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leaving certificate ORDINARY LEVEL
Active Maths 3
2π
r
Book 2
2
πr
m
1
= y2 y
x
1
2 x
Supplementary Material
for 2015 exam and
onwards
ISBN 978-1-78090-529-7
2714
Michael Keating, Derek Mulvany and James O’Loughlin
folensonline.ie
9 781780 905297
Special Advisors:
Oliver Murphy, Colin Townsend and Jim McElroy
Contents
Chapter 1 Statistics I
Exercise 1.2 ���������������������������������������������������������������������������������������������������������������������������������������������1
Chapter 4 Statistics II
4.3A Sampling���������������������������������������������������������������������������������������������������������������������������������������2
Biased Samples�����������������������������������������������������������������������������������������������������������������������������2
Sampling Variability�����������������������������������������������������������������������������������������������������������������������4
4.7 Margin of Error and Hypothesis Testing������������������������������������������������������������������������������������������8
Margin of Error������������������������������������������������������������������������������������������������������������������������������9
Hypothesis Testing�����������������������������������������������������������������������������������������������������������������������10
Activity 4X����������������������������������������������������������������������������������������������������������������������������������������������14
Chapter 9 Transformations and Enlargements
Rotations������������������������������������������������������������������������������������������������������������������������������������������������18
Answers����������������������������������������������������������������������������������������������������������������������������������������������������������� 20
Additional copies of this booklet are available to download from folens.ie
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ISBN: 978-1-78090-529-7
© Michael Keating, Derek Mulvany, James O’Loughlin and Colin Townsend, 2014
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1
Statistics I
9.The following back-to-back stem-and-leaf diagram compares the pulse rates of 25 people before
and after a 5-km run.
NEW
The aim of the study is to determine whether or not pulse rates increase after physical activity.
Before run
After run
7, 5, 2
5
8, 6, 4, 2, 1, 1, 0
6
8, 8, 8, 8, 6, 5, 3, 3, 1
7
3, 2, 1
8
0, 2, 5
8, 5
9
8, 6
9
10
0, 0, 1, 5, 6, 6, 8, 9
11
1, 2, 2, 6, 9
12
7, 8, 8
13
0, 1, 7
14
2 Key: 11|1 = 111 beats/min
(i) How many people had pulse rates of 100 or more beats per minute after the run?
(ii) How many people had pulse rates of 100 or more beats per minute before the run?
(iii) What conclusions can you draw from the stem-and-leaf diagram?
10. The following table shows the heights (in centimetres) of a group of men and a group of women.
NEW
The samples have been randomly selected. It is hoped to show that, in general, men are taller
than women.
Men
179, 183, 181, 186, 185, 175, 191, 171, 174, 176, 179, 184, 159, 160, 166,
170, 178, 175, 170, 161, 168, 174, 183
Women
157, 155, 148, 171, 151, 157, 167, 162, 174, 166, 165, 149, 169, 178, 158,
154, 153, 152, 155, 150, 161, 158, 163
(i) Copy and complete the back-to-back stem-and-leaf diagram to compare their heights:
Men
Stem
Women
14
15
16
17
18
19
(ii) What conclusion can be drawn from this diagram?
1
4
Learning Outcomes
ÂÂ Discuss populations and samples
ÂÂ Select a sample (simple random sample)
ÂÂ Recognise the importance of representativeness so as to avoid
biased samples
ÂÂ Recognise how sampling variability influences the use of sample
information to make statements about the population
(  )
ÂÂ Recognise the concept of a hypothesis test
1
___
   ​  ​for a population proportion
ÂÂ Calculate the margin of error ​ ​ ____
​ n ​ 
√
ÂÂ Conduct a hypothesis test on a population proportion using the
margin of error
Statistics II
4.3A Sam pling
Obtaining information from part of a population is called sampling. Collecting data from every member
of a population is not always practical as it might require too much time and effort, or may not even be
possible. So sampling is necessary. For example, a company producing batteries for TV remote controls
wants to find out the average lifetime of its batteries. If it tested the entire population, there would be no
batteries left to sell! So it needs to test a sample.
ce ns us
Biased Samples
Biased samples are samples that do not represent the population from which they are taken. A sample
that is not representative of the population is of no use. Such a sample should never be used in any
statistical investigation.
2
4
Surveys conducted through the Internet, telephone call-in surveys to radio stations and television
programmes, or mail-in surveys generate biased samples. This is because they produce ‘self-selected’
samples – people take part in the surveys on their own initiative rather than being selected by a researcher.
Generally speaking, people with strong opinions are more likely to take part in such surveys. People who
do not have strong opinions are less likely to respond, hence the sample is biased.
