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សាកលវិទ្យាល័យ ន័រតុន
ផ្នកែ កុព្ុំ យូទ្យរ័ វិទ្យា
Assignment A
Course: Introduction to Statistics
Lecturer Name: Mong Mara
1- In a Statistics course, the scores of 30 students on midterm test are
10.0
9.0
6.0
5.0
9.0
7.5
10.0
6.0
5.5
8.0
9.0
9.0
5.0
4.0
4.5
12.0
10.0
4.5
6.0
5.5
4.5
7.0
10.5
8.0
4.5
6.0
5.0
6.0
8.5
8.0
a) Compute the arithmetic mean, median and mode of the raw data.
b) Construct a grouped frequency distribution, including relative frequency and percentage for
the scores.
c) Construct Histogram, frequency polygon and (less-than) cumulative frequency polygon.
d) From the frequency table in b), again, estimate the arithmetic mean, median, and mode.
e) At least about 85% of students have their scores between what two values?
2- In a distribution of 300 values, the mean is 60 and the standard deviation is 5. Answer each. Use
Chebyshev’s theorem.
a) At least how many values will fall between 35 and 85?
b) At most how many values will be less than 40 or more than 80?
3- Given a frequency distribution as below
Class limits
frequency
20 -24
3
25 -29
5
30 -34
10
35 -39
7
40 -44
4
45 -49
1
a) Estimate the mean, median and mode for the data
b) Estimate standard deviation
c) Find estimate the value corresponding to the first and third quartiles and compute the quartile
deviation.
4- An aptitude test has a mean of 220 and a standard deviation of 10. Find the corresponding z score for
each exam score.
a. 200
b. 232
c. 218
d. 212
e. 225
5- Which of these exam grades has a better relative position?
a) A grade of 43 on a test with x  40 and s  3
b) A grade of 75 on a test with X  72 and s  5
6- Find the percentile rank for each test score in the data set 12, 28, 35, 42, 47, 49, 50
7- Given data set 12, 28, 35, 42, 47, 49, 50what values correspond to the 65th, 70th, 75th, 80th, 82nd
percentile?
ជុំនាញ វិទ្យាសាស្រ្កត ព្ុុំ យូទ្យរ័
ទ្យុំព្រ័ ទ្យី1នន13
សាកលវិទ្យាល័យ ន័រតុន
ផ្នកែ កុព្ុំ យូទ្យរ័ វិទ្យា
8- These data are number of inches of snow reported in randomly selected U.S. cities for September 1
through January 10. Construct a box plot and comment on the skewness of the data.
9.8
8.0
13.9
4.4
3.9
21.7
15.9
3.2
11.7
24.8
34.1
17.6
9- An important module is tested by three independent teams of inspectors. Each team detects a problem
in a defective module with probability 0.8. What is the probability that at least one team of inspectors
detects a problem in a defective module?
10- A building is examined by policemen with four dogs that are trained to detect the scent of explosives.
If there are explosives in a certain building, and each dog detects them with probability 0.6,
independently of other dogs, what is the probability that the explosives will be detected by at least one
dog?
11- A shuttle’s launch depends on three key devices that may fail independently of each other with
probabilities 0.01, 0.02, and 0.02, respectively. If any of the key devices fails, the launch will be
postponed. Compute the probability for the shuttle to be launched on time, according to its schedule.
12- Under good weather conditions, 80% of flights arrive on time. During bad weather, only 30% of
flights arrive on time. Tomorrow, the chance of good weather is 60%. What is the probability that
your flight will arrive on time?
ជុំនាញ វិទ្យាសាស្រ្កត ព្ុុំ យូទ្យរ័
ទ្យុំព្រ័ ទ្យី2នន13
សាកលវិទ្យាល័យ ន័រតុន
ផ្នកែ កុព្ុំ យូទ្យរ័ វិទ្យា
Answers of Assignment
Answer 1.
In a Statistics course, the scores of 30 students on midterm test are:
4
4.5
4.5
4.5
4.5
5
5
5
5.5
5.5
6
6
6
6
6
7
7.5
8
8
8
8.5
9
9
9
9
10
10
10
10.5
12
a) Compute the arithmetic mean, median and mode of the raw data.
Mean:

x
 x  x  4  4.5  4.5  4.5  ...  12  213.5

n n  30
x
213.5
 7.11
30
Median:
67
 6.5
2
Mode:
(4.5, 5, 5.5, 6, 8, 9, 10)
b) Construct a grouped frequency distribution, including relative frequency and percentage
for the scores.
2k  n STOP
 25  32  n  30 STOP
Highest value  Lowest value
k

