Transcript Section 1.3

Section 1.3
Density Curves and Normal
Distributions
Basic Ideas
• One way to think of a density curve is as a smooth
approximation to the irregular bars of a histogram.
• It is an idealization that pictures the overall pattern of the
data but ignores minor irregularities.
• Oftentimes we will use density curves to describe the
distribution of a single quantitative continuous variable
for a population (sometimes our curves will be based on a
histogram generated via a sample from the population).
– Heights of American Women
– SAT Scores
• The bell-shaped normal curve will be our focus!
Density Curve
Page 64
Shape?
Center?
Spread?
Sample Size =105
Page 65
Density Curve
Shape?
Center?
Spread?
Sample Size=72 Guinea pigs
Two Different but
Related Questions!
1. What proportion
(or percent) of seventh
graders from Gary,
Indiana scored below 6?
2. What is the probability
(i.e. how likely is it?)
that a randomly chosen
seventh grader from Gary,
Indiana will have a test
score less than 6?
Example 1.22
Page 66
Sample Size = 947
Relative “area under the
curve”
VERSUS
Relative “proportion of
data” in histogram
bars.
Page 67 of text
The classic “bell shaped”
density curve.
Shape?
Center?
Spread?
Median separates area under
curve into two equal areas
(i.e. each has area ½)
A “skewed” density curve.
What is the geometric interpretation
of the mean?
The mean as “center of mass” or “balance point” of the density curve
• We usually denote the mean of a density curve by m rather
than x .
• We usually denote the standard deviation of a density
curve by s instead of s.
The normal density curve!
Shape? Center? Spread?
Area Under Curve?
Assume Same Scale on
Horizontal and Vertical
(not drawn) Axes.
How does the standard
deviation affect the
shape of the normal
density curve?
How does the magnitude
of the standard deviation
affect a density curve?
(aka the “Empirical Rule”)
The distribution of heights of young women (X) aged 18 to 24 is
approximately normal with mean mu=64.5 inches and standard
deviation sigma=2.5 inches (i.e. X~N(64.5,2.5)). Lets draw the
density curve for X and observe the empirical rule!
Example 1.23, page 72
How many standard
deviations from the mean
height is the height of a
woman who is 68 inches?
Who is 58 inches?
Note: the z-score of an observation x is simply the
number of standard deviations that separates x from
the mean m.
The Standard Normal
Distribution
(mu=0 and sigma=1)
Notation:
Z~N(0,1)
Horizontal axis in units of z-score!
Let’s find some proportions
(probabilities) using normal
distributions!
Example 1.25 (page 75)
Example 1.26 (page 76)
(slides follow)
Let’s draw the
distributions by hand
first!
Example 1.25, page 75
TI-83 Calculator Command: Distr|normalcdf
Syntax: normalcdf(left, right, mu, sigma) = area under curve from left to right
mu defaults to 0, sigma defaults to 1
Infinity is 1E99 (use the EE key), Minus Infinity is -1E99
Example 1.26, page 76
On the TI-83: normalcdf(720,820,1026,209)
Let’s find the same probabilities using z-scores!
The Inverse Problem:
Given a normal density proportion or
probability, find the corresponding z-score!
What is the z-score such that 90% of the data has a z-score
less than that z-score?
(1) Draw picture!
(2) Understand what you are solving for!
(3) Solve approximately! (we will also use the invNorm key
on the next slide)
Now try working Example 1.30 page 79!
(slide follows)
Syntax:
invNorm(area,mu,sigma)
gives value of x with area
to left of x under normal
curve with mean mu and
standard deviation sigma.
TI-83: Use Distr|invNorm
How can we use our
TI-83s to solve this??
invNorm(0.9,505,110)=?
invNorm(0.9)=?
Page 79
How can we tell if our data is “approximately normal?”
Box plots and histograms should show essentially symmetric,
unimodal data. Normal Quantile plots are also used!
Histogram and Normal Quantile Plot for Breaking
Strengths (in pounds) of Semiconductor Wires
(Pages 19 and 81 of text)
Histogram and Normal Quantile Plot for Survival Time of
Guinea Pigs (in days) in a Medical Experiment
(Pages 38 (data table), 65 and 82 of text)
Using Excel to Generate Plots
• Example Problem 1.30 page 35
– Generate Histogram via Megastat
– Get Numerical Summary of Data via Megastat
or Data Analysis Addin
– Generate Normal Quantile Plot via Macro (plot
on next slide)
Normal Quantile plot for
Problem 1.30 page 35
Extra Slides from Homework
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Problem 1.80
Problem 1.82
Problem 1.119
Problem 1.120
Problem 1.121
Problem 1.222
Problem 1.129
Problem 1.135
Problem 1.80 page 84
Problem 1.83 page 85
Problem 1.119 page 90
Problem 1.120 page 90
Problem 1.121 page 92
Problem 1.122 page 92
Problem 1.129 page 94
Problem 1.135 page 95-96