Question

The annual sales of romance novels follow the normal distribution. However, the mean and the standard deviation are unknown. Forty percent of the time, sales are more than 470,000, and 10% of the time, sales are more than 500,000. What are the mean and the standard deviation?

Answer 1

Mean(m) = 462,536

sd = 29,268.29

Explanation:

Given the following:

P(sales > 470,000) = 40% = 0.4

P(sales > 500,000) = 10% = 0.1

Using the z - table, we can locate the corresponding P values

Z = 1 - p = 1 - 0.4 = 0.6; 1 - 0.1 = 0.9

Locating the closest value to 0.6 on the z table ;

(0.25 + 0.26) / 2 = 0.255

Locating the closest value to 0.9 on the z table ;

Z = 1.28

Recall;

z =( x - m) / sd

Where m = mean ; sd = standard deviation

First condition:

0.255 = (470,000 - m) / sd

0.255 × sd = (470,000 - m) - - - - - (1)

1.28 = (500,000 - m) / sd

1.28 × sd = (500,000 - m) - - - - (2)

We can solve for one of the unknowns y subtracting equation (1) FROM 2

1.28sd - 0.255sd = (500,000 - m) - (470000 - m)

1.025sd =500,000 - m - 470000 + m

1.025sd = 30,000

sd = 29,268.29

Substituting the value od SD into (1) or (2)

1.28 × 29,268.29 = 500000 - m

37463.41 = 50000 - m

m = 50000 - 37463.41

Mean(m) = 462,536