How can we eliminate bias? Bias can be eliminated by taking a random sample. Every member of the
population has an equal chance of being included in a random sample.
sa mp lin g fra me
Consider a classroom with 60 students arranged in six rows of 10 students each. A teacher selects
a sample of 10 students by rolling a die and selecting the row corresponding to the outcome. Every
student has an equal chance of being in the sample, but not every possible group of 10 has a chance
of being in the sample. For example, a student from Row 1 and a student from Row 6 can never be
together in a sample. Therefore, this is not a simple random sample.
Wo rke d E x a m p l e 4 . 7 A
Solution
Here is a random number table:
19
53
84
38
63
31
56
92
9
93
92
11
93
90
15
27
11
66
46
6
54
65
93
9
(i)As the population size is 50, John will
accept only numbers between 1 and 50.
The first number, 51, has to be discarded,
as it is greater than 50. So the first number
John could use is 19. The next two
numbers, 53 and 84, have to be discarded,
as they are greater than 50, so the second
number of use is 38. Continuing in this
fashion, we obtain the following list of
numbers: 19, 38, 9, 11, 31.
(ii)John could sort the list of 50 names in
alphabetical order and designate the
first person on the list as number 1, the
second person as number 2, and so on.
The people assigned the numbers 19, 38,
9, 11 and 31 will make up the sample.
John wants to take a random sample of five people
from a list of 50 people. He starts from the top left
of the table and works across the table row by row.
(i)Write down the set of numbers John
could use.
(ii)Explain how John could use these
numbers to get his sample.
Statistics II
51
3
4
Sampling Variability
Suppose the mean height of all 16-year-old males in the country is 176 cm. (In a real problem, of
course, you would never know exactly this value.) Then, suppose you take a random sample from this
population and compute the mean height of the sample. Will it be exactly 176 cm? Probably not. Most
likely it will be close to 176 cm, especially if the sample size is large. If another sample is taken, will the
mean height of the new sample be exactly the same as the mean of the original sample? Again, probably
not. The difference between the mean height of a sample and the mean of the population is known as
the sampling error. In practice we never know the sampling error, as we never know the mean of the
population. (After all, we are looking for an estimate of the mean of the population.)
Reducing the amount of sampling error helps us make more accurate generalisations about the whole population when using sample data.
One way in which this can often be done is by increasing the sample size.
ACTIVITY 4X
Statistics II
Wo rke d E x a m p l e 4 . 7 B
4
A group of five boys measured each other’s heights in
centimetres. The results are: 168, 175, 178, 182, 190.
(i)Calculate the mean height of the five
boys.
The boys are investigating the concept of sampling
variability. They decide to select all possible
samples of size 3 from the five measurements.
Their results are shown in the table below:
(ii)Calculate each sample mean height
(correct to two decimal places).
(iii)Calculate the mean value of the means.
What do you notice?
(iv)What is the probability of selecting a
random sample of size 3 whose mean is
within 1 cm of the true mean?
(v)What is the probability of selecting a
random sample of size 3 whose mean is
greater than 1 cm but less than 2 cm from
the true mean?
Sample 1
Sample 2
Sample 3
Sample 4
Sample 5
Sample 6
Sample 7
Sample 8
Sample 9
Sample 10
168
168
168
168
168
168
175
175
175
178
175
175
175
178
178
182
178
178
182
182
178
182
190
182
190
190
182
190
190
190
4
Solution
168 + 175 + 178 + 182 + 190
 ​
(i)Mean = ____________________________
​ 
   
   = 178.6 cm
5
(ii)
Sample 1
Sample 2
Sample 3
Sample 4
Sample 5
Sample 6
Sample 7
Sample 8
Sample 9
Sample 10
168
168
168
168
168
168
175
175
175
178
175
175
175
178
178
182
178
178
182
182
178
182
190
182
190
190
182
190
190
190
Mean (cm)
173.67
175
177.67
176
178.67
180
178.33
181
182.33
183.33
173.67 + 175 + 177.67 + 176 + 178.67 + 180 + 178.33 + 181 + 182.33 + 183.33
(iii) Mean = ___________________________________________________________________________
​ 
        
 
    ​
10
= 178.6 cm
Therefore, the mean of the means is equal to the mean of the population.
(iv)There are three samples whose mean is within 1 cm of the true mean (Samples 3, 5 and 7).
3
Therefore, the probability of selecting a sample within 1 cm of the true mean is __
​ 10
  ​ = 0.3.
