SOLUTIONS
University of Botswana
Department of Statistics
STA 116: Introduction to Statistics
Semester 1 Academic Year 2023-2024
November 4th, 2023
Test 3
Marks:
100
Duration: 90 minutes
Instruction: Answer All Questions in the spaces provided
Surname: ________________________
First name: _____________________
Student IDNO: __________________
LEC:________
Section
Mark
A
B
Total
1
SOLUTIONS
Question A1
Indicate TRUE or FALSE to the following statements.
(i)
If the Pearson’s product moment correlation coefficient between the variables 𝑋 and 𝑌 is
equal to zero, it means that there is no relationship between the variables. T
(ii)
Fisher’s Index takes into account both prices and quantities of base and current
periods. T
(iii)
In the simple linear regression model, the dependent variable is also called the
explanatory variable. F
(iv)
An index number is a measure that compares the price, quantity, or value of a
commodity over a time or place. T
(v)
The consumer price index number is based on the Paasche’s approach.T
(vi)
(vii)
The index number for the base year is always 100. T
If the Paasche’s price index is 96%, it means that there has been a decrease by 4%
in the price of the items over the two periods. T
(viii)
When constructing indices, the purpose of assigning weights to the individual
commodities is to reflect their relative importance. T
(ix)
Drobisch index is the geometric mean of the Paasche’s and Laspeyres index. F
(x)
The price relative index is the ratio of the price of a commodity in base period to
the price in current period. F
[1.5*10=15 Marks]
Question A2
Fill in the missing words/values for the following statements.
a) Pearson’s
correlation
measures
the
direction
as
well
as
the
_strength/magnitude_______ of a linear relationship between variables.
b) The ___slope/_𝜷𝟏 __ of a simple linear regression equation represents the change in
the value of the dependent variable whenever the independent variable changes by a
unit.
c) The_____Spearman_______ coefficient of correlation is computed based on the
ranks of the original observations.
d) Laspeyres index is constructed using the_base year quantities__as weights.
e)
The ____Simple price__ index gives equal importance to all items included in the
index
f) Fisher’s method of calculating the index number is based on the ___geometric
mean____.
2
SOLUTIONS
g) A price relative of 108% indicates that the price of a commodity increased by 8%
between the base period and the current period.
h) The Pearson’s Correlation coefficient lies between ___-1_______ and __1____.
i) ______Index numbers___ are useful for calculations of vital indicators such as cost
of living index.
[1*10=10 Marks]
Question B1
a) The following table shows the weekly labor costs paid (𝑥) and weekly revenues
received from bicycle repairs (𝑦) by a bicycle shop.
Labor costs (𝑥)
Repair revenues (𝑦)
7
4
9
6
12
7
15
9
20
15
The summary statistics are given as follows:
∑ 𝑥 = 63
∑ 𝑥𝑦 = 601
i.
∑ 𝑥 2 = 899 ∑ 𝑦 2 = 407
∑ 𝑦 = 41
Plot a scatter diagram of the data and comment on the relationship between labor costs
and repair revenues.
Labour costs vs repair revenue
repair revenue
20
15
10
5
0
0
5
10
15
20
25
Labour costs
[6 marks chart]
There is a positive relationship between labor costs and repair revenue. [1 mark]
3
SOLUTIONS
ii.
Calculate the value of the slope of the regression equation between labor costs and
repair revenue and interpret the coefficient.
𝑠𝑙𝑜𝑝𝑒 𝑏1 =
∑ 𝑥𝑦 − 𝑛𝑥̅ 𝑦̅ 601 − 5(12.6)(8.2)
84.4
=
=
= 0.802281368 ≈ 0.802
∑ 𝑥 2 − 𝑛𝑥̅ 2
899 − 5(12.62 )
105.2
[6 marks]
Interpretation: For a unit change in labor costs, repair revenue increases by 0.802. [1 Mark]
iii.
Calculate the value of the intercept of the regression equation between labor costs and
repair revenue and interpret the coefficient.
𝑏0 = 𝑦̅ − 𝑏1 𝑥̅ = 8.2 − (0.802 × 12.6) = −1.91
[4 Marks]
Interpretation: When the labor costs are equal to zero, repair revenue will be equal to zero. [1 Mark]
iv.
