Lebanese University
Faculty of Science
November 2018
Duration: 2h
Partial Exam - S1101
Exercise 1 (12 points)
For each of the following 8 questions below, choose only one of the answers:
Q1) A distribution of 8 values has a median of 23. If the highest value increases 3 points,
the median will become:
(a)
(b)
(c)
(d)
(e)
23
23.5
25
Cannot be determined without additional information
none of the above
Q2) The mean and the standard deviation of the sampling distribution are respectively
equal to 25 and 0. what must you conclude?
(a)
(b)
(c)
(d)
(e)
Someone has made a mistake
All the elements in the sample are null
There are no elements in the sample
All the elements in the sample are 25
none of the above
Q3) The histogram below shows the relative positions of median, mode and mean. What
is the proper ordering of these parameters?
(a)
(b)
(c)
(d)
(e)
I
I
I
I
I
= mean ; II = median ; III = mode
=mode ; II = median ; III = mean
= median ; II = mean ; III = mode
= mode ; II = mean ; III = median
= mean ; II = mode ; III = median
Q4) Which of the following measures is not a measure of dispersion?
(a)
(b)
(c)
(d)
(e)
The Range
Variance
Interquartile Range
50th Percentile
Standard Deviation
1/3
Q5) In a group of 10 students, the highest grade in statistics is increased by 30 points.
What effect will this have on the mean of grades?
(a)
(b)
(c)
(d)
(e)
It will be increased by 10 points
It will remain unchanged
It will be increased by 3 points
It will increase by 30 points
None of the above
The following information will be used in the next three questions:
The distribution of the age of 40 smokers is shown int he following table:
Age
Density
[10,20[
0.4
[20,30[
d2
[30,x[
1.5
[x,50[
0.5
[50,60[
0.4
Q6) Knowing that the ages of 60% of smokers belong to [30, 50[, then the value of d2 is
equal to:
(a)
(b)
(c)
(d)
(e)
0.1
0.2
0.4
0.8
None of the above
Q7) Suppose that d2 = 0.9. If the mode of this sample is equal to 31.5 years, then the
value of x is:
(a)
(b)
(c)
(d)
(e)
34
42
34
54
None of the above
Q8) Suppose that d2 = 0.9. The percentage of smoking people whose age is less than 28
years is:
(a)
(b)
(c)
(d)
(e)
28%
30%
32%
34%
None of the above
Exercise 2 (18 points)
The following table shows the distribution of 20 Lebanese University students according
to their study time per day (in minutes), preferred football team and number of times
per day they connect to a mobile game.
2/3
Study Time
Preferred team
20
23
21
22
30
30.5
30
40
44.5
41
27
28
37.5
38
32.5
32
33
38
31
37
AC Milan
AC Milan
AC Milan
Barcelona
Barcelona
AC Milan
Barcelona
Barcelona
AC Milan
AC Milan
AC Milan
Barcelona
AC Milan
AC Milan
AC Milan
Barcelona
AC Milan
Barcelona
AC Milan
Barcelona
Number of times
they connect
1
1
1
2
2
1
3
4
3
1
5
5
1
2
2
a
b
c
5
5
1. Precise the population, sample, and the variables and their types.
2. The results for the study time should be summarized in classes of a statistical table:
(a) Show that the number of classes and their width are equal to 5. Construct the
frequency distribution.
(b) Construct the polygon of increasing cumulative frequency for the variable study
time. Deduce the median graphically and by calculation.
(c) Calculate the study time of individual number 15 (rank =15).
3. We consider that a > 1, b > 3, and c > 3. Find the value of a if the median number
of times they connect is equal to 2.
4. In this part we suppose that a=2. Find the values of b and of c if the average
number of times they connect to a mobile game is equal to 3 and the third quartile
is equal to 4.5.
5. In this part consider that a = 2, b= 10 and c =4. Draw the boxplot for the number
of times they connect to a mobile game for students who support AC Milan.
3/3
Lebanese University
Faculty of Sciences
Midterm Exam
November 2018
S1101- Statistic
Solution
Exercise 1 (12 pts =(1.5*8))
Q1
(a)
Q2
(d)
Q3
(a)
Q4
(d)
Q5
(c)
Q6
(d)
Q7
(a) or (c)
Q8
(a)
Exercise 2 (18 pts)
1. (4 pts = 0.5*8)
Set of Students
Population
(0.5 pts)
20 Lebanese University Students
Sample
(0.5 pts)
Study Time
Preferred
Number ot times
Variables
team
they connect
(0.5*3 pts)
Quantitative – Qualitative
Quantitative Nature
Continuous
– Nominal
Discrete
(0.5*3 pts)
2.
(a) (3 pts = 1+1+1) Let
Let
number of classes then
then
the width (or amplitude) of classes then
Study Time
20 – 25
25 – 30
30 – 35
35 – 40
40 – 45
Total
(b) (4 pts=1+1+1+1)
Ni
4
6
13
17
20
1/2
ni
4
2
7
4
3
20
.
then
Ni
4
6
13
17
20
The median
(c) (2 pts) The study time of individual number 15 is
then
3. (1 pts) The number of times they connect: 1 1 1 1 1 1 2 2 2 2 3 3 4 5 5 5 5 a b c
then
We have
but we have
4. (2 pts) a=2 then
, but
or
then
so that to find
then we have
.
then
(
and
) or (
and
).
5. (2pts) a=2, b=10 then the distribution is: 1 1 1 1 1 1 2 2 3 5 5 10 then
and
and
*
Number of times
1 1.5
4
5
10
2/2
Supplementary Exercises: Continuous Variable
Exercise 1 - We consider the distribution of 200 employees of a bank in Beirut
based on their annual income in thousands of $.
Annual income
[30-50[
[50-70[
[70-90[
[90-110[
[110-130[
[130-150[
Total
ni
Ni
Ni
fi
10
10
200
20
30
190
50
80
170
60
140
120
20
160
60
40
200
40
200
0,05
0,05
1
40
20
-58
0,1
0,15
0,95
60
20
-38
0,25
0,4
0,85
80
20
-18
0,3
0,7
0,6
100
20
2
0,1
0,8
0,3
120
20
22
0,2
1
0,2
140
20
42
1
Fi
Fi
Midpoint : ci
Width ai :
ci-
1)
The population is the employees in a bank in Beirut; the sample is the group of;
the variable is the annual income in thousands of $. Its kind is Quantitative continuous.
2)
Determine graphically and analytically the mode, the first quartile Q1,
the second quartile Q2 and the third quartile Q3.
Step 1 – (Same Width) => 60>40>… => Modal Class = [90-110[
Step 2 –
Interpretation: The most observed income in employees of bank is 94 thousand of
dollars.
3) Draw the cumulative frequency polygon in an ascending and descending
direction. Fid Q1, Q2 and Q3. Interpret
Me: N/2 = 200/2 = 100 => Median class = [90-110[
Step1 –
[90-110[
Setp 2 –
Interpretation: 50% of the employees of the bank in Beirut have an income less than 96.67
thousands of dollars and 50% of employees of the bank in Beirut have an n more than
96.67 thousand of dollars.
Q1:
Step 1 –
[70-90[
Step 2 –
Interpretation: 25% of the employees of the bank in Beirut have an income less than 78
thousands of dollars and 75% of employees of the bank in Beirut have an n more than 78
thousand of dollars.
Q3:
Step 1 –
[110-130[
Step 2 –
Interpretation: 75% of the employees of the bank in Beirut have an income less than 120
thousands of dollars and 25% of employees of the bank in Beirut have an n more than 120
thousand of dollars.
4) Find the Inter-quartile range. Interpret.
IQR = Q3-Q1=120-78= 42
Interpretation: 50% of the employees of the bank in Beirut earn between 78 and 120
thousands of dollars. The annual income of 50% of the employees are spread over a range
of 42 thousands of $.
5) Determine the income of the employee number 150
We must find P75
Step 1 –
[110-130[
Step 2 –
6) Find the proportion of employees in this population whose income exceeds
120 thousand of $.
7) The distribution is homogeneous?

