Download AP Stats Chapter 2 Notes 2013-14

AP Statistics
Chapter 2 - The Normal Distribution
2.1 Describing
Location in a
Distribution
Objective:
 MEASURE position using percentiles
 INTERPRET cumulative relative frequency graphs
 MEASURE position using z-scores
 TRANSFORM data
 DEFINE and DESCRIBE density curves
Measuring Position:
Percentiles
The pth percentile of a distribution is the value with p percent of the observations less than it.
Examples:
Here are the scores of all 25 students in Mr. Pryor’s statistics class on their first test:
79
77
81
83
80
86
77
90
73
79
83
85
74
83
93
89
78
84
80
82
75
77
67
72
73
Problem: Use the scores on Mr. Pryor’s test to find the percentiles for the for the following students
(how did they perform relative to their classmates):
a) Jenny, who earned an 86.
b) Norman, who earned a 72.
c) Katie, who earned a 93.
d) the two students who earned scores of 80.
A cumulative relative frequency graph (or ogive) displays the cumulative relative frequency of each
class of a frequency distribution.
Cumulative Relative
Frequency Graphs
Age of First 44 Presidents When They Were Inaugurated
Age
Frequency
40-44
2
45-49
7
50-54
13
55-59
12
60-64
7
65-69
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Relative
frequency
Cumulative
frequency
Cumulative
relative
frequency
1
Was Barack Obama, who was inaugurated
at age 47, unusually young?
Was Ronald Reagan, who was inaugurated
at age 69, unusually old?
Example:
Measuring Position:
z-Scores
Check Your Understanding page 89
A z-score tells us how many standard deviations from the mean an observation falls, and in what
direction.
To compare data from distributions with different means and standard deviations, we need to find a
common scale. We accomplish this by using standard deviation units (z-scores) as our scale. Changing
to these units is called standardizing. Standardizing data shifts the data by subtracting the mean and
rescales the values by dividing by their standard deviation.
z  score 
datavalue  mean
st .dev.
or
z
x

Standardizing does not change the shape of the distribution. It changes the center (shifts it to zero) and
the spread by making the standard deviation one.
Example:
#11 page 106
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Transforming Data
Transforming converts the original observations from the original units of measurements to another scale.
Transformations can affect the shape, center, and spread of a distribution.
Effect of Adding (or
Subracting) a
Constant
Adding the same number a (either positive, zero, or negative) to each observation:
• adds a to measures of center and location (mean, median, quartiles, percentiles), but
• Does not change the shape of the distribution or measures of spread (range, IQR, standard
deviation).
Effect of Multiplying
(or Dividing) by a
Constant
Example:
Remember: Exploring
Quantitative Data
Multiplying (or dividing) each observation by the same number b (positive, negative, or zero):
• multiplies (divides) measures of center and location by b
• multiplies (divides) measures of spread by |b|, but
• does not change the shape of the distribution
page 107 # 20
To describe a distribution:
- Make a graph
- Look for overall patterns (shape, center, and spread) and outliers
- Calculate a numerical summary to describe the center (mean, median) and spread (minimum,
maximum, Q1, Q3, range, IQR, standard deviation)
 In addition to the above distributions sometimes the overall pattern of a large number of
observations is so regular that we can describe it by a smooth curve.
Density Curves
A density curve describes the
overall pattern of a distribution
o Is always on or above the
horizontal axis
o Has exactly 1 underneath it
o The area under the curve and
above any range of values is the
proportion of all observations
The overall pattern of this histogram of the scores of all 947 seventh-grade students in Gary, Indiana, on the vocabulary part of the
Iowa Test of Basic Skills (ITBS) can be described by a smooth curve drawn through the tops of the bars.
edian of a density curve is the equal areas point, the point that divides the are under the curve in half
Mean of a density curve is the balance point, at which the curve would balance if made of solid
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Median and Mean of
a Density Curve
Example:
material.
Use the figure shown to answer the following questions.
1. Explain why this is a legitimate density curve.
2. About what proportion of observations lie
between 7 and 8?
3. Mark the approximate location of the median.
4. Mark the approximate location of the mean.
Explain why the mean and median have the
relationship that they do in this case.
Page 108 #28,30
Examples:
Page 108 #32
Assignment page
105-109 #1-31 odd,
33-38 all
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2.2 Normal
Distribution
Objectives:
 DESCRIBE and APPLY the 68-95-99.7 Rule
 DESCRIBE the standard Normal Distribution
 PERFORM Normal distribution calculations
 ASSESS Normality
Normal
Distributions
N(μ,)







