Page 229 - DMTH404_STATISTICS

Unit 15: Exponential Distribution and Normal Distribution

3. The marks obtained in a certain examination follow normal distribution with mean 45 Notes
and standard deviation 10. If 1,000 students appeared at the examination, calculate the
number of students scoring (i) less than 40 marks, (ii) more than 60 marks and (iii) between
40 and 50 marks.
Hint : See example 30.

4. The ages of workers in a large plant, with a mean of 50 years and standard deviation of 5
years, are assumed to be normally distributed. If 20% of the workers are below a certain
age, find that age.

Hint : Given P(X < X ) = 0.20, find X .
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5. The mean and standard deviation of certain measurements computed from a large sample
are 10 and 3 respectively. Use normal distribution approximation to answer the following:
(i) About what percentage of the measurements lie between 7 and 13 inclusive?
(ii) About what percentage of the measurements are greater than 16?

Hint : Apply correction for continuity.
6. There are 600 business students in the post graduate department of a university and the
probability for any student to need a copy of a particular text book from the university
library on any day is 0.05. How many copies of the book should be kept in the library so
that the probability that none of the students, needing a copy, has to come back disappointed
is greater than 0.90? (Use normal approximation to binomial.)
Hint : If X is the required number of copies, P(X £ X )  0.90.
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7. The grades on a short quiz in biology were 0, 1, 2, 3, ...... 10 points, depending upon the
number of correct answers out of 10 questions. The mean grade was 6.7 with standard
deviation of 1.2. Assuming the grades to be normally distributed, determine (i) the
percentage of students scoring 6 points, (ii) the maximum grade of the lowest 10% of the
class.
Hint : Apply normal approximation to binomial.
8. The following rules are followed in a certain examination. "A candidate is awarded a first
division if his aggregate marks are 60% or above, a second division if his aggregate marks
are 45% or above but less than 60% and a third division if the aggregate marks are 30% or
above but less than 45%. A candidate is declared failed if his aggregate marks are below
30% and awarded a distinction if his aggregate marks are 80% or above. "
At such an examination, it is found that 10% of the candidates have failed, 5% have obtained
distinction. Calculate the percentage of students who were placed in the second division.
Assume that the distribution of marks is normal. The areas under the standard normal
curve from 0 to z are:
z : 1.28 1.64 0.41 0.47
Area : 0.4000 0.4500 0.1591 0.1808

Hint : First find parameters of the distribution on the basis of the given information.
9. For a certain normal distribution, the first moment about 10 is 40 and the fourth moment
about 50 is 48. What is the mean and standard deviation of the distribution?
Hint : Use the condition b = 3, for a normal distribution.
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10. In a test of clerical ability, a recruiting agency found that the mean and standard deviation
of scores for a group of fresh candidates were 55 and 10 respectively. For another experienced
group, the mean and standard deviation of scores were found to be 62 and 8 respectively.

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