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Semester test 1 memorandum provides answers to semester
test 1 question paper to assess the level of knowledge for
students
QUANTITATIVE TECHNIQUES I (Tshwane University of Technology)
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TSHWANE UNIVERSITY OF TECHNOLOGY
Semester test 1: Memorandum
QTS105D/OMT105D
TOTAL MARKS: 55
FULL MARKS: 50
Surname……………………… Initials…………..
Student number………………………..
Date: ………………..2020
EXAMINER: Mr KM Mampana
MODERATOR: Mr P Seragatlala
Please mark your lecturer:
KM
Mampana
What is your
T Maunye
KH Mnisi
P Masibigiri
P Seragatlala
Group?
Instruction:






Answer all questions
Round the answers to two(2) decimal place
Do not use a pencil on answers
Do not use a tipp-ex on answers
Only non-programmable calculators are allowed
Make a cross underneath the name of your lecturer
1
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Question 1
[6]
Certain company of 1000 workers, want to find out how their workers are saving for retirement.
They grouped the workers according to the age as follows:
20 - 29 years =
160 workers
30 – 39 years =
40 – 49 years =
220 workers
240 workers
50 - 59 years =
200 workers
60 + years =
180 workers
Use stratified sampling method to sample 50 workers from the 1000 workers. (6)
𝑛1 =
𝑛2 =
𝑛3 =
𝑛4 =
𝑛5 =
160
1000
220
1000
240
1000
200
1000
180
50 = 8√
50 = 11√
50 = 12√
50 = 10√
1000
50 = 9√
𝑛 = 8 + 11 + 12 + 10 + 9 = 50√
Question 2
[5]
For each of the following random variables, indicate the data type with an X
Type of data
The miles joggers run per week
The starting salaries of graduates
of MBA
The months in which a firm’s
employees choose to take their
vacations
The final letter grades received by
students in a statistics course
The size of soft drink (small,
medium or large )
Nominal
Qualitative
Ordinal Interval
Quantitative
Ratio Discrete Continuous
X
X
X
X
X
2
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Question 3
[8]
The following information represents the weight (in kg) of 10 first year students.
10
11
9
Arranged data: 3
4
12
5
14
7
3
9
5
9
10
9
11
7
12
12
12
14
15
4
15
If a frequency table must be constructed for the above data, answer the below questions:
3.1
Determine the data range
[1]
R = 15 – 3 = 12 √
3.2
If the data must be grouped into a frequency table, calculate the number of class
intervals (round your answer to the next integer)
[2]
3.3
k = 1+3.3log (12) √ = 4.6≈ 5 √
Calculate the class width (round your answer to the next integer
[2]
3.4
c = 12/5 √= 2.4 ≈ 3√
[2]
Determine the suitable first class interval
[3; 6)√√
3.5
Determine the possible frequency of the first class is:
[1]
[3; 6) = 3 √
Question 4
[5]
The table below shows the information about 500 young female readers and their magazine
preference
Magazine
Number of young female
readers
95
146
118
55
Sowetan
Daily Sun
Pretoria News
Junk-mail
Business News
86
3
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4.1.
Draw a bar chart.
[5]
Bar chart of magazines vs reader
No. of young famale reader
160
140
120
100
80
60
40
20
0
Sowetan
Daily Sun
Pretoria News
Junk-mail
Business News
Magazines
Heading √ x-axis√
plot√ √
y-axis√
Question 5
[8]
The demand for a product on each of 15 randomly selected days was as follows ( in units)
3
12
7
17
3
1
4
19
7
15
14
7
6
11
10
First ordered the data from the smallest to the largest
1
3
3
4
6
7
7
7
10
11
12
14
15
17
19
For the following questions, indicate the correct answer with an X.
5.1 Mean
A. 5.48
[2]
B. 9.07
C. 82.20
D. 19
E. NONE
5.2 Mode
A. 3
[2]
B. 19
C. 7
D. 1
5.3 Variance
E. NONE
[2]
4
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A. 5.48
B. 5.29
C. 28.06
D. 30.07
E. NONE
5.4 Median
[2]
A. 7
B. 6
C. 19
D. 1
E. NONE
Question 6
[9]
The following table represent the time taken (in minutes) by Statistics students to complete a
multiple choice question comprising of 5 questions.
