Download normal distribution

Chapter No. 10
1
Normal Distribution
NORMAL DISTRIBUTION
Short Questions
Q.1 Define normal distribution.
Ans: Normal distribution is defined as a limiting form of binomial distribution when n  30
and p  q. It is a distribution of continuous random variable. The p.d.f. or equation of normal
distribution is:
1 x
 (
)
1
e 2  ;   x  
 2
2
f ( x)  Y 
Q.2 Give background of Normal Distribution.
Ans:Normal distribution is also called Gaussian distribution in honor of Karl F Gauss who
derived its equation. Karl Pearson named it normal distribution, in 1830.
Q.3 What is the range of normal variable?
Ans: Normal variable “X” range from  to   .
Q.4 What are the parameters of Normal distribution?
Ans:It has two parameters  and  .
Q.5 What is the shape of normal distribution?
Ans: It is uni-modal and bell shaped distribution.
Q.6 Define normal frequency distribution.
Ans: If normal distribution is multiplied by number of experiments “N” then resulting
distribution is known as normal frequency distribution.
N f ( x)  N
1 x
 (
)
1
e 2 
 2
2
Q.7 What is standard normal variable?
Ans: The normal r.v. “X” expressed in terms of deviations of “X” from “  ” divided by “  ”
is called standard normal random variable. It is denoted by “Z”.
Z
X 

,
Z
N (0,1)
Q.8 What is standard normal distribution?
Ans: Probability distribution of standard normal variable “Z” is called standard normal
distribution. Its p.d.f. or equation is:
Senior Subject Specialist
03457412131
Chapter No. 10
2
Normal Distribution
1  12 Z
f ( z) 
e
,Where    Z   
2
2
Q.9 What is the importance of Normal distribution?
Ans: It is used in applied Statistics. Many variables follow the normal distribution. Accurate
approximation to probability laws.
Q.10 How you define normal curve?
Ans:The graph of a normal probability density function is called a normal curve.
Q.11 When the shape of normal distribution remain same?
Ans: If “  ” is held constant and “  ” is varied the shape of the density function remains the
same.
Q.12 What is the total area of normal curve?
Ans:Total area of normal curve is unity(1).
Q.13 What is the median and mode of the normal distribution?
Ans: Median and Mode of the normal distribution is equal to mean(  ).
Q.14 Normal distribution is mesokurtic, explain?
Ans: The normal curve is neither very peaked nor very flat-topped, and then it is called
mesokurtic.
Q.15 What are the points of inflexion in normal distribution?
Ans: These are two points equidistance from mean i,e    and    .
Q.16 Write the equation of maximum ordinate of normal distribution.
Ans: The equation of maximum ordinate is
1
 2
Q.17 Write the maximum ordinate of standard normal curve.
Ans: The maximum ordinate of standard normal curve is
1  12 (0)
 (0) 
e
 0.3989
2
2
Q.18 What is the relation between mean deviation and standard normal of curve?
Ans: The relation between mean deviation and standard normal of curve is
M.D = 0.7979 
Q.19 What is the relation between Quartile deviation and standard normal of curve?
Ans: The relation between Quartile deviation and standard normal of curve is
Senior Subject Specialist
03457412131
Chapter No. 10
3
Normal Distribution
Q.D = 0.6745 
Q.20 What is the lower and upper quartile of standard normal distribution?
Ans: The lower quartile is -0.6745 and upper quartile is0.6745.
Q.21 What is the lower and upper quartile of normal distribution?
Ans: Q1    0.6745
and Q3    0.6745
Q.22 What is the standard deviation of normal distribution?
Ans: The value of standard deviation is
 
X 0.75  X 0.25
1.349
Q.23 What is the value of quartile deviation of the normal distribution?
Ans: The value of quartile deviation of the normal distribution is
Q.D 
X 0.75  X 0.25
2
Q.24 What are the first 4 moments about mean of the normal distribution?
