Download AP Stats Chapter 2 Notes 2012-2013

AP Statistics
Chapter 2 - The Normal Distribution
2.1 Describing
Location in a
Distribution
Activity:
Where do I stand?
Objective:
 MEASURE position using percentiles
 INTERPRET cumulative relative frequency graphs
 MEASURE position using z-scores
 TRANSFORM data
 DEFINE and DESCRIBE density curves
1. Fill in dot plot of class heights
_______________________________________________________________________________
55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75
76 77 78
2. What percent of the students in the class have heights less than yours (percentile)? ___________
3. Calculate the mean class height. x = __________
Calculate the standard deviation of class heights. s x = ____________
4. Where does your height fall in relation to the mean: above or below? ____________
How far above or below the mean? ____________
How many standard deviations above or below the mean (z-score)? ____________
5a. What would happen to the class’s height distribution if you converted each data value from
inches to centimeters? (2.54 cm = 1 in)
5b. How would this change of units affect the measures of center, spread, and location (percentile
and z-score) that you calculates?
Measuring Position:
Percentiles
The pth percentile of a distribution is the value with p percent of the observations less than it.
Here are the scores of all 25 students in Mr. Pryor’s statistics class on their first test:
Examples:
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.
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Cumulative Relative
Frequency Graphs
A cumulative relative frequency graph (or ogive) displays the cumulative relative frequency of each
class of a frequency distribution.
Age of First 44 Presidents When They Were Inaugurated
Frequency
40-44
2
45-49
7
50-54
13
55-59
12
60-64
7
65-69
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Relative
frequency
Was Barack Obama, who was inaugurated
at age 47, unusually young?
Was Ronald Reagan, who was inaugurated
at age 69, unusually old?
Cumulative
frequency
Cumulative relative frequency (%)
Age
Cumulative
relative
frequency
100
80
60
40
20
0
40 45 50 55 60 65 70
Age at inauguration
Example:
Check Your Understanding page 89
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Measuring Position:
z-Scores
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:
Transforming Data
Effect of Adding (or
Subracting) a
Constant
Effect of Multiplying
(or Dividing) by a
Constant
Example:
#11 page 106
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.
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).
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
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Remember: Exploring
Quantitative Data
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.
Median and Mean of
a Density Curve
Example:
Median 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
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.
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Examples:
Page 108 #28,30
Page 108 #32
2.1 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 deviation 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 (1) of the mean,
95% of the values fall within 2 of the mean, and
99.7% (almost all) of the values fall
within 3 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
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.
Normal calculations
Women’s heights are approximately normal with N(64.5, 2.5).
Example: a. What proportion of all young women are less than 68 inches tall?
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b. In what percentile does a woman 60 inches tall fall?
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
Technology Corner page 123
Examples Check Your Understanding page 119
Assessing Normality


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.
Normal Probability
Plots
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.
Assignment 2.2
Pt. 1 #41-51 odd
Pt. 2 #53-59 odd
Pt. 3 #63,65,66,68-74
Example page 127
Technology Corner page 128
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Summary
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