Time taken
Frequency
(in seconds)
𝑓
𝑓𝑥
𝑓𝑥 2
𝑥
7.5
7.5
56.25
1
[5;10)
1
[10;15)
4
12.5
50
625
5
[15;20)
6
17.5
105
1837.5
11
[20;25)
4
22.5
90
2025
15
[25;30)
2
27.5
55
1512.5
17
[30;35)
3
32.5
97.5
3168.75
20
Total
20
120
405
9225
𝐹
6.1 Determine the following:
6.1.1 Mean
𝑥̅ =
[2]
405
√= 20.25√
20
6.1.2 Standard deviation
[3]
1
(405)2
9225
−
√
20
𝑠=
√
20 − 1
5
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= √53.88√= 7.34√
6.1.3 Median
𝐹≥
20
= 10√
2
[4]
𝑚𝑒𝑑𝑖𝑎𝑛 = 15 +
(20 − 15)(10 − 5)
√√ = 19.17√
6
Question 7
[14]
7.1 The following table shows South African average wholesale prices (in Rands / kg) and the
production (in 1000kg) of milk, butter and cheese for the years 2016, 2017 and 2018
Price
Quantity
2016
2017
2018
2016
2017
2018
Cream
20.1
19.1
19.5
116
108
100
Yogurt
33
41
52
76
79
98
Milk
7.9
9.3
11.2
917
967
1001
Calculate the following
7.1.1 Simple price index for yogurt for 2018, with 2017 as base year.
[2]
52
 100 √= 126.83√
41
P=
7.1.2 Laspeyres price index for 2010, with 2008 as the base year.
PL =
[4]
(19.5  116)  (52  76)  (11.2  917)
16484.4
√√×100 √= 136.42√
 100 =
12083.9
(20.1  116)  (33  76)  (7.9  917)
7.2 Consider the following two price index series.
2012
89
2013
54
2014=100
2014
2015
100
110
6
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2015
58
2016
89
2017
100
2017=100
2018
86
Splice the two price index series to form one continuous series with 2015 as the base year.
Base period of the new series: 2015
This means that we have to shift the base period of both series to 2015:
𝑨𝒅𝒋𝒖𝒔𝒕𝒎𝒆𝒏𝒕 𝒇𝒂𝒄𝒕𝒐𝒓 =
𝟏𝟎𝟎
𝒊𝒏𝒅𝒆𝒙 𝒐𝒇 𝟐𝟎𝟏𝟓
𝑨𝒅𝒋𝒖𝒔𝒕𝒎𝒆𝒏𝒕 𝒇𝒂𝒄𝒕𝒐𝒓 =
YEAR
INDEX
2012
89
2013
54
2014
100
2015
110
= 𝟎, 𝟗𝟎𝟗𝟏√
𝟏𝟎𝟎
𝟏𝟏𝟎
ADJUSTMENT
NEW INDEX
89× 𝟎, 𝟗𝟎𝟗𝟏
80,9099
100× 𝟎, 𝟗𝟎𝟗𝟏
90,91
54× 𝟎, 𝟗𝟎𝟗𝟏
49,0914
110× 𝟎, 𝟗𝟎𝟗𝟏
100,001
√√
Shift the base period of the second series to 2015
𝑨𝒅𝒋𝒖𝒔𝒕𝒎𝒆𝒏𝒕 𝒇𝒂𝒄𝒕𝒐𝒓 =
𝑨𝒅𝒋𝒖𝒔𝒕𝒎𝒆𝒏𝒕 𝒇𝒂𝒄𝒕𝒐𝒓 =
𝟏𝟎𝟎
𝒊𝒏𝒅𝒆𝒙 𝒐𝒇 𝟐𝟎𝟏𝟓
𝟏𝟎𝟎
= 𝟏, 𝟕𝟐𝟒𝟏√
𝟓𝟖
7
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2015=100
2012
80,91
2013
49,09
2014
90,91
YEAR
INDEX
2015
58
2016
89
2017
100
2018
86
2015
100,01
2016
153,45
2017
172,41
ADJUSTMENT
𝟓𝟖 × 𝟏, 𝟕𝟐𝟒𝟏
2018
148,27
NEW INDEX
99,9978
89× 𝟏, 𝟕𝟐𝟒𝟏
153,4449
86× 𝟏, 𝟕𝟐𝟒𝟏
148,2726
100× 𝟏, 𝟕𝟐𝟒𝟏
√√
Spliced series:
√√
8
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172,41