Page 317 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Quantitative Techniques – I

Notes 15.9 Review Questions

1. Under what conditions will a random variable follow a normal distribution? State some
important features of a normal probability curve.

2. What is a standard normal distribution? Discuss the importance of normal distribution in
statistical theory.
3. State clearly the assumptions under which a binomial distribution tends normal
distribution.
4. Find the probability that the value of an item drawn at random from a normal distribution
with mean 20 and standard deviation 10 will be between (i) 10 and 15, (ii) –5 and 10 and
(iii) 15 and 25.

5. In a particular examination an examinee can get marks ranging from 0 to 100. Last year,
1,00,000 students took this examination. The marks obtained by them followed a normal
distribution. What is the probability that the marks obtained by a student selected at
random would be exactly 63?
6. A collection of human skulls is divided into three classes according to the value of a
‘length breadth index’ x. Skulls with x < 75 are classified as ‘long’, those with 75 < x < 80 as
‘medium’ and those with x > 80 as ‘short’. The percentage of skulls in the three classes in
this collection are respectively 58, 38 and 4. Find, approximately, the mean and standard
deviation of x on the assumption that it is normally distributed.
7. In a large group of men, it is found that 5% are under 60 inches and 40% are between 60 and
65 inches in height. Assuming the distribution to be exactly normal, find the mean and
standard deviation of the height. The values of z for area equal to 0.45 and 0.05 between 0
to z are 1.645 and 0.125 respectively.
8. Packets of a certain washing powder are filled with an automatic machine with an average
weight of 5 kg. and a standard deviation of 50 gm. If the weights of packets are normally
distributed, find the percentage of packets having weight above 5.10 kg.
9. For a normal distribution with mean 3 and variance 16, find the value of y such that the
probability of the variate lying in the interval (3, y) is 0.4772.
10. The mean income of people working in an industrial city is approximated by a normal
distribution with a mean of 24,000 and a standard deviation of 3,000. What percentage
of the people in this city have income exceeding 28,500? In a random sample of 50
employed persons of this city, about how many can be expected to have income less than
19,500?
11. A batch of 5,000 electric lamps have a mean life of 1,000 hours and a standard deviation of
75 hours. Assume a Normal Distribution.
(a) How many lamps will fail before 900 hours?
(b) How many lamps will fail between 950 and 1,000 hours?

(c) What proportion of lamps will fail before 925 hours?
(d) Given the same mean life, what would the standard deviation have to be to ensure
that not more than 20% of lamps fail before 916 hours?

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