Class Limit
4.0 – 5.7
5.8 – 7.5
7.6 – 9.3
9.40 – 11.1
11.2 – 12.9
ជុំនាញ វិទ្យាសាស្រ្កត ព្ុុំ យូទ្យរ័
Frequency
10
7
8
4
1
12  4 8
  1.6  1.8
5
5
Relative frequency
10/30 = 0.33
7/30 = 0.23
8/30 = 0.26
4/30 = 0.13
1/30 = 0.03
Percentage
(10*100)/30 = 33.3
(7*100)/30 = 23.3
(8*100)/30 = 26.6
(4*100)/30 = 13.3
(1*100)/30 = 3.3
ទ្យុំព្រ័ ទ្យី3នន13
សាកលវិទ្យាល័យ ន័រតុន
ផ្នកែ កុព្ុំ យូទ្យរ័ វិទ្យា
c) Construct Histogram, frequency polygon and (less-than) cumulative frequency polygon.
Class Limit
4.0 – 5.7
5.8 – 7.5
7.6 – 9.3
9.4 – 11.1
11.2 – 12.9
Total:
ជុំនាញ វិទ្យាសាស្រ្កត ព្ុុំ យូទ្យរ័
Frequency
10
7
8
4
1
30
boundaries
Midpoint (Xm)
3.95 – 5.75
4.85
5.75 – 7.55
6.65
7.55 – 9.35
8.45
9.35 – 11.15
10.25
11.15 – 12.95
12.05
C.F
10
17
25
29
30
f. Xm
48.5
46.55
67.6
41
12.05
215.70
(Xm)2
23.52
44.22
71.40
105.06
145.20
f. (Xm)2
235.23
309.56
571.22
420.25
145.20
1681.46
ទ្យុំព្រ័ ទ្យី4នន13
សាកលវិទ្យាល័យ ន័រតុន
ផ្នកែ កុព្ុំ យូទ្យរ័ វិទ្យា
d) From the frequency table in b), again, estimate the arithmetic mean, median, and mode.
Mean:
 f .xm
215.70

 7.19
n
30
x
Median:
 LB  3.95
n
cf  0
 cf

2
median  LB 
*w
f
 f  10
 w  1.8

30
0
median  3.95  2
*1.8  6.65
10
Mode:
 LB  3.95
d1  10  0  10
d1

mode  LB 
*w
d1  d 2
d 2  10  7  3
 w  1.8

mode  3.95 
10
*1.8  5.33
10  3
e) At least about 85% of students have their scores between what two values?
Find K:
1
1
1
1
 0.85  2  1  0.85  k 2 
 k 
2
k
k
0.15
1
 2.58
0.15
Find Standard Deviation:
S

n *  f ( xm ) 2  ( fxm ) 2
30*1681.46  (215.70) 2
50443.8  46526.49


n(n  1)
30(30  1)
30*29
4836
 4.502  2.121
870

x  kS  7.19  2.58* 2.121  1.717

x  kS  7.19  2.58* 2.121  12.662
So at least about 85% of students,they have their scores between 1.717 and 12.662.
ជុំនាញ វិទ្យាសាស្រ្កត ព្ុុំ យូទ្យរ័
ទ្យុំព្រ័ ទ្យី5នន13
សាកលវិទ្យាល័យ ន័រតុន
ផ្នកែ កុព្ុំ យូទ្យរ័ វិទ្យា
Answer 2.
In a distribution of 300 values, the mean is 60 and the standard deviation is 5. Answer each. Use
Chebyshev’s theorem.
a) At least how many values will fall between 35 and 85?
ជុំនាញ វិទ្យាសាស្រ្កត ព្ុុំ យូទ្យរ័
ទ្យុំព្រ័ ទ្យី6នន13
សាកលវិទ្យាល័យ ន័រតុន
ផ្នកែ កុព្ុំ យូទ្យរ័ វិទ្យា
Answer 3.
Given a frequency distribution as below
Class limits Freq. C.F.
Xm f.Xm Xm2
f.Xm2
20 -24
3
3
22
66
484
1452
25 -29
5
8
27
135
729
3645
30 -34
10
18
32
320 1024 10240
35 -39
7
25
37
259 1369
9583
40 -44
4
29
42
168 1764
7056
45 -49
1
30
47
47 2209
2209
Total:
30
995
34185
a) Estimate the mean, median and mode for the data
Mean:
x
 f .xm
995

