Statistics and Probability Answers

A scrap metal dealer claims that the mean of its cash sales is ‘no more than $80’, although an Internal Revenue Service agent believes that the dealer is being dishonest. Observing a sample of 20 cash customers, the agent finds the mean cash sales to be $91, with a standard deviation of $21. Assuming the population is distributed approximately normally, and using the 0.05 level of significance, will the agent’s suspicion be confirmed?

10 (b) Find the mean and variance of binomial distribution.

10. (a) For a distribution, the mean is 10, variance is 16, the skewness 4

sk is +1 and kurtosis 

b2

 is 4. Obtain the first four moments about the origin i.e. zero. Comment upon the 

nature of the distribution.

9 b) For the given distribution: (5) 

 ; ,2,1,0 ,

3

1

3

2

( )  = K











P X = x = x

x

 find moment generating function, mean and 

variance of X.

9. (a) Let X be a binomial variate with n =100, p = .1.0 Find the approximate value of

P 10( ≤ X ≤12) using: (5) 

 (i) normal distribution 

 (ii)poisson distribution 

 [You may like to use the following values. 

 P(Z ≤ 67.0 ) = .0 7486, P(Z ≤ 33.0 ) = .0 6293, P(Z ≤ )0 = ]5.0

8(b) For 25 army personnels, line of regression of weight of kidneys (Y) on weight of 

heart (X ) is Y = .0 399X + .6 934 and the line of regression of weight of heart on 

weight of kidney is X − .1 212Y + .2 461= .0 Find the correlation coefficient between 

X and Y and their mean values.

8. (a) Let X X Xn

, , , 1 2 K be independently and identically distributed b ,1( p) random 

variables. Obtain a confidence internal for p using Chebychev’s inequality. 

7(b) Let X X Xn

, , , 1 2 K be random sample of size n from a distribution with probability 

density function 





θ < < θ >

=

θ−

,0 else where.

0, ,1 0

( )0,;

1 X X

f X Obtain a maximum likeyhood 

estimator of θ.

7. (a) A factory produces steel pipes in three plant with daily production volumes of 500, 

1000 and 2000 units respectively from each of the plants. From the past experience it 

is known that the fraction of defective outputs produced by three plants are 

respectively 0.005, 0.008 and 0.010. If a pipe is selected at random from a day’s total 

production and founded to be defective, from which plant is that likely to 

have came?

6(b) If a random variable u has t -distribution with n degree of freedom, find the 

distribution of .