### P. STATISTICS LESSON 13 – 1 (DAY 1)

```AP STATISTICS
LESSON 13 – 1
(DAY 1)
CHI-SQUARE PROCEDURES
TEST FOR GOODNESS OF FIT
ESSENTIAL QUESTION:
2
What is X , goodness of fit, and
how are they used in statistics?
Objectives:
• To use data to find X2
• To use X2 to find the probability of a
sample fitting a population.
Introduction
Sometimes we want to examine the distribution of
proportions in a single population.
The chi-square test for goodness of fit allows us to
determine whether a specified population
distribution seems valid.
We can compare two or more population
proportions using chi-square test for homogeneity
of populations.
Introduction (continued…)
In doing so, we will organize our data in a two-way table.
It is also possible to use the information provided in a twoway table to determine whether the distribution of one
variable has been influenced by another variable.
The chi-square test of association/independence helps us
decide this issue.
Tests for Goodness of Fit
There is a single test that can be applied to see if the
observed sample distribution is significantly different
from the hypothesized population distribution. It is
called the chi-square (χ2) test for goodness of fit.
Before proceeding
with a significance
test, it’s always a
good idea to plot the
data.
Χ2 Formula
We need way to determine how well the observed
counts ( O ) fit the expected counts ( E ) under Ho.
The procedure is to calculate the quantity
( O – E )2
E
for each category and then add up these terms.
The sum is labeled Χ2 and is called the chi-square
statistic.
X 2 Formula (continued…)
Degrees of freedom is determined by taking the
number of categories and subtracting 1.
The chi-square test statistic is a point on the
horizontal axis, and the area to the right is the Pvalue of the test. This P-value is the probability of
observing a value X 2 at least as extreme as the one
actually observed.
Use Table E (Chi-squared Distribution Critical
Values) to find the P-values.
The CHI-SQUARE DISTRIBUTIONS
The chi-square distributions are a family of
distributions that take only positive values
and are skewed to the right.
A specific chi-square distribution is
specified by one parameter, called the
degrees of freedom.
The Chi-square Density Curve Properties
1.
2.
The total area under a chi-square curve is equal to 1.
Each chi-square curve (except when df = 1) begins at 0 on the
horizontal axis, increases to a peak, and then approaches the
horizontal axis asymptotically from above.
3. Each chi-square curve is
skewed to the right. As
the number of degrees of
freedom increases, the
curve becomes more and
more symmetrical and
looks more like a normal
curve.
Example 13.1 Page 728
The Graying of America
With better medicine and healthier lifestyles, people are living
longer. Consequently, a larger percentage of the population is of
retirement age.
We want to determine if the distribution of age groups in the US
in 1996 as changed significantly from the 1980 distribution.
Test the following Hypothesis:
H0: age group dist. In 1996 is the
same as the 1980 dist.
Ha: age group dist. In 1996 is the
different from as the 1980 dist.
Goodness of Fit Test
The chi-square test for goodness of fit can be applied to
see if the observed sample distribution is significantly
different from the hypothesized population distribution.
A goodness of fit test is used to help determine whether a
population has a certain hypothesized distribution,
expressed as proportions of population members falling
into various outcome categories.
Goodness of Fit (continued…)
To test the hypothesis
• Ho : the actual population proportions are equal to
hypothesized proportions
– First calculate the chi-square test statistic
X2 = ∑ ( O – E )2 / E
– Then X2 distribution with ( n – 1 ) degrees of freedom.
• For a test of Ho against the alternative hypothesis
– Ha : the actual population proportion are different from
the hypothesized proportions the P-value is P( x2 ≥ X2 ).
Goodness of Fit: Conditions
Conditions: You may use this test with
critical values from the chi-square
distribution when all individual expected
counts are at least 1 and no more than 20%
of the expected counts are less than 5.
Example 13.2 Page 733
Red-eyed Fruit Flies
The most common application of the chisquare goodness of fit test is in the field of
genetics.
In this example, it is used to investigate the
genetic characteristics of offspring that
result from mating parents with known
genetic makeups.
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