Question

Normal Distribution

Create an example of each of the three types of normal distribution problems.

The project will be graded on:

(3 points each) Each example of one of the problem types.

(1 point) Clear writing style and examples

Answer 1

Normal distribution problems come in three basic types.

1. “Between”: Contain the phrase “between” and includes an upper and lower limit (i.e. “find the number of houses priced between $50K and 200K”).
2. “More Than” or “Above”: contain the phrase “more than” or “above”.
3. “Less Than”.

Example:-

1. X is a normally normally distributed variable with mean ? = 30 and standard deviation ? = 4. Find

a) P(x < 40)

b) P(x > 21)

c) P(30 < x < 35)

Ans-

1. For x = 40, the z-value z = (40 - 30) / 4 = 2.5
   
   Hence P(x < 40) = P(z < 2.5) = [area to the left of 2.5] = 0.9938
   
   b) For x = 21, z = (21 - 30) / 4 = -2.25
   
   Hence P(x > 21) = P(z > -2.25) = [total area] - [area to the left of -2.25]
   
   = 1 - 0.0122 = 0.9878
   
   c) For x = 30 , z = (30 - 30) / 4 = 0 and for x = 35, z = (35 - 30) / 4 = 1.25
   
   Hence P(30 < x < 35) = P(0 < z < 1.25) = [area to the left of z = 1.25] - [area to the left of 0]
   
   = 0.8944 - 0.5 = 0.3944

2.  The annual salaries of employees in a large company are approximateley normally distributed with a mean of $50,000 and a standard deviation of $20,000.

a) What percent of people earn less than $40,000?

b) What percent of people earn between $45,000 and $65,000?

c) What percent of people earn more than $70,000?

Ans-

1. For x = 40000, z = -0.5
   
   Area to the left (less than) of z = -0.5 is equal to 0.3085 = 30.85% earn less than $40,000.
   
   b) For x = 45000 , z = -0.25 and for x = 65000, z = 0.75
   
   Area between z = -0.25 and z = 0.75 is equal to 0.3720 = 37.20 earn between $45,000 and $65,000.
   
   c)For x = 70000, z = 1
   
   Area to the right (higher) of z = 1 is equal to 0.1586 = 15.86% earn more than $70,000.