. “right hand whisker is longer than LH one so positive skew”
\(Q_3 - Q_2 \lt Q_2 - Q_1\) followed by “positive skew” is B1B0

Mark scheme (c)

Scheme	Marks
not true since the lower quartile is 7000 and therefore 75% above 7000 not 10000 or 10 is inside the box or any other sensible comment	B1 dB1
	(2)
	(9 marks)

Notes

1st B1 for rejecting the company’s claim
2nd dB1 for an appropriate supporting reason. Dependent on rejecting company’s claim.