(v) There is just one sample (Sample 6) whose mean is greater than 1 cm but less than 2 cm from the
true mean. Therefore, the probability of selecting a sample whose mean is greater than 1 cm but
1
less than 2 cm from the true mean is __
​ 10
   ​= 0.1.
Exercise 4.5 A
1.A shop sells shoes. Use the most appropriate words from the list below to complete the sentences.
(i)The number of pairs of shoes sold by the shop is
(ii)The colour of the shoes is
data.
(iii)The weight of the shoes is
.
(iv)A selection of shoes chosen from the back of the shop is called a
Statistics II
sample ordered discrete numerical continuous categorical nominal
data.
.
2.The manager of a factory wants to carry out a survey to find out the workers’ views on the sale of
overalls at the factory. There are 1,800 workers in the factory.
(i)Give two reasons why the manager would take a sample rather than a census to find out the
workers' views.
(ii)The manager has decided to use a questionnaire to find out the workers’ views on the sale of
overalls at the factory. One of the questions on the questionnaire is shown below:
Give one reason why you support the idea of having overalls sold at the factory.
Write down one criticism of this question.
5
4
(iii)The manager’s assistant distributes the questionnaire to the first 30 workers to arrive at the
factory on a particular morning. Will the assistant’s sample be representative of all workers, or
will it be biased? Explain your answer in detail.
3.Dublin Bus wishes to obtain opinions on the quality of its service.
(i)Explain why a sample of its customers should be taken, rather than a census.
(ii)The general manager of Dublin Bus suggests that everyone on the 46A bus on Monday morning
should be surveyed. Give two possible sources of bias in this sample.
(iii)An afternoon radio programme asks listeners to call in with their views on the quality of the
Dublin Bus company’s services. Why might the views of those calling into the programme not
be representative of all Dublin Bus customers? Give two possible reasons.
(iv)Describe a suitable way in which Dublin Bus might select a representative sample of its customers.
4.Eoin wants to choose a sample of size 8 from his class. He writes down all the students’ names on
pieces of paper, puts all the boys’ names in one bag and all the girls’ names in another bag. He then
chooses, without looking, four pieces of paper from each bag.
(i) Explain why Eoin's sample is random.
(ii)Explain why Eoin’s sample is not a simple random sample.
(iii)Explain how Eoin could obtain a simple random sample of eight students from his class.
Statistics II
5. Here is an extract from a table of random numbers.
86
13
84
10
7
30
39
5
97
96
88
7
37
26
4
89
13
48
19
20
60
78
48
12
99
47
9
46
91
33
17
21
3
94
79
0
8
50
40
16
78
48
6
37
82
26
1
6
64
65
94
41
17
26
74
66
61
93
14
97
(i)Starting from the first line and the fifth column with the number 7, and reading across the table,
from left to right, write down ten random numbers betweeen 0 and 69.
(ii)Explain how you could use the above table of random numbers to select a sample of
12 students from 80 students.
6.Thomas is investigating if distance from a polling station affects whether people usually vote in his
town. He decides to select a simple random sample of 50 people from the register of electors, use
the telephone directory to find the telephone numbers of the people in the sample and then carry
out a telephone survey.
(i) Define the population for this survey.
(ii)What obstacles could Thomas encounter in carrying out this survey?
(iii)One question he uses is, ‘What is the distance from your house to the polling station?’
Give one criticism of this question.
7.Bob wants to find out what is the most popular PlayStation game in his school. He randomly selects a
group of First Year boys and asks them to name their favourite PlayStation game. Using his results, he
concludes that Call of Duty is the most popular game in the school.
6
(i) Define the population for this survey.
(ii)Why is the sample that Bob uses non-representative of the school population?
(iii)Describe an improved way in which Bob could select a suitable sample for his study.
4
8.A researcher determines that she needs results from at least 400 subjects to conduct a study.
To compensate for low return rates, she mails the survey to 5,000 subjects. She receives 750
responses. Is the sample of 750 responses a good sample for her to use? Explain.
9. Give one example of each of the following:
(i) A non-random sample
(ii) A simple random sample
(iii)A random sample that is not a simple random sample
10.Using stopwatches, a group of five girls asked each other to estimate 80 seconds. The results
(in seconds) are: 68, 75, 78, 82, 87.
The girls are investigating the concept of sampling variability. They decide to select all possible
samples of size 3 from the five estimates. Their results are shown in the table below:
Sample 1
68
75
78
Sample 2
68
75
82
Sample 3
68
75
87
Sample 4
68
78
82
Sample 5
68
78
87
Sample 6
68
82
87
Sample 7
75
78
82
Sample 8
75
78
87
Sample 9
75
82
87
Sample 10
78
82
87
(i)Calculate the mean estimated time for the five girls.
