1. Lesson 15-5 The Normal Distribution
Today’s Date:
5/4/17
Lesson 15-5 The Normal Distribution
Pg #1 – 11, 17 – 23, 25 – 28
Pre-calculus
Normal Distribution (Bell Curve)
Z-Table

2. A frequency polygon can be replaced by a smooth curve
A data set that is normally distributed is called a normal curve.
The normal curve:
- has the general shape of a bell (aka bell curve)
- is symmetric about the vertical line through the mean
- approaches the horizontal axis at both extremes
- mean = median = mode at the line of symmetry
Standard deviation measures how the data varies from the middle.
The total area under a normal curve is 1
- because the curve represents the total probability distribution.
* Note µ (mu) will represent the mean of the data set.

3. For any normal distribution, the following holds (empirical rule):
About 68% of the data lies within 1 standard deviation of the mean
About 95% of the data lies within 2 standard deviations of the mean
About 99.7% of the data lies within 3 standard deviations of the mean
68%
95%
99.7%
*Good strategy is to sketch the normal curve & shade the area you are looking for

4. Ex 1) For the general population, IQ scores are normally distributed with a mean of 100 and standard deviation of 15. Approximately what percent of the population have IQ scores:
a) Between 85 and 115? 
µ = 100
 = 15
µ –  = 100 – 15 = 85
µ +  = = 115
Both scores are within 1 SD of mean  68% 85
115
b) Above 115?
µ +  = = 115 50 34
100 – 84 = 16%
115

5. Ex 1) For the general population, IQ scores are normally distributed with a mean of 100 and standard deviation of 15. Approximately what percent of the population have IQ scores:
c) Below 130?
µ + 2 = (15) = 130
100 – 2.5 = 97.5%
130
2.5%
95%
2.5%

6. Ex 2) In her first year at college, Sandy received a grade of 80 in math and 74 in psychology. The 80 in math was in a class with a mean of 72 and a standard deviation of 4. The 74 in psychology was in a class with a mean of 66 and a standard deviation of 8. Which grade is relatively better?
*Determine how many standard deviations above the mean each score is
math 
psychology 
Math grade is relatively better

7. What if the value we wanted to investigate was not exactly at a standard deviation?
We can use a table of values using the z-score.
Normal Distribution Tables
Table shows value of area under the normal curve to the left of z
For example, if z = then area under the curve =
What if your z-score value is negative?
Use symmetry and draw that picture!

8. Ex 3) Using the table, find the indicated areas under the standard normal curve.
a) between z = 0 and z = 1.03
want:
table:
z = 1.03
.8485 – .5 = .3485
.8485
b) between z = – 0.9 and z = 0.9
want:
table:
z = 0.9
.8159 – .5 = .3159
.8159
= .6318

9. Ex 3) Using the table, find the indicated areas under the standard normal curve.
c) to the right of z = 0.81
want:
table:
z = .81
1 – = .2090
.7910
want
d) to the right of z = – 0.81
want:
table:
z = .81
.7910
same area!

10. Ex 4) An automobile manufacturer claims that a new car gets an average 42 mpg and that the mileage is normally distributed with a standard deviation of 4.5 mpg. If this claim is true, what percent of these new cars will get less than 40 mpg?
table:
z = .44
want:
use symmetry
1 – = .3300
33%
.6700
***NOTE: This question could have been phrased, “What is the probability that one of these cars will have mileage less than 40 mpg?”
(answer would be )

11. Homework
Pg #1 – 11, 17 – 23, 25 – 28