Download The Standard Normal Distribution

The Standard Normal
Distribution
Section 4.3.1
Example 1
Distribution of head circumference for male
soldiers is approximately normal with
mean = 22.8 inches and std. dev. 1.1 inches.
a) What percent of soldier have head
circumference greater than 23.9 inches?
b)
Between 21.7 and 23.9 inches?
b)
Less than 25.1 inches?
UH OH…..what do we do now?????!?!?!?!
Standardizing and Z - Scores

All normal distributions are the same if we
measure in units of size σ about the mean μ as center.

Z – score: x is an observation that has μ and σ, so the
standardized value is:

ALWAYS ROUND THE Z-SCORE TO 2 DECIMAL PLACES!

The z-score tells us how many standard deviations the
original observation falls away from the mean.
 If x > μ, the z-score will be positive.
x 
z=

 If
x < μ, the z-score will be negative.
Example 1 continued . . .
Distribution of head circumference for male
soldiers is approximately normal with
mean = 22.8 inches and std. dev. 1.1 inches.
Find the z-score for head sizes less than 25.1 in.
x 
z=

Standard Normal Distribution


N(0, 1)
Find areas under any standard normal curve
using Table A or the z- table.
Example 1 almost complete . . .
Distribution of head circumference for male
soldiers is approximately normal with
mean = 22.8 inches and std. dev. 1.1 inches.
The z-score for head size less than 25.1 in is 2.09.
Using Table A find the probability of head sizes
less than 25.1 inches.
Example 2
N(64.5, 2.5) is the normal distribution of
women’s heights in inches. What proportion
of all young women are less than 68 inches tall?
Find the z-score for height less than 68 in.
x 
z=

Example 2 continued . . .
Understand that the area under the N(64.5, 2.5) curve to
the left of x = 68 is equal to the area to the left of z = 1.4
under the standard normal curve N(0, 1).
Use Table A to find the percent of women that are less
than 68 inches tall.
REMEMBER:
The z- value in Table A is the
area/percentage to the left of the
z-score!!!!!!!!
Examples Using Table A
1. Find the proportion of observations from
the standard normal distribution that are
less than 3.3
2. Find P(Z > -2.15).
3. Find the proportion of observations from the
standard normal distribution that are between
-2.15 and 1.4.
Practice Using Table A
4. Find the proportion of observations from
the standard normal distribution that are less
than 2.2.
98.61%
5. Find the proportion of observations from the
standard normal distribution that are greater
than -1.35.
1 - .0885 = .9115 = 91.15%
6. P(-1.30 < Z < 2.65)
.9960 - .0968 = .8992 = 89.92%
Normal Curve & Table A
Practice Answers
1.
3.
5.
7.
50%
99.83%
4.66%
94.78%
2. 70.88%
4. 95.73%
6. 91.46%
FINDING NORMAL
PROPORTIONS
&
FINDING A VALUE
GIVEN A PROPORTION
Section 4.3.2
FINDING NORMAL PROPORTIONS
1.
Standardize x to a standard normal variable z.
2.
Draw a picture and shade the area of interest
under the standard normal curve.
3.
Find area using Table A and the fact that the
total area under the curve is 1.
4.
Write a conclusion in context of the problem.
Example 1
Height for women is N(64.5, 2.5).
a. Find P (x< 68)
b.
Find P (x > 63)
c.
Find P (63 < x < 68)
Example 2
For 14-year old boys the distribution of blood
cholesterol levels is normal with N(170, 30).
a. What percent of 14 year old boys have
more than 240 mg/dl of cholesterol?
b.
P (x ≤ 225)
c.
P( 170 ≤ x ≤ 240)
We can standardize different
distributions so that we can
easily compare them.
Example 3
Jessica took the SATs and the ACTs. On
the SATs she got a 610. The SATs are
normally distributed with a mean of 500
and standard deviation of 100. On the
ACTs she got a 27. The ACTs are
normally distributed with a mean of 18
and standard deviation of 6.
On which test did she get a better score?
FINDING A “Z” VALUE GIVEN A PROPORTION
A.K.A. Working Backwards with Table A
Example 4
Find the value z on the standard normal
curve that is less than only the top 30%
of the data.
FINDING A “X” VALUE GIVEN A PROPORTION
A.K.A. Working Backwards with Table A
Example 5
Find the value on the curve X = N(60, 2)
that is greater than 20% of the population.
Normal Curves Practice Answers
1. a. 50%
b. 97.5%
d. 16%
e. 0%
g. 97.36%
2. z = .675
3. x = 13.425
4. x = 19.935
5. -1.645 < x < 1.645
6. 72.2125 inches or taller
7. a. z = 0.525 b. z = .845
8. a. 5.16%
b. 54.71%
c. x = 279.52 days or more
c. 95%
f. 98.68%