# IB DP Maths AA Topic 4 : Statistics and probability-SL 4.3-Measures of dispersion Question Bank SL Paper 1

### Question

Maya harvested 49 mangoes from her farm. The weights of the mangoes, w , in grams,Â are shown in the following grouped frequency table.

 Weight (g) 100 â‰¤ w < 200 200 â‰¤ w < 300 300 â‰¤ w < 400 400 â‰¤ w < 500 500 â‰¤ w < 600 Frequency 4 7 14 16 8
1. Write down the modal group for these data. [1]

2. Use your graphic display calculator to find an estimate of the standard deviation of theÂ weights of mangoes from this harvest. [2]

3. On the grid below, draw a histogram for the data in the table. [3]

Ans:

(a) 400 â‰¤ w < 500

(b) 115 (115.265… ( g))Â

(c)

## Question

The following box-and-whisker plot represents the examination scores of a group of students.

Write down the median score.

The range of the scores is 47 marks, and the interquartile range is 22 marks.

[1]
a.

Find the value of

(i) Â  Â  $$c$$;

(ii) Â  Â  $$d$$.

[4]
b.

## Markscheme

60 Â  Â  A1 Â  Â  N1

a.

(i) Â  Â  valid approach Â  Â  (M1)

eg $$\max – \min = {\rm{range}},c = 40 + 47$$

$$c = 87$$ Â  Â  A1 Â  Â  N2

(ii) Â  Â  valid approach Â  Â  (M1)

eg$$\;\;\;Q3 – Q1 = IQR,{\text{ }}74 – 22$$

$$d = 52$$ Â  Â  A1 Â  Â  N2

b.

## Question

A school collects cans for recycling to raise money. Samâ€™s class has 20 students.

The number of cans collected by each student in Samâ€™s class is shown in the following stem and leaf diagram.

The following box-and-whisker plot also displays the number of cans collected by students in Samâ€™s class.

There are 80 students in the school.

The mean number of cans collected is 39.4. The standard deviation is 18.5.

Each student then collects 2 more cans.

Find the median number of cans collected.

[2]
a.

(i) Â  Â  Write down the value of $$a$$.

(ii) Â  Â  The interquartile range is 14. Find the value ofÂ $$b$$.

[3]
b.

Samâ€™s class collected 745 cans. They want an average of 40 cans per student.

How many more cans need to be collected to achieve this target?

[3]
c.

The students raise $0.10 for each recycled can. (i) Â Â Find the largest amount raised by a student in Samâ€™s class. (ii) Â Â The following cumulative frequency curve shows the amounts in dollars raised by all the students in the school. Find the percentage of students in the school who raised more money than anyone in Samâ€™s class. [5] d. (i) Â Â Write down the new mean. (ii) Â Â Write down the new standard deviation. [2] e. Answer/Explanation ## Markscheme valid approach Â Â (M1) eg$$\,\,\,\,\,$$between 10th and 11th,Â $$\frac{{8 + 8}}{2}$$ median $$= 38$$Â Â Â A1 Â Â N2 [2 marks] a. (i) Â Â $$a = 20$$Â Â Â A1 Â Â N1 (ii) Â Â valid approach Â Â (M1) eg$$\,\,\,\,\,$$$${Q_3} – {Q_1},{\text{ }}{Q_1} + 14,{\text{ }}b – 30 = 14$$ $$b = 44$$ Â Â A1 Â Â N2 [3 marks] b. valid approach Â Â (M1) egx$$\,\,\,\,\,$$$$40 \times 20,{\text{ }}\frac{{x + 745}}{{20}},{\text{ }}40 – \frac{{745}}{{20}}$$ correct working Â Â (A1) eg$$\,\,\,\,\,$$$$800 – 745,{\text{ }}20 \times 2.75$$ 55 (more cans) Â Â A1 Â Â N2 [3 marks] c. (i) Â Â most cans in Samâ€™s class $$= 50$$Â Â Â (A1) 5 ($) Â  Â  A1 Â  Â  N2

(ii) Â  Â  correct value of 64 or 16 Â  Â  A1

valid approach Â  Â  (M1)

eg$$\,\,\,\,\,$$$$\frac{{64}}{{80}},{\text{ }}80\% ,{\text{ }}80 – 64,{\text{ }}\frac{{16}}{{80}}$$

20% Â  Â  A1 Â  Â  N2

[5 marks]

d.

(i) Â  Â  41.4 (exact) Â  Â  A1 N1

(ii) Â  Â  18.5 Â  Â  A1 Â  Â  N1

[2 marks]

e.

## Question

There are 10 items in a data set. The sum of the items is 60.

The variance of this data set is 3. Each value in the set is multiplied by 4.

Find the mean.

[2]
a.

(i) Â  Â  Write down the value of the new mean.

(ii) Â  Â  Find the value of the new variance.

[3]
b.

