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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