 31.71
n
30
Median:
 LB  29.5
n
cf  8
 cf

2
median  LB 
*w
f
 f  10
 w  5

30
8
median  29.5  2
*5  33
10
Mode:
 LB  29.5
d1  10  5  5
d1

mode  LB 
*w
d1  d 2
d 2  10  7  3
 w  5

mode  29.5 
5
*5  32.165
53
b) Estimate standard deviation
n *  f ( xm )2  ( fxm ) 2
30*34185  (995) 2
102550  990025
S


n(n  1)
30(30  1)
30* 29

ជុំនាញ វិទ្យាសាស្រ្កត ព្ុុំ យូទ្យរ័
35525
 40.83  6.4
870
ទ្យុំព្រ័ ទ្យី7នន13
សាកលវិទ្យាល័យ ន័រតុន
ផ្នកែ កុព្ុំ យូទ្យរ័ វិទ្យា
c) Find estimate the value corresponding to the first and third quartiles and compute the
quartile deviation.
QD  (Q3  Q1) / 2
Find Q1:
n / 4  30 / 4  7.5  Q1 place in class 25  29
 LB  24.5
n
 cf

Q1  LB  4
* w cf  3
f
f 5

 w  5
7.5  3
Q1  24.5 
*5  29
5
Find Q3:
3n / 4  3*30 / 4  22.5  Q3 place in class 35  39
 LB  34.5
3n
 cf


Q3  LB  4
* w cf  18
f
f 7


w  5
22.5  18
Q3  34.5 
*5  37.71
7
Find QD:
Q1  29
QD  (Q3  Q1) / 2 
Q3  37.71
37.71  29
QD 
 4.355
2
ជុំនាញ វិទ្យាសាស្រ្កត ព្ុុំ យូទ្យរ័
ទ្យុំព្រ័ ទ្យី8នន13
សាកលវិទ្យាល័យ ន័រតុន
ផ្នកែ កុព្ុំ យូទ្យរ័ វិទ្យា
Answer 4.
An aptitude test has a mean of 220 and a standard deviation of 10. Find the corresponding z score
for each exam score.
 x  value
xx 
z
 x  mean
s 
 s  Standard Deviation
a) 200
z
200  220
 2
10
z
232  220
 1.2
10
z
218  220
 0.2
10
z
212  220
 0.8
10
z
225  220
 0.5
10
b) 232
c) 218
d) 212
e) 225
Answer 5.
Which of these exam grades has a better relative position?
a) A grade of 43 on a test with x  40 and s  3
z
x  x 43  40

