Question

Suppose that the average annual revenue of a small business is $150,000 with a standard deviation of $40,000.

Assume that the revenue distribution is normal.

1. What is the probability that one business selected at random makes less than $120,000?

2. What is the probability that the average annual revenue of a random sample of 4 businesses is less than $120,000?

3. Why are your answers to the previous two questions different?

Answer 1

We are given the distribution here as:

Note that we are keeping everything in thousands here.

a) The probability that one business selected at random makes less than $120,000 is computed here as:

Converting it to a standard normal variable, we have here:

Getting it from the standard normal tables, we get:

Therefore 0.2266 is the required probability here.

b) For 4 businesses, the distribution of sample mean is given as:

Therefore the probability now is computed here as:

Getting it from the standard normal tables, we get here:

Therefore 0.0668 is the required probability here.

c) The answers to the above two parts is different because as we increase the sample size, and look at the distribution of sample mean, the standard deviation of the sample decreases, and therefore the probability of being in extreme point decreases.

Therefore the main reason was the change in standard deviation for the distribution of sample mean. The standard deviation for sample mean distribution is always lower than that for the original distribution.