1. Finding Probabilities
Section 5.3
Normal Distributions
Finding Probabilities

2. Probabilities and Normal Distributions
If a random variable, x is normally distributed, the probability that x will fall within an interval is equal to the area under the curve in the interval.
IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. Find the probability that a person selected at random will have an IQ score less than 115.
Recall that in a discrete probability distribution, we could use the area of the bar in the probability histogram to obtain the probability of the event. Here we can only find the probability that x will lie in a given interval.
100
115
To find the area in this interval, first find the standard score equivalent to x = 115.

3. Probabilities and Normal Distributions
Find P(x < 115).
100
115
Standard Normal Distribution
SAME
SAME
The area is the same
Find P(z < 1). 1
P(z < 1) = , so P(x <115) =

4. Application
Monthly utility bills in a certain city are normally distributed with a mean of $100 and a standard deviation of $12. A utility bill is randomly selected. Find the probability it is between $80 and $115.
Normal Distribution
P(80 < x < 115)
P(–1.67 < z < 1.25)
– =
The probability a utility bill is between $80 and $115 is

5. From Areas to z-Scores z –1 1 2 3 4
Find the z-score corresponding to a cumulative area of
z = 2.06 corresponds
roughly to the
98th percentile.
0.9803
Be sure to emphasize that here, the area is given. Tell students to choose the z score closest to the given area. The only exception is if the area falls exactly at the midpoint between two z-scores, use the midpoint of the z=scores. –4 –3 –2 –1 1 2 3 4 z
Locate in the area portion of the table. Read the values at the beginning of the corresponding row and at the top of the column. The z-score is 2.06.

6. Finding z-Scores from Areas
Find the z-score corresponding to the 90th percentile.
.90 z
The closest table area is The row heading is 1.2 and column heading is .08. This corresponds to z = 1.28.
A z-score of 1.28 corresponds to the 90th percentile.

7. Finding z-Scores from Areas
Find the z-score with an area of .60 falling to its right.
.40
.60 z z
With .60 to the right, cumulative area is .40. The closest area is The row heading is 0.2 and column heading is .05. The z-score is 0.25.
A z-score of 0.25 has an area of .60 to its right. It also corresponds to the 40th percentile

8. Finding z-Scores from Areas
Find the z-score such that 45% of the area under the curve falls between –z and z.
.275
.275
.45 –z z
The area remaining in the tails is .55. Half this area is
in each tail, so since .55/2 = .275 is the cumulative area for the negative z value and = .725 is the cumulative area for the positive z. The closest table area is and the z-score is The positive z score is 0.60.
Because the normal distribution is symmetric, the z scores will have the same absolute value. As a result, you can find one z-score and use its opposite for the other.

9. From z-Scores to Raw Scores
To find the data value, x when given a standard score, z:
The test scores for a civil service exam are normally distributed with a mean of 152 and a standard deviation of 7. Find the test score for a person with a standard score of:
(a) (b) – (c) 0
(a) x = (2.33)(7) =
Show students that the formula given is equivalent to the z-score formula. Some students prefer to use only one formula and others like to use both. Have students work these through before displaying the answers. Emphasize the meaning of z-scores. A z-score of 2.33 is a 2.33 standard deviations above the mean.
(b) x = (–1.75)(7) =
(c) x = (0)(7) = 152

10. Finding Percentiles or Cut-off Values
Monthly utility bills in a certain city are normally distributed with a mean of $100 and a standard deviation of $12. What is the smallest utility bill that can be in the top 10% of the bills?
$ is the smallest
value for the top 10%.
90%
10% z
Students find these “cut-off” problems easier if they think in terms of percentiles, which in turn are interpreted as cumulative areas.
Find the cumulative area in the table that is closest to (the 90th percentile.) The area corresponds to a z-score of 1.28.
To find the corresponding x-value, use
x = (12) =

11. Homework : 1-37 (odd) pgs
42-46 even pgs Day 2:Homework : 2-36 (even) pgs
41-45 odd pgs