Ans: 1  0, 2   2 , 3  0, 4  3 4
Q.25 Give co-efficient of skewness and kurtosis of the normal distribution.
Ans: Co-efficient of skewness is 1  0
Co-efficient of kurtosis is 2  3
Q.26 Give all odd order moment about mean of normal distribution?
Ans: All odd order moment about mean are
1  3  5  ..................  0
Q.27 What are the area under these values    ,   2 ,   3 of normal distribution?
Ans:The area under the    is 68.27% ,the area of   2 is 95.44% and the area of
  3 is 99.73%.
Practice Questions
1 – In normal distribution M.D= 16. Find standard deviation.
2 – In normal distribution mean is 100 and S.D. is 10.Find MD and QD.
3 – In normal distribution mean is 6 and the value of upper quartile is 171.094.Find SD.
4 – Second mean moment of normal distribution is 4. Find third and fourth moment.
Senior Subject Specialist
03457412131
Chapter No. 10
4
Normal Distribution
5 – If Z is normally distributed with mean 0 and variance 1. Find P(Z>-1.64 or Z< 1.09).
6 – If X
N (50, 25) ,find P(X< 45)
7 – Calculate quartile deviation of standard normal distribution.
8 – Calculate inter-quartile range of standard normal distribution.
9 – If lower and upper quartiles are 8 and 17 respectively, then find median and Standard
deviation.
10 – If mean is 50 and sd is10 find the value of “a” if P(X<a)= 0.45.
Multiple Choice Questions
Each question has four possible answers. Select correct answer.
1
Total area under normal curve is
(a) 0
2
(c) fixed
(d) very small
(b) mean
(c) MD
(d) QD
(b) M
(c) U
(d) bell
(b) platykurtic
(c) mesokurtic
(d) none
(b) unimodal
(c) bimodal
(d) trimodal
(c) 4
(d) 1
The normal distribution has parameters:
(a) 2
9
(b) large
Normal distribution is:
(a) multimodal
8
(d) none of these
Normal distribution is
(a) leptokurtic
7
(c ) 0.5
Shape of normal curve is:
(a) circle
6
(b) 2.0
The shape of normal curve depends upon:
(a) SD
5
(d) 1
Normal distribution is used when “n” is:
(a) small
4
(c) >1
Area under the normal curve on either side of mean is:
(a) 1
3
(b) -1
(b) 3
The range of normal distribution is:
Senior Subject Specialist
03457412131
Chapter No. 10
(a) 0 to 1
10
If X ,
(b) zero
If X
(b) 50
(b) 50
If Y=5X +10 and X
(a) 60
14
(b) 5
(b) 40
(b) 3
If X
(c) 40
(d) 25
(c) 25
(d) 5
(c) 2
(d) 1
(b) MD
(c) SD
(d) variance
(b) 100
(c) 12
(d) 144
(b) 0.7979 
(c) 0.6745 
(d) 0.50 
(b) 1
(c) 0.7979 
(d) none
The value of lower quartile in standard normal distribution is:
(b) -0.7979
(c) +0.6745
(d) -0.6745
The value of upper quartile in standard normal distribution is:
(a) +0.7979
23
(d) 260
The value of quartile deviation in normal distribution is:
(a) +0.7979
22
(c) 250
The value of mean deviation in normal distribution is:
(a) 0.6745 
21
(d) 2
N (100,144) ,then standard deviation is :
(a) 1.6745 
20
(c) 5
In normal distribution E ( X   )2
(a) 244
19
(d) 2
The median of standard normal distribution is:
(a) QD
18
(c) 5
If in normal distribution  = 40 and  2 =25 ,then mode is
(a) 0
17
(d) negative
If in normal distribution  = 40 and  2 =25 ,then median is
(a) 65
16
(c) positive
N (50, 25) ,then the mean of Y is:
(b) 50
(a) 65
15
(d)  to 
N (50, 25) ,then variance is :
(a) 25
13
(c) 0 to 
N (50, 25) then mean is:,
(a) 25
12
(b) -  to 0
Normal Distribution
In normal distribution  is always:
(a) one
11
5
(b) -0.7979
(c) +0.6745
(d) -0.6745
(c) 
(d) 
Standard normal random variable has mean:
(a) 1
(b) 0
Senior Subject Specialist
03457412131
Chapter No. 10
24
(d) 0.8545
(c) 0.7973
(d) 0.7399
(c) maximum ordinate
(d) asymptotic curve
(c) (a)
(d) 2 (a) -1
(c) (a)
(d) 2 (a) -1
(c)  (b) - (a)
(d)  (b) + (a)
(c) 1- 2 (a)
(d) 1- (a)
(c) 1- 2 (a)
(d) 1- (a)
(b) 0.6827
(b) point of inflexion
(b)1 - (a)
(b)1 - (a)
(b) (a) +  (b)
(b) 2 (a)
(b) 2  (  a )
(b)  2
(c)2  2
(d) 1
In normal distribution fourth moment about mean is :
(a) 0
38
(c) 0.6445
(b) 0.6827
In normal distribution second moment about mean is :
(a) 0
37
(d) 0.2768
P( Z  a) is equal to :
(a) 2 (a) -1
36
(c) 0.8627
(b) 0.6827
P( Z  a) is equal to:
(a) 2 (a) -1
35
(d) X0.25
P(a  X  b) is equal to
(a) (a) -  (b)
34
(c) X0.75
In standard normal distribution, P(Z  a)is
(a) 2 (a)
33
(b) X0.05
In standard normal distribution, P(Z  a)is
(a) 2 (a)
32
(d) X0.25
   and    are called:
(a) moment ratio
31
(c) X0.5
P(   3  X    3 ) is equal to
(a) 0.9973
30
(b) X0.75
P(   2  X    2 ) is equal to
(a) 0.9545
29
(d) 
P(     X     ) is equal to
(a) 0.7979
28
(c) 
Q3 is equal to:
(a) X0.10
27
(b) 1
Q1 is equal to:
(a) X0.10
26
Normal Distribution
Standard normal random variable has variance:
(a) 0
25
6
(b) 3  2
(c)2  2
(d) 3 4
In normal distribution  2 = 5 then  4 is
Senior Subject Specialist
03457412131
Chapter No. 10
(a) 25
39
(b) 75
45
(d) 99.73%
(c) Q1
(d) 
(d) 
(c) 
(d) 
(c) 
(d) 
(b) a – zb
(c) a+z b
(d) a – z b
(b) 1
(c) -1
(d) none
(b) 1
(c) -1
(d) none
(b )1
If X
N (a, b) , then the value of X is
() is equal to:
() is equal to:
Which is correct regarding the normal distribution?
(b)  2 =3
(c) 1 =1
(d)  3 = 3 4
(c) Standard deviation
(d) skewness
Second mean moment is called:
(a) mean
50
(c) 95.45%
(c) 
(a) 0
(a) 1 =3
49
(d)1
Standard normal distribution has maximum ordinate at Z =
(a) 0
48
(b) variance
(b )1
(a) 0
47
(c) 0.75
The maximum ordinate of standard normal curve is at:
(a) a+zb
46
(b) 68.27%
(b )1
(a) 0
44
(b) 0.50
Normal distribution has maximum ordinate at X is equal to :
(a) 0
43
(d) 625
In a normal curve, ordinate is highest at:
(a) Mean
42
(c) 125
In normal distribution   0.6745 covers area:
(a) 50%
41
Normal Distribution
The probability that the value of standard normal variate exceeds 0 is
(a) 0.25
40
7
(b) variance
The normal distribution is --------------- distribution
(a) discrete
(b) continuous
Senior Subject Specialist
(c) a and b
(d) none
03457412131