(ii)Calculate the mean estimated time of each sample.
(iii)Calculate the mean value of the means. What do you notice?
(iv)What is the probability of selecting a random sample of size 3 whose mean is within 1 second
of the true mean?
(v)What is the probability of selecting a random sample of size 3 whose mean is greater than
1 second but less than 2 seconds from the true mean?
Statistics II
11.A company making light bulbs advertises that less than 1% of its bulbs have defects. To guarantee
this claim the company tests a percentage of bulbs produced every day.
There are three different testing methods currently in use to check daily production:
Method 1: Test every 400th bulb produced.
Method 2: Test 200 randomly selected bulbs at the end of the day. A computer program selects the
bulbs at random by batch number. Each bulb has a unique batch number.
Method 3: Test every bulb produced between 12:59 and 13:00 that day.
7
4
(i) Complete the table below:
Method 1
Method 2
Number of bulbs tested
Number of defects
292
3
Proportion of defects
Statistics II
Method 3
Total daily production:
4
2
____
​     ​ (= 1%)
200
140,000
(ii)Does any method of testing provide some evidence that contradicts the company’s claim?
Say which method(s) and explain the evidence.
(iii)Which of the three methods is the only method that involves simple random sampling?
Give reasons for your answer.
(iv)In your opinion, which method is most likely to generate a biased sample of daily production?
Explain your answer.
(v)Excluding the method you outlined in part (iv) above, which of the remaining two methods
would you choose as the most suitable method of testing to use? Why?
(vi)How could you adjust the method of testing you outlined in part (v) above so that the
company’s claim could be accepted/rejected with more certainty?
(vii) What disadvantages could such an adjustment lead to?
4.7 Margin of Error and Hypot hesis
Testing
In this section, we begin working with inferential statistics, as we use sample data to make inferences
(draw conclusions) about populations.
The two main applications of inferential statistics that we will study involve the use of sample data to:
(1) Estimate the value of a population proportion
(2)Test some claim (hypothesis) about a population proportion
To begin, you will need to know what statisticians mean by a population proportion and a sample
proportion. Suppose you want to know the percentage of Leaving Certificate students in Ireland who
will go on to third-level education next year. To find out, you decide to survey your class and use the
result to infer the percentage of students for the whole country. (Not a great sample! Why?)
Suppose the survey reveals
. that 25 out of the 30 students in your class are going on to third-level education
25
 
 
​
or
0.83
is the sample proportion. We use the symbol p̂ to denote the sample proportion.
next year. Then ​ ___
30
The population proportion is the proportion of the whole population (in this case, this year’s Leaving
Certificate students) who will go on to third-level education. We use p to denote the population
proportion.
We can never find the exact value of p from p̂. We can only estimate p.
8
4
Margin of Error
To understand the margin of error, we need to know about confidence intervals.
Out of 50 randomly selected students in an all girls school, 15 said they liked rock music, i.e. the sample
15
proportion p̂ is ​ ___
50  ​= 0.3 The school population is 500. Using only information from the sample, can we
give the exact proportion of girls in the school who like rock music? The answer is no, but we can give a
range within which we can state, with a certain degree of confidence, the proportion of students who like
rock music lie.
A statistician might say, ‘I can say with a confidence level of 95% that the proportion of girls in this school
who like rock music lies in the interval 0.1586 < p < 0.4414.’ This interval is called a confidence interval.
How does the statistician determine the interval and what does he mean by ‘a confidence level of 95%’?
First, we calculate the margin of error using the formula shown, where n is the size of the sample.
There are other ways of calculating the margin of error, but we will use this formula on our course.
FO RM UL A
1
___
In this example, n = 50, so E = ​ ____
   ​ = 0.1414.
Margin of error
1
___
  ​ 
E = ​ ​√__
n ​ 
​√ 50 ​ 
This means that the population proportion p will differ from the sample proportion p̂ by at most 0.1414,
95% of the time. In other words, if we were to take 100 random samples of size 50 from any population,
95 of the samples would have proportions that would differ from the population proportion by at most
0.1414. When we use this formula, our level of confidence is always 95%.
The confidence interval for the proportion is: FO RM UL A
1
___
1__
___
p̂ – ​ ​√__n   ​ ​ < p < p̂ + ​ ​√n   ​​  
Statistics II
Wo rke d E x a m p l e 4 . 1 4
A company wishes to estimate the proportion, p, of its employees who went on sick leave during the
past year. A random sample of 20 employees was taken. Nine of the sample went on sick leave during
the past year. Construct a 95% confidence interval for p.
