Page 219 - DMTH404_STATISTICS

Unit 15: Exponential Distribution and Normal Distribution

Notes

Example 31:
The average monthly sales of 5,000 firms are normally distributed with mean Rs 36,000 and
standard deviation Rs 10,000. Find:
(i) The number of firms with sales of over Rs 40,000.
(ii) The percentage of firms with sales between Rs 38,500 and Rs 41,000.

(iii) The number of firms with sales between Rs 30,000 and Rs 40,000.
Solution.
Let X be the normal variate which represents the monthly sales of a firm. Thus X ~ N(36,000,
10,000).
-
æ 40000 36000ö
P z >
(i) P (X > 40000 ) = ç ÷ = P (z > 0.4 )
è 10000 ø
-
= 0.5000 P (0 £ £ 0.4 ) = 0.5000 0.1554 = 0.3446.
z
-
Thus, the number of firms having sales over Rs 40,000
= 0.3446 × 5000 = 1723

-
-
æ 38500 36000 41000 36000ö
(ii) P (38500 £ X £ 41000 ) = ç £ z £ ÷
P
è 10000 10000 ø
z
z
z
= P (0.25 £ £ 0.5 ) = P (0 £ £ 0.5 ) P- (0 £ £ 0.25 )
= 0.1915 0.0987 = 0.0987.
-
Thus, the required percentage of firms =0.0987 × 100 = 9.87%.
-
-
æ 30000 36000 40000 36000ö
P
(iii) P (30000 £ X £ 40000 ) = ç £ z £ ÷
è 10000 10000 ø
z
z
= P ( 0.6- £ z £ 0.4 ) = P (0 £ £ 0.6 ) + P ( 0 £ £ 0.4 )
= 0.2258 0.1554 = 0.3812.
+
Thus, the required number of firms = 0.3812 ´ 5000 = 1906

Example 32: In a large institution, 2.28% of employees have income below Rs 4,500 and
15.87% of employees have income above Rs. 7,500 per month. Assuming the distribution of
income to be normal, find its mean and standard deviation.

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