Question

1. A bank has kept records of the checking balances of its 4000 customers and determined that the average daily balance of its customers is $300 with a standard deviation of $48. A random sample of 144 checking accounts is selected. Please answer the following questions.

(a)	What is the probability that the sample mean will be more than $309?
(b)	What is the probability that the sample mean will be between $292 and $308?
(c)	Suppose the number of customers decreases from 4000 to 2400. What is the probability that the sample mean will be between $296 and $305?

Answer 1

Given,

= 300 , = 48

Using central limit theorem, .

P( < x) = P (Z < x - / / sqrt(n) )

a)

P( > 309) = P( Z > 309 - 300 / 48 / sqrt(144) )

= P( Z > 2.25)

= 0.0122

b)

P(292 < < 308) = P( < 308) - P( < 292)

= P( Z < 308 - 300 / 48 / sqrt(144) ) - P( Z < 292 - 300 / 48 / sqrt(144) )

= P( Z < 2) - P( Z < -2)

= 0.9772 - 0.0228

= 0.9545

c)

P(296 < < 305) = P( < 305) - P( < 296)

= P( Z < 305 - 300 / 48 / sqrt(144) ) - P( Z < 296 - 300 / 48 / sqrt(144) )

= P( Z < 1.25) - P( Z < -1)

= 0.8944 - 0.1587

= 0.7357