Solution
Step 1
Step 2
Calculate the sample proportion, p̂. 9
p̂ = ​ ___  ​ = 0.45
20
Find E, the margin of error.
1
E = ____
​  ___
   ​ = 0.2236
​ 20 ​ 
√
Step 3 Construct the confidence interval.
0.45 – 0.2236 < p < 0.45 + 0.2236
0.2264 < p < 0.6736
9
4
The margin of error in Worked Example 4.14 is quite big, 0.2236 or 22.36%. Margins of error that are
this big are of little use. However, we can quite easily reduce the margin of error, simply by increasing
the size of our sample.
Wo rke d E x a m p l e 4 . 1 5
What size sample is required to have a margin of error of 0.04 or 4%?
Solution
Let n be the sample size.
1__
​ ___
  ​ = 0.04
​ n ​ 
√
1
​ __
n ​ = 0.0016
0.0016n = 1
1
n = ​ _______
   ​ 
0.0016
n = 625
Hypothesis Testing
Hypothesis testing is an important aspect of statistics.
hy po the sis te st
The following statement is an example of an hypothesis about a population: ‘65% of all heavy smokers will
contract a serious lung or heart ailment before the age of 60.’ To accept or reject this statement, you must
collect a random sample of medical records of heavy smokers under the age of 60 and find the proportion
of the sample who have contracted serious heart or lung ailments. Then you must set up a hypothesis test
to prove or disprove the claim. In our course, we will only test claims about a population proportion.
Statistics II
Procedure for hypothesis testing for a population proportion
Step 1State clearly the null hypothesis, H​0​,
and the alternative hypothesis, H
​ ​ 1​.
nu ll hy po the sis
For example, ‘65% of all heavy smokers
will contract a serious lung or heart ailment
before the age of 60.’ Usually you are hoping to show that the null hypothesis is not true and
so the alternative hypothesis, H
​ ​ 1​, is often the hypothesis you want to establish. In the example
just given, the alternative hypothesis, H
​ ​ 1​, is that ‘65% of all heavy smokers will not contract a
serious lung or heart ailment before the age of 60.’
Step 2 Calculate p̂, the sample proportion.
Step 3 Set up a confidence interval for p, the population proportion.
„„ If
the proportion for the population stated in the null hypothesis is within the confidence
interval, then accept H
​ ​ 0​, the null hypothesis.
„„ If
the population proportion is outside the confidence interval, then reject the null hypothesis
and accept ​H​ 1​.
10
4
Wo rke d E x a m p l e 4 . 1 6
Irish Express Newspapers surveyed 800 randomly selected voters to find out whether or not they would
vote for the government parties in the next election. Of the sample, 40% indicated that they would vote
for the government in the next election. A month later, a rival newspaper surveyed 900 voters and 315
said they would support the government. Had support for the government changed during this month?
Solution
Step 1​H​ 0​– Government support has remained
unchanged.
​H​ 1​– Government support has changed.
Step 2Calculate sample proportions for both
samples.
315
Sample proportion p̂1 = ____
​   ​ = 0.35
900
Sample proportion p̂2 = 0.4
1
≈ 0.03
Step 3 Margin of error E1 = _____
​  ____
   ​ 
​ 900 ​ 
√
1
Margin of error E2 = _____
≈ 0.04
​  ____
   ​  
​ 800 ​ 
√
Step 4Construct confidence intervals.
Confidence Interval (1) = 0.32 < p1 < 0.38
Confidence Interval (2) = 0.36 < p2 < 0.44
As there is an overlap between Confidence
Interval (1) and Confidence Interval (2), we
fail to reject the null hypothesis and accept
that support for the government has not
changed.
Wo rke d E x a m p l e 4 . 1 7
A local newspaper is investigating a claim made by the CEO of a large multinational company. The CEO
claimed that 80% of the company’s 500,000 customers are satisfied with the service they receive. Using
simple random sampling, the newspaper surveyed 200 customers. Among the sampled customers, 146
said they were satisfied with the company’s service. Based on these findings, can we reject the CEO’s
claim that 80% of customers are satisfied with the company’s service?
Solution
Step 1 H0 – The company’s satisfaction rating is 80%.
H1 – The company’s satisfaction rating is not 80%.
146
Step 2 Sample proportion p̂ = ​ ____ ​ = 0.73 or 73%.
200
1
_____
Step 3 Margin of error E = ​  ____
≈ 0.071.
   ​  
​ 200 ​ 
√
Step 4 CI for the population proportion is 0.73 – 0.071 < p < 0.73 +.071.