Write the regression line and fit the predicted line on the graph obtained in (i) above.
The line:
𝑌̂ = −1.91 + 0.802𝑥
[1 marks]
Calculation of any two points to obtain the line.
[2 marks]
See scatter plot for the line.
[1 marks]
v.
Calculate the predicted repair revenue when the labor cost is 10 for (iv) above.
𝑌̂ = −1.91 + 0.802(10) = 6.11
[3 Marks]
vi.
Calculate the estimated error given that the actual repair revenue is 11.
∈𝑖 = 𝑌 − 𝑌̂ = 11 − 6.11 = 4.89
[4 marks]
vii.
i.
Compute the Pearson correlation coefficient between the repair revenue and labor cost
and interpret the coefficient.
𝑟=
∑ 𝑥𝑦−𝑛𝑥̅ 𝑦̅
√[∑ 𝑥 2 −𝑛𝑥̅ 2 ][ ∑ 𝑦 2 −𝑛𝑦̂ 2 ]
=
601−5(12.6)(8.2)
√[899−5(12.62 )][407−5(8.22 )]
=
84.4
√105.2×70.8
=
0.97795291 ≈ 0.98
[9 Marks]
Interpretation: There is very strong positive relationship between labor costs and repair revenue.
[1 Mark]
4
SOLUTIONS
Question B2
a) A group of students were given a take home test. After marking the test, the instructor
decided to give the students the same test again but under supervision. The marks (both out of
50) obtained for take home and supervised tests were as follows:
Student No
1
2
3
4
5
6
7
Take Home( 𝑥)
41
27
32
21
44
24
36
Supervised (𝑦)
22
25
12
11
19
7
14
Rank 𝑥
2
5
4
7
1
6
3
[2 marks]
Rank 𝑦
2
1
5
6
3
7
4
[2 marks]
0
6
1
1
4
1
1
[2 marks]
𝐷𝑖 2
i.
Calculate the Spearman rank correlation coefficient take home and supervised
tests.
𝑟𝑠 = 1 −
=1−
6 ∑ 𝑑𝑖
𝑛3 − 𝑛
6(24)
(73 − 7)
= 1 − 0.428571428
= 0.551428571 = 0.57
[4 Marks]
b)The following table gives the wholesale prices and quantities of a number of commodities
in a certain district of Botswana for the years 2000 and 2005.
Calculate the following indices:
(i) Price Index for the year 2000 with 2005 as base using Simple Aggregative method.
Price per Kg (P)
Quantity (000 Kg)
P0
Q0
Commodity
Wheat
Rice
Meat
Maize
TOTAL
Pn
2000
6.25
5
15
3.25
29.5
2005
8.75
6.25
20
6.75
41.75
Qn
2000
1500
5000
6000
2000
Price
2005 relative
2000
140.00
8000
125.00
4000
133.33
3000
207.69
5 marks
(ii) Composite price index for 2005 and interpret the value.
∑𝑝
Composite index=∑ 𝑝𝑛 × 100 =
0
41.75
29.5
× 100 = 141.52
5
SOLUTIONS
[5 Marks]
Interpretation: Prices increased by 41.52% in 2005 for the concerned basket of goods.
[1 Mark]
(iii) Laspeyeres index for the year 2005.
Price
per Kg
Commodity (P)
P0
Wheat
Rice
Meat
Maize
TOTAL
Quantity
(000 Kg)
Q0
Qn
Pn
2000
6.25
5
15
3.25
29.5
2005
8.75
6.25
20
6.75
41.75
2000
1500
5000
6000
2000
14500
Price
2005 relative
PnQ0
P0Q0
2000
140.00
13125
9375
8000
125.00
31250
25000
4000
133.33
120000
90000
3000
207.69
13500
6500
17000 606.02564
177875
130875
2 marks
𝐿𝑝 =
2marks
∑ 𝑄𝑛 𝑃0
177875
× 100 =
× 100 = 135.91%
∑ 𝑄0 𝑃0
130875
[5 Marks]
(iv) - provide a value useful for comparing magnitudes of aggregates of related variables to
each other, and to measure the changes in these magnitudes over time.
- Used as economic indicator, e.g CPI.
[2.5 Each]
6