Download Standard Normal Distribution

2.2 Normal Distributions
Normal Distribution and Normal Curve
A Normal Distribution is described by a Normal density
curve. Any particular Normal Distribution is completely
specified by 2 numbers: its mean and standard deviation.
The mean of a Normal Distribution is at the center of the
symmetric Normal Curve. The standard deviation is the
distance from the center to the change of curvature
points on either side. We abbreviate the Normal
Distribution with mean 𝜇 and standard deviation 𝜎
𝑁(𝜇, 𝜎)
68-95-99.7 Rule
In the Normal Distribution with mean 𝜇 and standard
deviation 𝜎:
 Approximately 68% of the observations fall within 𝜎
of the mean 𝜇
 Approximately 95% of the observations fall within
2𝜎 of the mean 𝜇
 Approximately 99.7% of the observations fall within
3𝜎 of the mean 𝜇
Check Your Understanding
The distribution of heights of young women aged 18 to
24 is approximately 𝑁(64.5, 2.5)
1. Sketch a Normal density curve for the distribution of
young women’s heights. Label the points one, two, and
three standard deviations from the mean.
2. What percent of young women have heights greater
than 67 inches? Show you work.
3. What percent of young women have heights between
62 and 72 inches? Show your work.
Standard Normal Distribution
The standard Normal distribution is the Normal
distribution with mean 0 and standard deviation 1.
If a variable x has a Normal distribution 𝑁(𝜇, 𝜎)
Then the standardized variable 𝑧 =
𝑥−𝜇
𝜎
has the standard
Normal distribution.
Standard Normal Table
Table A is a table of areas under the standard Normal
curve. The table entry for each value z is the area under
the curve to the left of z.
Check Your Understanding
Use Table A in the back of the book to find the
proportion of observations from a standard Normal
distribution that fall in each of the following regions. In
each case, sketch a standard Normal curve and shade
the area representing the region.
1. 𝑧 < 1.39
2. 𝑧 > −2.15
3. −.56 < 𝑧 < 1.81
Use Table A to find the value 𝒛 from the standard
Normal distribution that satisfies each of the following
conditions. Sketch a curve representing your findings.
4. The 20th percentile
5. 45% of observations are > 𝑧
How to Solve Problems Involving Normal Distributions
State: Express the problem in terms of the observed
variable x.
Plan: Draw a picture of the distribution and shade the
area of interest under the curve.
Do: Perform calculations
 Standardize and
 Use Table A (or calculator)
Conclude: Write your conclusion in the context of the
problem.
Check Your Understanding
1. Cholesterol levels above 240 mg/dl may require
medical attention. What percent of 14 yr old boys have
more than 240 mg/dl of cholesterol, if the distribution of
cholesterol levels for 14 yr old boys is 𝑁(170, 30)?
2. People with cholesterol levels between 200 and 240
mg/dl are at considerable risk for heart disease. What
percent of 14 yr old boys have blood cholesterol
between 200 and 240 mg/dl?
3. What distance would a ball have to travel to be at the
80th percentile of Tiger Woods’s drive lengths, if driving
lengths had a mean of 304 yds and standard deviation of
8 yrds?
Assessing Normality
To determine whether your data is normally distributed,
make a Normal probability plot.
This is just a graph of the value of your variable in the
problem against the z-score.
If the Normal probability plot is approximately linear,
your data is approximately Normally distributed. If not, it
is not.