CI: 0.659 < p < 0.801
Statistics II
The population proportion is within the confidence interval. Therefore, we accept the null hypothesis that
the satisfaction rating is 80%, and hence, we are not rejecting the CEO’s claim.
E xercise 4.8
1. A manufacturer of computer components wishes to estimate the proportion, p, of its present stock of
components that are defective. A random sample of 500 components is selected, and 10 are found to
be defective.
Construct a 95% confidence interval for p, the proportion of all components that are defective.
11
4
2. A bank randomly selected 120 customers
with savings accounts and found that 90 of
them also had cheque accounts.
If the maximum margin of error the company
will accept for the proportion is 4%, then find
the necessary sample size at the 95% level.
Construct a 95% confidence interval for the
true proportion of savings-account customers
who also have cheque accounts.
6. In a study of 10,000 car crashes, it was
found that 5,550 of them occurred within
10 kilometres of the driver’s home. Test the
hypothesis that 50% of car crashes occur
within 10 kilometres of home.
3. In a survey of 1,000 people, 740 said that
they voted in a recent general election. Voting
records show that 71% of eligible voters
actually did vote.
7. A coin is tossed 100 times and heads occur
57 times. Test the hypothesis that the coin is
biased.
(i)Find a 95% confidence interval estimate
of the proportion of people who say they
voted.
8. A six-sided die is thrown 180 times and a 6
occurs 40 times. Can we conclude that the
die is biased?
(ii)Are the survey results consistent with the
actual voter turnout? Explain.
9. A pharmaceutical company is replacing one
of its pain-killing drugs with another drug
that the company has developed and tested.
The company’s records show that the old
drug provided relief for 70% of all patients
who were administered it. A random sample
of 625 patients was administered the new
drug, and 450 of these claimed that the new
drug provided relief.
4. In a recent poll, 1,000 randomly selected
adults were surveyed and 27% of them said
that they use the Internet for shopping at least
once a year. Construct a 95% confidence
interval for the true proportion of adults who
shop on the Internet.
5. A company wishes to estimate the proportion
of its employees who would accept an extra
week of holidays instead of the annual rise in
salary.
Test the hypothesis that the new drug is
different to its old counterpart.
10. The World Health Organisation percentile chart below gives the heights or lengths of baby girls aged
between zero and two years.
Statistics II
Length-for-age GIRLS
Birth to 2 years (percentiles)
95
95
97th
90
85th 90
50th
85
85
15th
3rd
Length (cm)
80
80
75
75
70
70
65
65
60
60
55
55
50
50
45
Months
1
Birth
2
3
4
5
6
7
8
9
10
11
1 year
1
2
3
4
5
6
7
8
9
10
11
45
2 years
Age (completed months and years)
WHO Child Growth Standards
12
4
(i)Copy and complete the following table:
Age (months)
0
1
2
3
4
5
6
7
8
9
Median height (cm)
(ii)If a baby girl aged two months has a height of 57 cm, what percentage of baby girls of the same
age are taller than her?
(iii)A baby girl aged one year has a height of 80 cm. Is this height unusual? Explain your reasoning.
(iv)Calculate the range of heights between the 3rd and 97th percentiles for a baby girl aged 11 months.
(v)Give one similarity and one difference between a length-for-age girls percentile chart and a
length-for-age boys percentile chart.
Revision Exercises – Extra Questions
NEW 12. A
group of five boys measured each other’s
heights in centimetres. The results are: 168,
175, 178, 182, 190.
T
he boys are investigating the concept of
sampling variability. They decide to select
all possible samples of size 3 from the five
measurements. Their results are shown in
the table below:
168
168
168
168
168
168
175
175
175
178
175
175
175
178
178
182
178
178
182
182
(i)Calculate the mean height of the five
boys.
(ii)Calculate each sample mean height
(correct to two decimal places).
(iii)Calculate the mean value of the means.
What do you notice?
178
182
190
182
190
190
182
190
190
190
(iv)What is the probability of selecting a
random sample of size 3 whose mean is
within 1 cm of the true mean?
(v)What is the probability of selecting a
random sample of size 3 whose mean
is greater than 1 cm but less than 2 cm
from the true mean?
13. Using stopwatches, a group of five girls asked
each other to estimate 80 seconds. The
results (in seconds) are: 68, 75, 78, 82, 87.
The girls are investigating the concept of
sampling variability. They decide to select
all possible samples of size 3 from the five
estimates. Their results are shown in the
table below:
Sample 1
68
75
78
Sample 2
68
75
82
Sample 3
68
75
87
Sample 4
68
78
82
Sample 5
68
78
87
Sample 6
68
82
87
Sample 7
75
78
82
Sample 8
75
78
87
Sample 9
75
82
87
Sample 10
78
82
87
(i)Calculate the mean estimated time for
the five girls.
