MARCH 2022
ST. JOSEPH’S CONVENT, P.O.S.
APPLIED MATHEMATICS
MATHEMATICAL APPLICATIONS
UNIT 1 – PAPER 02
2 hours 30 minutes
READ THE FOLLOWING INSTRUCTIONS CAREFULLY
1. This paper consists of THREE sections. Each section consists of TWO questions.
2. Answer ALL questions.
3. Write your answers in the spaces provided in this booklet.
4. Do NOT write in the margins.
5. Unless otherwise stated in a question, all numerical answers MUST be given exactly
OR to three significant figures as appropriate.
6. If you need to rewrite any answer and there is not enough space to do so on the original
page, you must use the extra lined page(s) provided at the back of this booklet.
Remember to draw a line through your original answer.
7. If you use the extra page(s) you MUST write the question number clearly in the
box provided at the top of the extra page(s) and, where relevant, include the
question part beside the answer.
Examination Materials
Mathematical formulae and tables (Revised 2012)
Mathematical instruments
Silent, non-programmable electronic calculator
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO
SECTION A
MODULE 1: COLLECTING AND DESCRIBING DATA
Answer BOTH questions.
1. a.
State ONE similarity and ONE difference between a stratified random sample and a
quota sample.
[2]
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b.
A director wishes to select 20 contestants from among 500 applicants for a beauty
pageant. Sampling methods T, U, V and W are possible ways of selecting the sample of
20 contestants.
Name EACH of the following sampling methods described below.
(i)
T:
Select equal numbers of male and female contestants, even though there
are twice as many female applicants as make applicants.
[1]
………………………………………………………..…………………
(ii)`
U:
Group the applicants by age and select contestants at random from
within each group in proportion to the number of applicants who make
up that group.
[1]
………………………………………………………..…………………
(iii):
V:
Assign a unique number from 001 to 500 to each applicant and use
random numbers to select the contestants.
[1]
………………………………………………………..…………………
2
(iv)
W:
First, select a contestant randomly from among the first k applicants on a
numbered list; thereafter select every kth applicant on the list.
[1]
………………………………………………………..…………………
(v)
From the sampling methods described above, identify (by letter), the sampling
method which would result in a sample that is MOST representative of the 500
applicants.
[1]
………………………………………………………..…………………
(vi)
If the sampling method U is used, 4 of the 20 contestants selected from the 500
applicants would be over the age of 50 years. Determine the number of
contestants who are older than 50 years of age.
[2]
(vii)
Using Table 4: Random Sampling Numbers provided in the List of Formulae
and statistical Tables, start at the first (1st) row and third (3rd) column with 36,
and move along the row from the left to right. List the FIVE numbers in the
order in which they would be selected, if sampling method V is used to select
the first 5 of the 20 contestants.
[3]
(viii)
Calculate the value of k in the sampling method W.
3
[2]
c.
The following table shows the results of a survey of the time taken (in minutes) by a
group of children to complete a reading assignment.
(i)
(ii)
Complete the cumulative frequency row in the table above.
[1]
On the grid provided on page 5, draw a cumulative frequency curve to represent
the data in the table above. Use a scale of 2 cm to represent 10 minutes on the
x-axis and 2 cm to represent 10 children on the y-axis.
[4]
Using the cumulative frequency curve drawn in (c) (ii) on page 5,
(iii)
Estimate the median time taken to complete a reading assignment.
[1]
………………………………………………………..…………...………………
(iv)
Estimate the interquartile range.
[3]
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(v)
Estimate the proportion of children who took less than 35 minutes to complete a
reading assignment.
[2]
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Total 25 marks
4
5
2. a.
Identify the measure of variability described in EACH of the following statements.
(i)
It is influenced most by extreme values.
[1]
………………………………………………………..…………...………………
(ii)
It may be obtained by squaring the standard deviation.
