Chapter 6, part B

Chapter 6, continued...

C. Standard Normal Probability Distribution The standard normal is a normal distribution where all values of x have been converted to z-scores. Thus =0 and =1 and the RV is referred to as z, which represents how many standard deviations a particular x falls from the mean. How do we find the probability that z falls within 1 standard deviation of the mean? Need to use tables.

-1

0

+1

z

Using the standard normal probability table Look at a table like that found in the inside cover of your textbook. Go down to z=1.0, then across to .00. The result is P(0z1.00)=.3413 Z .9

.00

.01

.02

.3159 .3186 .3212

1.0 .3413 .3438 .3461 1.1 .3643 .3665 .3686

This is an excerpt from the larger table.

Example Suppose the starting salaries of individuals with an MBA degree is normally distributed with a mean of $40,000 and a standard deviation of $5000. What is the probability that a randomly selected individual with an MBA degree will get a starting salary of between $40,000 and $48,000?

Steps for solving 1. Diagram the situation

This is the area we need to calculate.

=40K

48K

x

Steps, continued 2. Convert to z-scores.

This is the area we need to calculate.

Z=(48,000-40,000)/5000= 1.60

0

1.6

z

Steps, continued... We’ve converted the original normal distribution to the standard normal. 3. Go to the table and find z=1.60. The table gives P(0z1.60)=.4452. Since this is exactly what we’re looking for, we’re done. Thus the probability of a salary between $40,000 and $48,000 is .4452.

Another Example What is the probability that a randomly selected individual with an MBA degree will get a starting salary of at least $47,500? Another way of asking the same question is to ask the probability of a person making $47,500 or more.

Solving... 1. Diagram the situation.

This is the area we need to calculate.

40K

47.5K

x

Solving... 2. Convert to z-scores.

This is the area we need to calculate.

Z=(47,500-40,000)/5000=1.50 Look up z=1.50 in the table. 0

This is the area that the table will provide.

1.5

z

Solving... 3. From the table we find P(0z1.50)=.4332, but we actually need the P(z>1.50). Since we know that the area under the entire right side of the curve is .50, and we know the area between 0 and 1.50 is .4332, the area beyond 1.50 must be: P(z>1.50) = .5 - P(0z1.50)=.0668

A diagram of what we did.

.5-.4332=.0668

.4332

0

1.50

Entire area = .5

z