(ii)Calculate the mean estimated time of
each sample.
(iii)Calculate the mean value of the
means. What do you notice?
(iv)What is the probability of selecting a
random sample of size 3 whose mean
is within 1 second of the true mean?
(v)What is the probability of selecting a
random sample of size 3 whose mean
is greater than 1 second but less than
2 seconds from the true mean?
Statistics II
Sample 1
Sample 2
Sample 3
Sample 4
Sample 5
Sample 6
Sample 7
Sample 8
Sample 9
Sample 10
NEW
13
4
Ac t i v i t y 4 X
One hundred discs are shown below.
1. Label the discs from 1 to 100.
Statistics II
2. Complete the table.
Colour
Red
Proportion
0.4
Blue
Green
Purple
3. What number do you get when you add up all four proportions? 4. Using the random number generator in your calculator, select 10 discs and record the proportion of red,
blue, green and purple discs in the sample. For example, if the calculator generates the number 67 and this
corresponds to a blue disc, then record the result in the ‘Blue’ column.
Collect six such samples. Note: No disc should be selected more than once within a single sample.
Sample 1
Colour
Tally
Proportion
14
Red
Blue
Green
Purple
4
Sample 2
Colour
Red
Blue
Green
Purple
Red
Blue
Green
Purple
Red
Blue
Green
Purple
Red
Blue
Green
Purple
Red
Blue
Green
Purple
Tally
Proportion
Sample 3
Colour
Tally
Proportion
Sample 4
Colour
Tally
Proportion
Sample 5
Colour
Tally
Proportion
Sample 6
Tally
Proportion
5. Do all six samples have the same proportion of red, blue, green and purple discs?
Statistics II
Colour
6. What term is used to describe this variation in the proportions in each sample?
15
4
7. Complete the table below to show the sampling error for the red discs for each sample.
Sample
1
2
3
4
5
6
Sample proportion
Sampling error
8. Using the random number generator in your calculator, select 20 discs and record the proportion of red, blue,
green and purple discs in the sample.
Collect six such samples. Note: No disc should be selected more than once within a single sample.
Sample 1
Colour
Red
Blue
Green
Purple
Red
Blue
Green
Purple
Red
Blue
Green
Purple
Tally
Proportion
Sample 2
Colour
Tally
Proportion
Statistics II
Sample 3
Colour
Tally
Proportion
Sample 4
Colour
Red
Blue
Green
Purple
Red
Blue
Green
Purple
Tally
Proportion
Sample 5
Colour
Tally
Proportion
16
4
Sample 6
Colour
Red
Blue
Green
Purple
Tally
Proportion
9. Complete the table below to show the sampling error for the red discs for each sample.
Sample
1
2
3
4
5
6
Sample proportion
Sampling error
10. What was the sample size for each sample in Question 4? 11. What was the sample size for each sample in Question 8? 12. By comparing your answers to Question 7 and Question 9, what conclusion can you reach about the
relationship between the sampling error for red discs and the sample size?
Statistics II
17
9
Learning Outcome
ÂÂ Recognise images of points and objects under rotations
Rotations
Another type of transformation is a rotation.
The amount the shape rotates is called the angle of rotation.
This is given either as an angle or as a fraction of a complete
turn, for example, 270° or _​ 34 ​turn.
The direction of rotation is given as clockwise (negative) or
anti-clockwise (positive).
The fixed point about which the object is rotated is called the point (centre) of rotation.
Therefore, when describing the rotation of an object, we should include, if possible:
(i) The centre of rotation
(ii) The angle of rotation
Transformations
and Enlargements
(iii) The direction of rotation (positive or negative)
Every point on this object has
been rotated anti-clockwise
through an angle of 60° about
the point P.
18
Positive (anti-clockwise) rotation
of 90° about the point A. This is
denoted as R90°.
Negative (clockwise) rotation
of 90° about the point A. This
is denoted as R–90°.
9
Exercise 9.1 – Extra Questions
NEW
NEW
4A. In each case, identify a rotation that maps:
(i) A onto B
(ii) C onto D
(iii) A onto D
(iv) A onto C
Make sure in your answer to include:
(i) The point of rotation
(ii) The angle of rotation
(iii) Whether the rotation is positive or negative
5A. In each question below, three images labelled A, B and C are the images of the object under a
transformation. The transformation could be a translation, an axial symmetry, a central symmetry
or a rotation. For each image, state which transformation is used, and in the case of a rotation,
state the angle and direction.
