Page 310 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Unit 15: Normal Probability Distribution

4. Since the area between 1.65 and z < the area between 1.65 and 0 (which, from table, is 0.4505), Notes
z is negative. Further z can be to the right or to the left of the value 1.65. Thus, when z lies
to the right of 1.65, its value, corresponds to an area (0.4505 0.0173) = 0.4332, is given by
z = 1.5 (from table). Further, when z lies to the left of 1.65, its value, corresponds to an area
(0.4505 + 0.0173) = 0.4678, is given by z = 1.85 (from table).
5. Since the area between 0.5 to z > area between 0.5 to 0 ( which, from table, is 0.1915), z is
positive. The value of z, located corresponding to an area (0.5376 0.1915) = 0.3461, is given
by 1.02.

Task If X is a random variate which is distributed normally with mean 60 and standard
deviation 5, find the probabilities of the following events:

(i) 60 X 70, (ii) 50 X 65, (iii) X > 45 , (iv) X 50.

Self Assessment

State whether the following statements are true or false:
1. The normal probability distribution occupies a place of central importance in Ancient
Statistical Theory.
2. This distribution was first observed as the normal law of errors by the statisticians of the
eighteenth century.

3. An observed value of X is the sum of its true value and the net effect of a large number of
independent errors which may be positive or negative each with equal probability.
4. The observed distribution of a random variable was found to be in close conformity with
a continuous curve, which was termed as the normal curve of errors or simply the normal
curve.

5. Various factors operate in such a way that the deviations of observations above and below
mean are balanced with regard to their magnitude as well as their number.
6. p and e are absolute constants with values 3.14159.... and 2.71828.... respectively.

7. Normal probability distribution is a bell shaped symmetrical curve about the ordinate at
X.
8. Normal probability distribution is unimodal curve and its tails extend infinitely in both
directions.
9. In NPD, All the three measures of central tendency coincide, i.e.,
mean + median + mode=0

10. The total area under the curve gives the total probability of the random variable taking
values between - to .
11. In Normal distribution, since the distribution is symmetrical, all odd ordered central
moments are zero.
12. The value of p(X) is always non-negative for all values of X, i.e., the whole curve lies above
X axis.
13. The quartiles are equidistant from median, i.e., Md + Q = Q – Md , by virtue of symmetry.
1 3
14. The value of moment coefficient of skewness is never zero.

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