Question

2. The following probability distribution of the monthly incomes (X in $1000) of
account executives has been estimated for a big accounting firm.
a. Find the expected value and the standard deviation of the monthly
incomes.
b. A random sample of 64 executives is taken. What is the probability
that the sample mean value exceeds $20,500?

X   12 14 20 22 30
P(X) 0.15 0.15 0.4 0.2 0.1

Answer 1

Given Probability Distribution of the monthly incomes (X) is as follows,

x	f(x)
12	0.15
14	0.15
20	0.4
22	0.2
30	0.1

Part a

Expected value of X is obtained as,

            

To calculate the variance, we need,

Answer : E(X)=19.3 ; V(X)=25.31

Part (b)

Given,
a random sample of size n = 64 is taken.

i.e. are identically and independently distributed with the above probability distribution function.

By the central limit theorem,

To Calculate,


, where is the CDF of a standard normal distribution.

, (Obtained from standard normal tables)

Required probability is 0.0282