Object
A
B
C
(ii)
Object
A
B
C
(iii)
Object
A
B
C
Transformations
and Enlargements
(i)
19
Answers
Chapter 1
Exercise 1.2
9. (i) 20 (ii) 1 10. (ii) The tallest person is a man.
Chapter 4
Exercise 4.5A
1. (i) Discrete numerical (ii) Nominal categorical (iii) Continuous (iv) Sample 2. (i) Time (quicker), Economy (cheaper) (ii) Biased question (iii) Biased – not a random sample 3. (i) Time (quicker), Economy (cheaper), Practicality (not possible to
survey all customers) (ii) Only representing a single route; Only on a Monday (iii) Only represents customers who listen to
the radio; Only represents customers who are willing to speak on radio (iv) Take a sample from each of its bus routes each
day of the week at both peak and off-peak times. 4. (i) The selection of each student is not influenced by anything (all names
selected without looking). (ii) Every possible sample within the population cannot occur, e.g. we cannot get a sample with all
eight being girls. (iii) Put all the names (girls and boys) into a single hat and select eight. 6. (i) Registered voters who have
phones and are in the telephone directory (ii) People may not answer or may not be willing to take part in the survey. (iii) Answers are likely to be estimated and hence subject to bias and error. 7. (i) Male First Year students who play
PlayStation (ii) Only boys, only First Years were sampled. (iii) Take a sample from all the students in the school. 8. No 10. (i) 78 (seconds) (ii) Sample 1: 73.66, Sample 2: 75, Sample 3: 76.66, Sample 4: 76, Sample 5: 77.66, Sample 6: 79,
Sample 7: 78.33, Sample 8: 80, Sample 9: 81.33, Sample 10: 82.33 (iii) 78 (same as the mean estimated time for the 5 girls) (iv) 0.3 (v) 0.1 11. (ii) Method 3 suggests that 1.37% of bulbs have defects, which is greater than the company’s claim of < 1%. (iii) Method 2 is the only method that gives every possible sample of the population the same chance of being selected. (iv) Method 3: The sample only considers a very small part of the whole population and hence is biased. (v) Method 2 (vi) Increase the size of the sample taken (vii) Takes more time; Costs more money; Wastes produce (bulbs)
Exercise 4.8
1. –0.0247 < p < 0.0647 2. 0.6587 < p < 0.8413 3. (i) 0.7084 < p < 0.7716 (ii) Yes 4. 0.2384 < p < 0.3016
5. 625 6. Accept the alternative hypothesis that p ≠ 50% 7. Accept the null hypothesis that the coin is fair.
8. No, the die is fair. 9. The new drug is not different. 10. (ii) 50% (iii) Yes (iv) Range: 9.5 cm
Revision Exercises – Extra Questions
12. (i) 178.6 cm (ii) Sample 1: 173.67 cm; Sample 2: 175 cm; Sample 3: 177.67 cm; Sample 4: 176 cm; Sample 5: 178.67 cm;
Sample 6: 180 cm; Sample 7: 178.33 cm; Sample 8: 181 cm; Sample 9: 182.33 cm; Sample 10: 183.33 cm (iii) 178.6 cm;
The mean of the means is equal to the mean of the population. (iv) 0.3 (v) 0.1 13. (i) 78 s (ii) Sample 1: 73.66 s;
Sample 2: 75 s; Sample 3: 76.66 s; Sample 4: 76 s; Sample 5: 77.66 s; Sample 6: 79 s; Sample 7: 78.33 s; Sample 8: 80 s;
Sample 9: 81.33 s; Sample 10: 82.33 s (iii) 78 s (iv) 0.3 (v) 0.1
Chapter 9
Exercise 9.1 – Extra Questions
4A. (i) A onto B; Positive 90° rotation about the origin (ii) C onto D; Negative 90° rotation about the origin
(iii) A onto D; Positive/negative 180° rotation about the origin (iv) A onto C; Negative 90° rotation about the origin
5A. (i) A: Translation, B: Axial symmetry, C: Negative rotation of 90° (ii) A: Central symmetry, B: Negative rotation of 90°,
C: Positive rotation of 90° (iii) A: Positive rotation of 90°, B: Axial symmetry, C: central symmetry
20
leaving certificate ORDINARY LEVEL
Active Maths 3
2π
r
Book 2
2
πr
m
1
= y2 y
x
1
2 x
Supplementary Material
for 2015 exam and
onwards
ISBN 978-1-78090-529-7
2714
Michael Keating, Derek Mulvany and James O’Loughlin
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