[1]
………………………………………………………..…………...………………
(iii)
It is the middle 50 percent of the distribution.
[1]
………………………………………………………..…………...………………
b.
The data below shows the distribution of scores obtained by 20 candidates in an
examination.
(i)
Calculate the 10% trimmed mean.
[4]
(ii)
Determine the modal score(s)
[1]
………………………………………………………..…………...………………
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(iii)
Calculate the interquartile range.
[3]
(iv)
Calculate the standard deviation of the scores.
[4]
(vi)
Interpret the value of the standard deviation calculated in (b) (iv).
[2]
………………………………………………………..…………...………………
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c.
The following table shows the frequency distribution of the number of eggs per tray, 𝑥,
in a random sample of 50 trays.
Number of Eggs
Per Tray (𝑥)
Frequency (𝑓)
(i)
12
13
14
15
16
4
10
20
11
5
Determine the median number of eggs per tray.
[4]
………………………………………………………..…………...………………
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(ii)
Determine the mode of the distribution.
[1]
………………………………………………………..…………...………………
(iii)
Calculate the mean number of eggs per tray.
[4]
(iv)
Describe the shape of the distribution.
[1]
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Total 25 marks
8
SECTION B
MODULE 2: MANAGING UNCERTAINTY
Answer BOTH questions.
3. a.
Event 𝐴 and Event 𝐵 are such that 𝑃(𝐴) = ¼ and 𝑃(𝐴|𝐵) = 5/8 and
𝑃(𝐵|𝐴) = 3/5.
Calculate the following probabilities:
(i)
𝑃(𝐴 ∩ 𝐵)
[3]
(ii)
𝑃(𝐵)
[3]
(iii)
𝑃(𝐴 ∪ 𝐵)
[2]
(iv)
State whether Event 𝐴 and Event 𝐵 are independent. Give ONE reason for your
answer.
[2]
………………………………………………………..…………...………………
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b.
The discrete random variable 𝑋 has the probability distribution shown in the table
below.
1
2
3
4
5
𝑋
0.15
0.25
0.2
0.15
𝑃(𝑋 = 𝑥)
𝑏
(i)
Calculate the value of the constant 𝑏.
[3]
(ii)
Determine the expected value of 𝑋.
[3]
(iii)
Determine the variance of 𝑋.
[3]
10
c.
A bag contains 5 blue counters and 7 yellow counters. Two counters are drawn from
the bag, one after the other, without replacement.
(i)
Draw a probability tree diagram to represent the scenario above.
[4]
(ii)
Use the diagram drawn in c. (i) to determine the probability that the counters
drawn are different colours.
[2]
Total 25 marks
11
4. a.
A credit union found that 25% of its customers do not make their monthly payments on
time. A random sample of 20 customers is taken.
(i)
Identify the distribution that may be used to model this situation.
[1]
………………………………………………………..…………...………………
(ii)
State the parameters of the distribution.
[1]
………………………………………………………..…………...………………
(iii)
Determine the expected number of customers in the sample who do not make
their monthly payments on time.
[2]
(iv)
Calculate the probability that exactly 5 customers do not make their monthly
payments on time.
[2]
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(v)
Calculate the probability that at most 3 customers do not make their monthly
payments on time.
[4]
(vi)
A new director was appointed and decided that a larger random sample of 100
customers would give better results. Use a suitable approximation to determine
the probability that no more than 35 customers make their monthly payments on
time. State all your assumptions.
[7]
13
b.
The length of a javelin is normally distributed with a mean of 240 cm and a variance of
64 cm2.
(i)
Determine the probability that a javelin selected at random will measure more
than 255 cm.
[4]
(ii)
Javelins that are less than 220 cm are cut and sold as fence posts. Calculate the
percentage of javelins that will have to be cut and sold as fence posts.
[4]
Total 25 marks
14
SECTION C
MODULE 1: ANALYZING AND INTERPRETING DATA
Answer BOTH questions.
