Homework on Random sampling and stratifying samples
1.
The manager of a cinema wants to find out how often teenagers attend the cinema.
He uses a questionnaire.
(a)
Here is one part of the questionnaire.
Question
How often do you attend the cinema?
Response
Sometimes
Occasionally
Regularly
Write down two criticisms of his response section.
Criticism 1 .................................................................................................................
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Criticism 2 .................................................................................................................
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(2)
(b)
Explain how the manager could distribute 50 questionnaires randomly to pupils from a
school of 1000 pupils.
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(1)
(Total 3 marks)
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2.
(a)
Jane stood outside the school library and collected the views of 50 pupils.
Explain why her method of obtaining data will not give a random sample.
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(1)
(b)
Sahima decides to take a stratified sample, of 50 pupils, by year group from the 1000
pupils in the school.
The table shows the number of pupils in each year group at the school.
Year
7
8
9
10
11
Number of
pupils
167
200
230
210
193
Calculate the number of pupils that Sahima should choose from each year group.
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Answer
Year 7 ........…...................................................
Year 8 .......…....................................................
Year 9 .......…....................................................
Year 10 .............................................................
Year 11 .............................................................
(4)
(Total 5 marks)
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3.
Each day 65 trains pass through Bristol. Of these, 23 go North, 21 go South, 13 go East
and 8 go West.
For a survey, Paul decides to take a 20% sample of these trains. He plans to choose every fifth
train that passes.
(a)
Explain why the sample obtained may not represent the different routes fairly.
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(1)
(b)
Paul decides to take a 20% stratified sample.
Calculate the number of trains going each way that he should choose.
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North: ………, South: ………, East: ………, West:………
(4)
(Total 5 marks)
4.
Annie wants to choose a sample of people from three neighbouring towns to find out what they
think about a plan to close a local hospital.
Town A
Town C
Town B
(a)
State one advantage of using a sample stratified by town.
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(1)
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(b)
Approximately 20 000 people live in Town A, 5 000 in Town B and 16 000 in Town C.
Annie wants a stratified sample of size 100.
(i)
Show that she should choose 49 people from Town A.
...…………………………………………………………………………........
...…………………………………………………………………………........
(1)
(ii)
Calculate how many people she should choose from each of the other towns.
...…………………………………………………………………………........
...…………………………………………………………………………........
Town B ........……................ Town C ………………………
(2)
(Total 4 marks)
5.
The table shows the number of boys and the number of girls in each year in a school of
2000 pupils.
(a)
Year 7
Year 8
Year 9
Year 10
Year 11
Boys
163
186
198
196
187
Girls
201
204
234
218
213
A 10% sample, stratified by year and gender, is needed.
Complete the table to show how many boys and how many girls should be chosen from
each year.
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Year 7
Year 8
Year 9
Year 10
Year 11
Boys
Girls
(3)
(b)
The sample chosen will be asked for their opinions on a number of school related issues.
Explain why a stratified sample will be more useful than a random sample.
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(2)
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(c)
There are 930 boys and 1070 girls in the school altogether.
Compare the numbers of children in your sample from part (a) with the numbers that you
would have in a 10% sample stratified by gender only.
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(2)
(Total 7 marks)
6.
Mr Singh wants to choose four pupils from a class list to attend a meeting.
There are 10 boys and 18 girls in the class.
(a)
Explain why a stratified sample may be better than a random sample in this situation.
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(1)
(b)
Calculate the number of boys and the number of girls who should be chosen for a
stratified sample of four pupils.
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Boys ...................... Girls ........................
(2)
(c)
Explain how Mr Singh should choose the girls to send to the meeting.
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(2)
(Total 5 marks)
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1.
(a)
No box for never
Sometimes/occasionally/regularly mean different people/mean the same thing
Doesn’t ask whether per week, month etc
B1 for each valid reason
(b)
B2
Number each pupil and pick 50 numbers from hat
Or systematic sampling with a random sampling with a random starting point
A valid random method
eg Ran on calc
eg Use random number tables etc
B1
[3]
2.
(a)
Biased towards towards library users or people in that area
(b)
167 ×
50
1000
B1
M1
any correct method seen
8.35
or
10 11.5 10.5 9.65
All 5 correct decimals to seen
Accept 8.4 and/or 9.7
8
10
12
11
10
Rounding their decimals to integers correctly or fully correct
8
10
8
10
11
11
10
From fully correct working
(if shown)
Ensuring sum is 50
NB 8 10 12 11 9
Scores 2 marks
12
10
A1
A1
10
A1
[6]
3.
(a)
Could all be going North, for example.
B1
(b)
23÷5=4.6, 21÷5=4.2, 13÷5=2.6, 8÷5=1.6
4.6, 4.2, 2.6, 1.6 or 5, 4, 3, 2 or 4, 4, 2, 1
B1
A1
5+4+3+2 = 14 need to lose 1
e.g. 4, 4, 3, 2
Also accept 5, 4, 2, 2 or 5, 4, 3, 1
M1
A1 cao
[5]
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4.
(a)
It will have the correct proportion of people from each town
Will include people from all the towns
(bod)
B1
(b)
(i)
100 × 20 ÷ 41 = 48.78
B1
(ii)
B = 12, C = 39
B1 B1
[4]
5.
(a)
(b)
Calculating 10%
Any one correct decimal or integer
M1
16 X X X X
20 20 23 X 21
X 19 20 20 19
X X X 22 X
Correctly rounding down
A1
Correctly rounding up
A1
A stratified sample will be proportional to all year groups
and/or boys and girls
Must say proportional to one item
Takes a relative sample …
A random sample could be biased in favour of boys or girls or year groups
Must say biased to one item
Cannot get both marks by implication
(c)
Part (a) gives 94 boys and 106 girls
Their row totals from (a)
By gender 93 boys and 107 girls
Part (a) gives one extra boy and one girl too few
B1
B1
B1
B1 ft
B1
[7]
6.
(a)
Because of the need to get boys and girls represented in proportion
Cannot be all of one type
(b)
10 ×
B1
4
4
 1.43 or 18 
28
28
= 2.57M1
Either calculation
(c)
Boys: 1, Girls: 3
A1
Give each girl a number
Labelling (eg ‘names in hat’, or ‘use register’)
B1
Use random numbers to choose three
Random choice (eg ‘draw from hat’)
B1
[5]
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