Download HW Guide for Chapter 2

Topic: Section 2.1-
Name:
Density Curves
Subject: AP Statistics
Date:
Density Curves
How do you construct a density
(smooth) curve for a distribution?
Define mathematical model
Why is it easier to work with a smooth
curve than with a histogram?
What do the areas of the bars in a
histogram represent?
What value should the total area under
the curve equal? What percentage of
the data values does this represent?
Definition of density curve:
A density curve is a _____________________ for a distribution. The area
under a density curve is ______, representing _____% of the data values in
the distribution and shaded areas under the curve represent
________________________________________________________.
Example:
Interpreting a density curve
The histogram below represents the final grades for Miss Paul’s Geometry
classes.
1. What proportion of students had
scores below 70?
Answer:
2. What proportion of students had
scores between 75 and 100?
Answer:
As data sets become larger, we can
draw a curve which describes the
overall pattern of the distribution.
Draw a smooth curve on the diagram
of the histogram above.
3. About what proportion of this
curve lies below 70?
Answer:
4. About what proportion of this
curve lies between 70 and 75?
Answer:
5. If the proportions represented by
each bar in the histogram were
equal to the area of each bar,
then what is the total area under
this curve?
Answer:
Shape of a Density Curve
What shape-attributes can a density
curve have?
A density curve that is symmetric is called a _________________ curve.
Picture:
If the tail of a density curve has been pulled to the right, the curve is said to
be _______________________.
Picture:
If the tail of a density curve has been pulled to the left, the curve is said to
be _______________________.
Picture:
True or false: outliers are described
by a density curve.
Explanation:
Mean & Median of a Density
Curve
What does the median of a density
curve represent?
How do you find the quartiles of a
density curve?
What does the mean of a density
curve represent?
Locate the mean and median in these
density curves:
What shape does a density curve take
if its median is equal to its mean?
True or false: we can identify the
standard deviation of a
density curve by eye.
Explanation:
Notation for mean & median:
The Greek symbol (mu) _____ represents ______________, while the
symbol (sigma) _____ represents ________________ for a density curve.
Summary of Density Curves:
Highlight 5-10 critical words and
phrases from the notes above and use
them to write a 3-4 sentence summary
of density curves.
Homework: Exercises 2.1 and 2.4 pg 83
Topic: Section 2.1-
Name:
The Normal Distribution
Subject: AP Statistics
Normal Distributions
Define normal distribution.
How do you describe the exact density
curve for a particular normal
distribution?
What happens if you change
changing
σ?
Draw a picture.
What happens if you change
changing
μ?
μ without
σ without
Draw a picture.
Which is more spread out: a
distribution with a large σ or a figure
with a small
σ?
Draw a picture.
How can you locate μ by eye on a
normal distribution?
How can you locate σ by eye on a
normal distribution?
What are inflection points?
Are all density curves described by
their μ and σ?
Why are normal distributions
important in statistics?
Date:
The Empirical Rule (68-95-99.7)
State the empirical rule (or, as it is
also called, the 68-95-99.7 rule)
Draw a picture of the 68-95-99.7 rule:
Give the short notation for a normal
distribution with mean, μ and standard
deviation,
σ.
Write the short notation describing the
following distribution:
The scores on the math portion of the SAT exam are normally distributed
with a mean score (  ) of 500 points and a standard deviation (  ) of 50
points
Describe each quartile of a distribution
in terms of percentiles.
The 1st quartile is the ______-th percentile.
The median is the ______-th percentile.
The 3rd quartile is the ______-th percentile.
Summary of Normal Distributions:
Highlight 5-10 critical words and
phrases from the notes above and use
them to write a 3-4 sentence summary
of normal distributions.
Homework: Exercises 2.6 and 2.7 pg 89, and Exercises 2.15 and 2.16 pg 91
Topic: Section 2.2-
Name:
Standard Normal Calculations
Subject: AP Statistics
Date:
Standard Normal Calculations
What does it mean to standardize a
normal distribution?
Define z-score:
What is the significance of a z-score?
A z-score tells us how many ___________________________ the original
observation falls away from the _________________.
Observations larger than the mean are ________________, while
observations smaller than the mean are ________________.
Example:
Use z-scores to answer the question to
the right. Recall that the larger the
z  score , the less “usual” the
observation, since the mean
represents the average measurement.
1. Former NBA star Michael Jordan is 78” tall, while WNBA player Rebecca
Lobo is 76” tall. Men’s heights have a mean of 69 in, and a standard
deviation of 2.8 in. Women’s heights have a mean of 63.6 in, and a
standard deviation of 2.5 in. Which player is relatively taller? In other
words, which player is taller in comparison to their gender?
*Note*
We use absolute value because a zscore of -1.5 is farther from the mean
than a z-score of 1.1)
2. Former NBA player Mugsy Bogues is 63 in. tall. Whose height is more
“unusual”, Mugsy’s or Michael’s?
Define standard normal distribution:
The standard normal distribution is the normal distribution with
mean____ and standard deviation_____, shown below.


