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Homework 3 – Solutions
Chapter 6

11.
a. P(z ≤ -1) = .1587
b. P(z ≥ -1) = 1 - P(z < -1) = 1 - .1587 = .8413
c. P(z ≥ -1.5) = 1 - P(z < -1.5) = 1 - .0668 = .9332
d. P(-2.5 ≤ z) = 1 - P(z < -2.5) = 1 - .0062 = .9938
e. P(-3 < z ≤ 0) = P(z ≤ 0) - P(z ≤ -3) = .5000 - .0013 = .4987
These probability values are read directly from the table of areas for the cumulative standard normal probability distribution. See Table B in my notes or the relevant table in your book.

17.
Let x = debt amount

μ = 15,015, σ = 3540
a.

z=

18,000 −15,015
= .84
3540
P(x > 18,000) = 1- P(z ≤ .84) = 1 - .7995 = .2005

b.

z=

10,000 − 15,015
= −1.42
3540
P(x < 10,000) = P(z < -1.42) = .0778

c. At 18,000, z = .84 from part (a)

At 12,000, z =

12,000 −15,015
= −.85
3540

P(12,000 < x < 18,000) = P(-.85 < z < .84) = .7995 - .1977 = .6018
d.

z=

14,000 −15,015
= −.29
3540

P(x ≤ 14,000) = P(z ≤ -.29) = .3859
1

22.
Use μ = 32.62 and σ = 2.32
a. We want to find P(30 ≤ x ≤ 35)

At x = 35:

z=

35 − 32.62
= 1.03 .
2.32

At x = 30:

z=

30 − 32.62
= −1.13
2.32

P(30 ≤ x ≤ 35) = P(-1.13 ≤ z ≤ 1.03) = P(z ≤ 1.03) - P(z ≤ -1.13) = .8485 - .1292 = 0.7193
The probability a financial manager earns between $30 and $35 per hour is .7193.
b. Must find the z-value that cuts off an area of .10 in the upper tail. Using the normal tables, we find z = 1.28 cuts off approximately .10 in the upper tail.

So, x = μ + zσ = 32.62 + 1.28(2.32) = 35.59
An hourly pay rate $35.59 or above will put a financial manager in the top 10%.
c. At x = 28: z =

28 − 32.62
= −1.99 . Thus, P(x < 28) = P(z < -1.99) = .0233
2.32

The probability a randomly selected financial manager earns less than $28 per hour is .0233.

24.
a.

xi
∑( xi − x )2
= 200 , s= = 26.04 n n −1
We will use x as an estimate of μ and s as an estimate of σ in

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