## Markscheme

correct approach Â  Â  (A1)

eg$$\,\,\,\,\,$$$$\frac{{60}}{{10}}$$

$${\text{mean}} = 6$$ Â  Â A1 Â  Â  N2

a.

(i) Â  Â  new mean $$= 24$$Â Â  Â  A1 Â  Â  N1

(ii) Â  Â  valid approach Â  Â  (M1)

eg$$\,\,\,\,\,$$$${\text{variance}} \times {(4)^2},{\text{ }}3 \times 16$$, new standard deviationÂ $$= 4\sqrt 3$$

new variance $$= 48$$Â Â  Â  A1 Â  Â  N2

[3 marks]

b.

## Question

A city hired 160 employees to work at a festival. The following cumulative frequency curve shows the number of hours employees worked during the festival.

The city paid each of the employees Â£8 per hour for the first 40 hours worked, and Â£10 per hour for each hour they worked after the first 40 hours.

Find the median number of hours worked by the employees.

[2]
a.i.

Write down the number of employees who worked 50 hours or less.

[1]
a.ii.

Find the amount of money an employee earned for working 40 hours;

[1]
b.i.

Find the amount of money an employee earned for working 43 hours.

[3]
b.ii.

Find the number of employees who earned Â£200 or less.

[3]
c.

Only 10 employees earned more than Â£$$k$$. Find the value of $$k$$.

[4]
d.

## Markscheme

evidence of median position Â  Â  (M1)

eg$$\,\,\,\,\,$$80th employee

40 hours Â  Â  A1 Â  Â  N2

[2 marks]

a.i.

130 employees Â  Â  A1 Â  Â  N1

[1 mark]

a.ii.

Â£320 Â  Â  A1 Â  Â  N1

[1 mark]

b.i.

splitting into 40 and 3 Â  Â  (M1)

eg$$\,\,\,\,\,$$3 hours more, $$3 \times 10$$

correct working Â  Â  (A1)

eg$$\,\,\,\,\,$$$$320 + 3 \times 10$$

Â£350 Â  Â  A1 Â  Â  N3

[3 marks]

b.ii.

valid approach Â  Â  (M1)

eg$$\,\,\,\,\,$$200 is less than 320 so 8 pounds/hour, $$200 \div 8,{\text{ }}25,{\text{ }}\frac{{200}}{{320}} = \frac{x}{{40}}$$,

18 employees Â  Â  A2 Â  Â  N3

[3 marks]

c.

valid approach Â  Â  (M1)

eg$$\,\,\,\,\,$$$$160 – 10$$

60 hours worked Â  Â  (A1)

correct working Â  Â  (A1)

eg$$\,\,\,\,\,$$$$40(8) + 20(10),{\text{ }}320 + 200$$

$$k = 520$$ Â  Â  A1 Â  Â  N3

[4 marks]

d.

## Question

The following box-and-whisker plot shows the number of text messages sent by studentsÂ in a school on a particular day.

Find the value of the interquartile range.

[2]
a.

One student sent k text messages, where k > 11 . Given that k is an outlier, find theÂ least value of k.

[4]
b.

## Markscheme

recognizing Q1 or Q3 (seen anywhere)Â  Â  Â (M1)

egÂ  Â  4,11 , indicated on diagram

IQR = 7Â  Â  Â A1 N2

[2 marks]

a.

recognizing the need to find 1.5 IQRÂ  Â  Â (M1)

egÂ  Â 1.5 Ã— IQR, 1.5 Ã— 7

valid approach to find kÂ  Â  Â (M1)

egÂ  Â 10.5 + 11, 1.5 Ã— IQR + Q3

21.5Â  Â  Â (A1)

k = 22Â  Â  Â A1 N3

Note: If no working shown, award N2 for an answer of 21.5.

[4 marks]

b.

## Question

A data set has n items. The sum of the items is 800 and the mean is 20.

The standard deviation of this data set is 3. Each value in the set is multiplied by 10.

Find n.

[2]
a.

Write down the value of the new mean.

[1]
b.i.

Find the value of the new variance.

[3]
b.ii.

## Markscheme

correct approachÂ  Â  Â Â (A1)

egÂ Â $$\frac{{800}}{n} = 20$$

40Â  Â  Â  A1 N2

[2 marks]

a.

200Â  Â A1 N1

[1 mark]

b.i.

METHOD 1

recognizing variance =Â Ïƒâ€‰2Â  Â  Â  (M1)

egÂ  32 = 9

correct working to find new varianceÂ  Â  Â  (A1)

egÂ  Ïƒâ€‰2Â Ã— 102, 9 Ã— 100

900Â  Â  Â  A1 N3

METHOD 2

new standard deviation is 30Â  Â  Â  (A1)

recognizing variance =Â Ïƒâ€‰2Â  Â  Â Â (M1)

eg 32Â = 9, 302

900Â  Â  Â Â A1 N3

[3 marks]

b.ii.