5. a.
The weight of bags of apples follows a normal distribution with mean 𝜇 and an
unknown variance 𝜎2. A random sample of 150 bags of apples gave ∑ 𝑥 = 1600 and
∑ 𝑥 ! = 18 040.
(i)
Calculate unbiased estimates for the mean and standard deviation of 𝑋.
[5]
(ii)
State an appropriate distribution for the sample mean, 𝑋;.
[2]
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(iii)
Determine an approximate 95% confidence interval for the unknown population
mean, 𝜇.
[3]
An inspector wishes to test whether 𝜇 = 11.0 at the 10% significance level.
(iv)
Formulate the null and alternative hypotheses.
[2]
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(v)
Determine the critical value.
[1]
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(vi)
Determine the rejection region.
[1]
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(vii)
Given that the test statistic is -1.59, state clearly a valid conclusion for the test.
[2]
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b.
A simple random sample of 14 people, taken from a certain normal population, resulted
in body mass indices (BMI) shown in the following table.
Person 1
BMI
23
2
25
3
21
4
37
5
39
6
21
7
23
8
24
9
32
10
57
11
23
12
26
13
31
14
45
A test was carried out at the 5% significance level to determine whether the average
BMI of the population is less than 35.
The null and alternative hypotheses are
𝐻" : 𝜇 = 35
𝐻# : 𝜇 < 35
(i)
Identify the distribution that can be used to model this situation. Give ONE
reason for the distribution identified.
[2]
………………………………………………………..…………...………………
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(ii)
Determine the critical value.
[2]
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(iii)
Determine the rejection region.
[1]
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………………………………………………………..…………...………………
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(iv)
Given that the mean is 30.5 and the standard deviation is 10.6, calculate the test
statistic.
[2]
(v)
State clearly a valid conclusion for the test.
[2]
………………………………………………………..…………...………………
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Total 25 marks
18
6. a.
A sociologist selected at random, 140 leaders of firms to investigate the effect of
different leadership styles on the satisfaction of employed. As assessment was made of
leadership styles (strict or moderate) and worder satisfaction levels (low, acceptable, or
high). The test yielded the following results.
Leadership Styles
Worker Satisfaction Levels
Strict
Moderate
Low
30
15
Acceptable
24
36
High
18
17
A 𝜒2 test is carried out to determine whether leadership styles affect the satisfaction
level of workers.
(i)
Formulate the null and alternative hypotheses for this test.
[2]
………………………………………………………..…………...………………
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(ii)
Complete the following table to show the missing expected and total
frequencies.
[4]
Leadership Styles
Strict
Moderate
Worker
Satisfaction
Levels
O
Low
30
15
Acceptable
24
36
High
18
17
Total
72
68
E
19
O
E
Total
(iii)
Calculate the 𝜒2 test value.
[3]
(iv)
Determine the critical region of the test at the 5% level of significance.
[3]
(v)
State clearly the conclusion that may be drawn from this test.
[2]
………………………………………………………..…………...………………
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b.
A random sample of 384 provided a 95% confidence interval for a population
proportion, 𝑝, with a margin of error of 0.05. Approximate the value of 𝑝 based on this
information.
[4]
20
c.
A random sample from a population with mean 𝜇 and standard deviation 𝜎 yielded the
values 16.0, 14.2, 14.4, 14.3, 14.6.
(i)
Calculate the sample mean and sample variance for these observations.
[3]
(ii)
Determine the critical region for a test of the null hypothesis 𝐻" : 𝜇 = 14.0
against the alternative hypothesis 𝐻# : 𝜇 ≠ 14.0 at the 5% significance level. [4]
Total 25 marks
END OF TEST
21
IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.
EXTRA SPACE
If you use this extra page, you MUST write the question number clearly in the box provided.
Question No.
22
EXTRA SPACE
If you use this extra page, you MUST write the question number clearly in the box provided.
Question No.
23