The 68-95-99.7 Rule
All Normal curves have the same overall shape: symmetric, single-peaked, bell shaped.
A Normal distribution is described by a Normal density curve.
A Normal distribution can be fully described by two parameters, its mean μ and standard deviation σ
The mean, µ, of a Normal distribution is at the center of the symmetric Normal curve and is the same
as the median.
The standard deviation σ controls the spread of a Normal curve. Curves with larger standard
deviations are more spread out.
The standard deviation, σ, is the distance from the center to the change-of-curvature points on either
side.
A short-cut notation for the normal distribution in N(μ,).
All normal curves obey the 68-95-99.7% (Empirical) Rule.
This rule tells us that in a normal distribution approximately
68% of the data values fall within one standard
deviation (1σ) of the mean,
95% of the values fall within 2σ of the mean, and
99.7% (almost all) of the values fall
within 3σ of the mean.
Application of the
68-95-99.7 Rule
Distribution of the heights of young women aged 18 to 24
What is the mean μ?
What is the ?
What is the height range for 95% of young women?
What is the percentile for 64.5 in.?
What is the percentile for 59.5 in.?
What is the percentile for 67 in.?
What is the percentile for 72 in.?
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The Standard
Normal Distribution
The standard Normal distribution is the Normal distribution with mean 0 and standard deviation 1.
If a variable x has any Normal distribution N(µ,σ) with mean µ and standard deviation σ, then the
standardized variable
Z-Score Table
z
x

has the standard Normal distribution, N(0,1).
Because all Normal distributions are the same when we standardize, we can find areas under any Normal
curve from a single 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.
Example Check Your Understanding page 119
4-Step Process
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 x to restate the problem in terms of a standard Normal variable z.
• Use Table A and the fact that the total area under the curve is 1 to find the required area
under the standard Normal curve.
Conclude: Write your conclusion in the context of the problem.
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Normal calculations
Example: a.
Women’s heights are approximately normal with N(64.5, 2.5).
What proportion of all young women are less than 68 inches tall?
On the driving range, Tiger Woods practices his swing with a particular club by hitting many,
many balls. When tiger hits his driver, the distance the balls travels follows a Normal distribution
with mean 304 yards and standard deviation 8 yards. What percent of Tiger’s drives travel at least
290 yards?
What percent of Tiger’s drives travel between 305 and 325?
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High levels of cholesterol in the blood increase the risk of heart disease. For 14 year old boys, the
Using Table A in
distribution of blood cholesterol is approximately Normal with mean µ = 170 milligrams of
Reverse
Example: cholesterol per deciliter of blood (mg/dl) and standard deviation σ = 30 mg/dl. What is the first
quartile off the distribution of blood cholesterol?
z-scores on the
calculator
Assessing Normality
Technology Corner page 123


Normal Probability
Plots
Plot the data.
• Make a dotplot, stemplot, or histogram and see if the graph is approximately symmetric
and bell-shaped.
Check whether the data follow the 68-95-99.7 rule.
• Count how many observations fall within one, two, and three standard deviations of the
mean and check to see if these percents are close to the 68%, 95%, and 99.7% targets
for a Normal distribution.
If the points on a Normal probability plot lie close to a straight line, the plot indicates that the data are
Normal. Systematic deviations from a straight line indicate a non-Normal distribution. Outliers appear as
points that are far away from the overall pattern of the plot.
Example page 127
Technology Corner page 128
Assignment 2.2
#41-65 odd, 68-74
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Summary
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