1
s
3
b) A grade of 75 on a test with X  72 and s  5
z
x  x 75  72

 0.6
s
5
 The exam grade of 43 has a better relative position than grade of 75.
Answer 6.
ជុំនាញ វិទ្យាសាស្រ្កត ព្ុុំ យូទ្យរ័
ទ្យុំព្រ័ ទ្យី9នន13
សាកលវិទ្យាល័យ ន័រតុន
ផ្នកែ កុព្ុំ យូទ្យរ័ វិទ្យា
Find the percentile rank for each test score in the data set 12, 28, 35, 42, 47, 49, 50.
Percentile 
number of values below X  0.5
*100%
total number of value
+ Score 12
Percentile 
0  0.5
*100%  7.1%
7
+ Score 28
Percentile 
1  0.5
*100%  21.4%
7
+ Score 35
Percentile 
2  0.5
*100%  35.7%
7
+ Score 42
Percentile 
3  0.5
*100%  50%
7
+ Score 47
Percentile 
4  0.5
*100%  64.2%
7
+ Score 49
Percentile 
5  0.5
*100%  78.5%
7
+ Score 50
Percentile 
ជុំនាញ វិទ្យាសាស្រ្កត ព្ុុំ យូទ្យរ័
6  0.5
*100%  92.8%
7
ទ្យុំព្រ័ ទ្យី10នន13
សាកលវិទ្យាល័យ ន័រតុន
ផ្នកែ កុព្ុំ យូទ្យរ័ វិទ្យា
Answer 7.
Given data set 12, 28, 35, 42, 47, 49, 50what values correspond to the 65th, 70th, 75th, 80th, 82nd
percentile?
12, 28, 35, 42, 47, 49, 50
C
n.P
100
+ 65th percentile
C
7 * 65
 4.55  5
100
The value correspond to the 65th percentile is 47.
+ 70th percentile
C
7 *70
 4.9  5
100
The value correspond to the 65th percentile is 47.
+ 75th percentile
C
7 * 75
 5.25  6
100
The value correspond to the 65th percentile is 49.
+ 80th percentile
C
7 *80
 5.6  6
100
The value correspond to the 65th percentile is 49.
+ 82nd percentile
C
7 *82
 5.71  6
100
The value correspond to the 65th percentile is 49.
ជុំនាញ វិទ្យាសាស្រ្កត ព្ុុំ យូទ្យរ័
ទ្យុំព្រ័ ទ្យី11នន13
សាកលវិទ្យាល័យ ន័រតុន
ផ្នកែ កុព្ុំ យូទ្យរ័ វិទ្យា
Answer 8.
These data are number of inches of snow reported in randomly selected U.S. cities for September
1 through January 10. Construct a box plot and comment on the skewness of the data.
9.8
3.2
3.2
3.9
4.4
8.0
11.7
8.0
13.9
24.8
9.8
4.4
34.1
11.7
3.9
17.6
13.9
21.7
15.9
15.9
17.6
21.7
24.8
34.1
+ Q1 = P25
c
nP 12* 25

3
100
100
 Q1 = 3rd value = 4.4
+ Q2 = P50
c
nP 12 *50

6
100
100
 Q2 = 6th value = 11.7
+ Q3 = P75
c
nP 12* 75

9
100
100
 Q3 = 9th value = 17.6
We obtain the lowest value = 3.2, Q1 = 4.4, Median = Q2 = 11.7, Q3 = 17.6 and the highest value
= 34.
ជុំនាញ វិទ្យាសាស្រ្កត ព្ុុំ យូទ្យរ័
ទ្យុំព្រ័ ទ្យី12នន13
សាកលវិទ្យាល័យ ន័រតុន
ផ្នកែ កុព្ុំ យូទ្យរ័ វិទ្យា
Answer 9:
An important module is tested by three independent teams of inspectors. Each team detects a
problem in a defective module with probability 0.8. What is the probability that at least one team of
inspectors detects a problem in a defective module?
At lease one team of inspectors detects a problem.
A = each team detects a problem
Ac = no team detects a problem
P( A)  0.8
P( Ac )  1 - 0.8  0.2
P(3 Ac )  [ P( Ac )]3  (0.2)3  0.008

P(At lease one team of inspectors detects a problem) =1-0.008= 0.992
Answer 10:
A building is examined by policemen with four dogs that are trained to detect the scent of
explosives. If there are explosives in a certain building, and each dog detects them with probability 0.6,
independently of other dogs, what is the probability that the explosives will be detected by at least one
dog?
At lease one dog detects the dxplosive.
A = each dog detects the dxplosive
Ac = no dog detects the dxplosive
P( A)  0.6
P( Ac )  1 - 0.6  0.4
P(3 Ac )  [ P( Ac )]4  (0.4) 4  0.0256

ជុំនាញ វិទ្យាសាស្រ្កត ព្ុុំ យូទ្យរ័
P(At lease one dog detects the dxplosive) =1-0.0256 = 0.9744
ទ្យុំព្រ័ ទ្យី13នន13