Standard normal distribution: N 0,1
Notation:
If the variable x has any normal distribution N  ,   with mean,  , and
standard deviation,

, then the standardized variable, z has the standard
normal distribution. Here, z =
x

.
Normal Distribution Calculations
How can we find what proportion of
observations lie in some range of
values in a normal distribution?
Shade the portion of the area under
the standard normal curve given by
the z-value in a z-table:
z
What is the best strategy (key) to
doing a normal calculation (page 100)?
Examples:
Find the proportion of observations
from the standard normal distribution
less than 1.7 and sketch a picture of
the area you have found.
Area:
Picture:
Find the proportion of observations
from the standard normal distribution
greater than -2.0 and sketch a
picture of the area you have found.
Area:
Picture:
Find the proportion of observations
from the standard normal distribution
between -1.2 and 2.57, and then
sketch a picture of the area you have
found.
Area:
Picture:
Beware!! Avoid this common mistake:
Finding Normal Proportions
Step 1:
To get full credit on any normal
calculation, you must include each of
the following steps:
Step 2:
Step 3:
Step 4:
What do you do of you need to find a
z-value that falls outside the range of
the z-table?
What do you do if you are given a
value for z and need to find the
observation that matches your zscore?
Summary of Normal Calculations:
Highlight 5-10 critical words and
phrases from the notes above and use
them to write a 3-4 sentence summary
of normal distributions.
Homework: Exercises 2.20 pg 95, and Exercises 2.23-2.25 pg 91
Topic: Section 2.2Assessing Normality
Name:
Subject: AP Statistics
Date:
Assessing Normality
Describe two methods for determining
if a distribution is approximately
normal.
Method 1:
Method 2:
How do you construct a normal
probability plot?
Step 1:
Step 2:
Step 3:
Draw an example of a normal
probability plot drawn for a
distribution that is approximately
normal and for a distribution that is
not normal.
If a distribution is not normal:
We can still get information about the
shape of the distribution from the
normal probability plot.
Approximately normal:
Not close to normal:
What we know about shape:
If the bulk of the x-y values of the
normal plot lie on an imaginary,
straight line with only a few data
values falling above or below the
line, then those points are likely
outliers.
What it looks like:
Possible
outliers
Possible
outliers
If both ends of the normality plot
bend above an imaginary line
passing through the bulk of the x-y
values of the probability plot, then
the population may be skewed
right.
If both ends of the normality plot
bend below an imaginary line
passing through the bulk of the x-y
values of the probability plot, then
the population may be skewed left.
Summary of Assessing Normality:
Highlight 5-10 critical words and
phrases from the notes above and use
them to write a 3-4 sentence summary
of normal distributions.
Homework: Exercise 2.27 pgs 108-109 and Exercise 2.35 pg 111: Please ensure responses are complete!