Question

Excess revenue (total revenue minus operating expenditures) in the nonprofit sector are normally distributed with a mean of $1.5 million and a standard deviation of $1 million.

(a) What is the probability that a randomly selected nonprofit has negative excess revenues?

(b) What is the probability that a randomly selected nonprofit has excess revenue between $1 million and $2 million?

(c) If 10% of nonprofits are expected to exceed a certain excess revenue level, what is that revenue level?

Answer 1

Given that excess revenue (total revenue minus operating expenditures) in the nonprofit sector are normally distributed with a mean of $1.5 million and a standard deviation of $1 million.

Before we go on to solve the problems let us know a bit about Normal Distribution.

Normal Distribution

A continuous random variable X is said to have a normal distribution if its PDF(Probability Density Function) is given by

its CDF(Cumulative Distribution Function) is given by,

Notation:

Standard Normal Distribution

A continuous random variable X is said to have a standard normal distribution if its PDF(Probability Density Function) is given by

its CDF(Cumulative Distribution Function) is given by,

Exact evaluation of ?(x) is not possible but numerical method can be applied. The values of ?(x) has been tabulated extensively in Biometrika Volume I.

Notation:

Coming back to our problem,

Let,

X=Excess revenue in the nonprofit sector.

(a) Here we need to find the probability that a randomly selected nonprofit has negative excess revenues.

[Now ?(x)=1-?(-x)]

Hence the probability that a randomly selected nonprofit has negative excess revenues is 0.0668

(b) Here we need to find the probability that a randomly selected nonprofit has excess revenue between $1 million and $2 million.

[Now ?(x)=1-?(-x)]

Hence the probability that a randomly selected nonprofit has excess revenue between $1 million and $2 million is 0.383

(c) Here we need to find if 10% of nonprofits are expected to exceed a certain excess revenue level, what is that revenue level.

Hence the revenue level is $2.78 million