Download 2.2 A Notes - Vista Peak Prep Math

AP Stats
2.2A Notes
Name
Date
Period
Chapter 2: Modeling Distributions of Data
2.2A: Describing a Location in a Distribution
 Density Curve
A density curve is a curve that:
 is always on or above the _________________ axis, and
 has area exactly ___ underneath it.
A density curve describes the __________ pattern of a distribution. The ______ under the curve and above any
interval of values on the horizontal axis is the ________________ of all observations that fall in that interval.
Mean vs. Median in a Density Curve
Mean: still the balancing point
Median: still the equal-areas point.
Sketch the mean and median for each of the following graphs…
Chapter 2: Modeling Distributions of Data
2.2: Normal Distributions
 Normal Distributions
One particularly important class of density curves are the Normal curves, which describe Normal distributions.
 All Normal curves are ___________________, single-peaked, and ________-shaped.
 A specific Normal curve is described by giving its mean ___ and standard deviation ____.
Definition:
A Normal distribution is described by a Normal density curve. Any particular normal distribution is completely
specified by two numbers: its mean  and standard deviation σ.
 The mean of a Normal distribution is the __________ of the symmetric normal curve.
AP Stats


2.2A Notes
The standard deviation is the distance from the center to the _________-of-__________ points on either
side.
We abbreviate the normal distribution with mean  and standard deviation σ as N(, σ).
 The 68-95-99.7 Rule (“The Empirical Rule”)
Although there are many Normal curves, they all have properties in common.
Definition:
In the Normal distribution with mean  and standard deviation σ:
 Approximately 68% of the observations fall within σ of .
 Approximately 95% of the observations fall within 2σ of ..
 Approximately 99.7% of the observations fall within 3σ of .
Helmet Sizes:
The army reports that the distribution of head circumference among male soldiers is approximately normal with
mean 22.8 inches and standard deviation 1.1 inches, (that is, N(22.8, 1.1)).
 What percent of soldiers have head circumference between 21.7 inches and 23.9 inches?

What percent of soldiers have a head circumference greater than 23.9 inches?

What percent of soldiers have a head circumference less than 20.6 inches?

Between what head circumferences do the middle 95% of soldiers fall?
 The Standard Normal Distribution
All Normal distributions are the same if we measure in units of size σ from the mean  as center.
Definition:
The standard Normal distribution is the Normal distribution with mean ___ and standard deviation ___. If a
variable x has any Normal distribution N(,σ) with mean  and standard deviation σ, then the standardized
variable
z=
has the standard Normal distribution, N(0, 1).
 The Standard Normal Table
Because all Normal distributions are the same when we standardize, we can find areas under any Normal curve
from a single table.
Definition:
Table A is a table of areas under the standard Normal curve. The table entry for each value z is the ________
under the curve to the _______ of z.
AP Stats
Use Table A to find:
 z < –2.2

z > 1.34

z > 5.2
2.2A Notes

0.58 < z < 1.79

Find z that corresponds
with the 20th percentile.

Find z such that 45% of
all observations are
greater than z.