Interpretation: the average income of employees of the bank to Beirut is 98 thousands of $


; Interpretation: The dispersion around the average value of
is equal to 28.2

>15% => the distribution is heterogeneous
8) The distribution is symmetric? Interpret
Me ≈ M 0 ≈
=> the distribution is symmetric

~ 0: Distribution is symmetric
Exercise 2- The graph below shows the age (in months) of 100 children treated for
respiratory physiotherapy spastic bronchitis at the Hôtel-Dieu in Beirut.
27%
23%
14%
9%
9%
7%
7%
2%
2%
0%
0%
3
8
13
18
23
28
33
38
43
48
0%
53
0%
58
63
68
Âge
Age (en
(in mois)
months)
1) Determine: The Population: children treated for respiratory physiotherapy spastic
bronchitis at the Hôtel-Dieu in Beirut.
Individual: 1 child
The variable: age (in months)
Type: Quant. Continuous
Age
[3-8[
[8-13[
[13-18[
[1823[
[2328[
[2833[
[3338[
[3843[
[4348[
[6368[
T
fi
ni = fi×N
Ni
ai
Midpoint: Ci
0.27
27
27
5
5.5
0.23
23
50
5
10.5
0.07
7
57
5
15.5
0.09
9
66
5
20.5
0.09
9
75
5
25.5
0.07
7
82
5
30.5
0.14
14
96
5
35.5
0
0
96
5
40.5
0.02
2
98
5
45.5
0.02
2
100
5
65.5
1
100
---------
2) Find the mode by 2 methods
Step 1: Same width => 27>23>… => Modal class = [3-8[
Step 2:
3) Calculate the median (Me) and the first quartile (Q1). Interpret.
Median:
Step1:
[13-18[
Step 2:
Interpretation: 50% of children have an age less than 13 months and 50% of children
have an age more than 13 months
First Quartile Q1:
Step 1:
[3-8[
Step 2:
Interpretation: 25% of children have an age less than 7.7 months and 75% of children
have an age more than 7.7 months
4) Calculate the average (mean) age of the children. Interpret.
5) The statistician at the hospital has found that the standard deviation of the
variable studied is 4.3. Interpret this value.
The dispersion of observations around the average value of
is equal to 4.3