Unit 5

6 marks · Hard

A cumulative frequency diagram shows the results of a survey measuring the heights, in cm, of 80 students. The median height is 162 cm and the upper quartile is 170 cm. The heights have been recorded to the nearest centimetre.

(a)

Write down the lower and upper bounds of the median height.

[1 mark]

(b)

If a box plot is drawn for the data, what is the minimum range of heights that could possibly be represented by the interquartile range? Give your answer to 1 decimal place.

[2 marks]

(c)

Calculate the lower and upper bounds for the proportion of students with a height strictly greater than 170 cm.

[3 marks]

Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.

Show answer

(a) Lower bound: 161.5 cm, Upper bound: 162.5 cm

(b) Minimum interquartile range: 170.5 - 161.5 = 9.0 cm

(c) Lower bound: 0 students (if upper bound is 170.5 cm), proportion = 0/80 = 0; Upper bound: up to just under 170 students, so possible 79/80 = 0.9875; Accept range 0 to 0.9875

Common mistakes: Using rounded values instead of bounds; Incorrectly applying strict inequality; Mismatch between cumulative frequency and bounds

Show mark scheme and worked solution

Marking points

A1 (a) · 1 mark · Correct lower and upper bound for a value rounded to nearest cm

M1 (b) · 1 mark · Finds IQR from bounds correctly

A2 (b) · 1 mark · Correctly gives minimum IQR as 9.0 cm to 1 d.p.

M2 (c) · 1 mark · Considers how bounds affect the count of students above 170 cm

A3 (c) · 1 mark · Calculates lower and upper possible number of students > 170 cm and the proportion

B1 (c) · 1 mark · Expresses answer as a range or correct inequalities

Worked solution

a) Height recorded to nearest cm, bounds = value ± 0.5 cm

Median lower: 162 - 0.5 = 161.5 cm, upper: 162 + 0.5 = 162.5 cm

b) IQR = upper quartile - lower quartile. Range of possible values is from lower bound of median (161.5) to upper bound of quartile (170.5).

c) 'Strictly greater than 170 cm' means heights must be more than 170.5 cm, as 170.5 is not included. If upper bound for an individual is <= 170.5, must be 170 or less.

So, minimum possible is 0 students (if everyone is less than or equal to 170.5), maximum is if only one student is just above 170.5: could be up to 79/80 = 0.9875.