Page 384 - DMTH404_STATISTICS

Statistics

Notes 27.4 Keywords

n
Mean: Mean of the random variable t is zero and standard deviation is , for n > 2.
n - 2
T - distribution: The probability curve of t - distribution is symmetrical about the ordinate at t
= 0. Like a normal variable, the t variable can take any value from -  to .

27.4 Self Assessment

1. State whether the following statements are true or false:
2
(i) Both, t and c distributions depend only one parameter.
(ii) Total number of samples of size 4, with replacement, from a population of 15 units
is 1365.

(iii) F - statistic is equal the ratio of two c 2 variates.

(iv) In sampling with replacement if N = n, the standard error of X is equal to zero.
2
(v) c - distribution depends upon two parameters.
27.5 Review Questions

1. A population consists of 4 families consisting of 2, 3, 4 and 5 children. By considering all
possible random samples of size two, with replacement, find mean and standard error of
X . Show that S.E. of X depends upon the sample size.
2. If X , X , X is a simple random sample of size three from a large population with mean
1 2 3
5 and variance 4, evaluate the expected value and standard error of the statistics T = (2X +
1
X – 3X ).
2 3
3. If X , X and X constitute a random sample of size 3 from a normal population with mean
1 2 3
2
X 1 + X 2 + X 3 X + X + X
2
m and the variance s , find the efficiency of relative to 1 2 3 .
4 3
4. The diameter of a component produced on a semi-automatic machine is known to be
distributed normally with mean of 10 mm. and a standard deviation of 0.1 mm. If we pick
up a random sample of size 25, what is the probability that the sample mean will be
between 9.95 and 10.05 mm?
5. It is known that 10% of the bolts manufactured by a factory are defective. If a random
sample of 100 bolts is chosen at random from a day's production, construct the sampling
distribution of (i) the number of defective bolts, (ii) the proportion of defective bolts.
6. The mean and standard deviation of per capita consumption of wheat in rural and urban
areas of Delhi are estimated to be 450 gms, 75 gms and 410 gms, 100 gms respectively.
Assuming that the per capita consumption of wheat is distributed normally, construct the
sampling distribution of the difference between two sample means obtained from random
samples of sizes 80 and 60 from rural and urban populations respectively.

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