Question

A sociologist finds that for a certain segment of the population, the numbers of years of formal education are normally distributed with a mean of 13.20 years and a standard deviation of 2.95 years. (Remember to draw the distribution)

a) For a person randomly selected from this group, find the probability that he or she has between 13.20 and 13.50 years of education.

b) For a person randomly selected from this group, find the probability that he or she has at least 12.00 years of education.

c) Find the first quartile, Q₁.  That is, find the value separating the lowest 25% from the highest. 75%.

d) If an employer wants to establish a minimum education requirement, how many years of education would be required if only the top 5% of this group would qualify?

Answer 1

Given,

= 13.20 , = 2.95

We convert this to standard normal as

P( X < x) = P( Z < x - / )

a)

P( 13.20 < X < 13.50) = P( X < 1350) - P( X < 13.20)

= P( Z < 13.50 - 13.20 / 2.95) - P( Z < 13.20 - 13.20 / 2.95)

= P( Z < 0.1017) - P( Z < 0)

= 0.5405 - 0.5

= 0.0405

b)

P( X >= 12) = P( Z >= 12 - 13.20 / 2.95)

= P( Z >= -0.4068)

= P( Z < 0.4068)

= 0.6579

c)

First quartile Q1 = 25th perecntile

Q1 = + Z * , Where Z is critical value at 0.25 level,

Q1 = 13.20 + (-0.6745) * 2.95

= 11.2102

d)

We have to calculate x such that P( X > x) = 0.05

That is P( X < x) = 0.95

P( Z < x - / ) = 0.95

From the Z table, z-score for the probability of 0.95 is 1.645

x - / = 1.645

Put the values of and in above equation and solve for x

x - 13.20 / 2.95 = 1.645

x = 18.053