AP Statistics
13.1 Test for Goodness of Fit
Read page 728, paragraph 2, and explain the use of the chi-square (  2 ) test for goodness of fit.
Goodness of Fit Test
A ________________ is used to help determine whether a population has a certain hypothesized
________________, expressed as proportions of population members falling into various
outcome categories. Suppose that the hypothesized distribution has ____ outcome categories. To
test the hypothesis
first calculate the chi-square test statistic
Then _____ has approximately a _____ distribution with ________ degrees of freedom.
For a test of H o against the alternative hypothesis
The P-value is ____________.
Conditions:
You may use this test with critical values form the chi-square distribution when all
_____________________ are at least 1 and _______________ if the ______________ are less
than _____.
The Graying of America
In recent years, the expression “the graying of America” has been used to refer to the belief that
with better medicine and healthier lifestyles, people are living longer, and consequently a larger
percentage of the population is of retirement age. We want to investigate whether this perception
is accurate. The distribution of the U.S. population in 1980 is shown. We want to determine if the
distribution of age groups in the United States in 1996 has changed significantly from the 1980
distribution.
U.S. Population by age group, 1980
Age Group
Population (in thousands)
0 to 24
93,777
25 to 44
62,716
45 to 64
44,503
65 and older
25,550
Total
Percent
41.39
27.68
19.64
11.28
We will test the following hypothesis:
The idea of this test is this: We compare the observed counts for a sample from the 1996
population with the counts that would be expected if the 1996 distribution were the same as the
1980 distribution, that is if H o were in fact true. The 1980 distribution is the population. The more
the observed counts differ from the expected counts; the more evidence we have to reject H o and
to conclude that the population distribution in 1996 is significantly different form that of 1980.
A random sample of 500 U.S. residents in 1996 is selected and the age of each subject is
recorded.
Sample results for 500 randomly selected individuals in 1996
Age group
Count
0 to 24
177
25 to 44
158
45 to 64
101
65 and older
64
Total
Percent
35.4
31.6
20.2
12.8
Before proceeding with a significance test, its always a good idea to _____________________.
The next step in the test is to calculate the expected counts for each age category.
Expected Counts
Age Group
0 to 24
25 to 44
45 to 64
65 and older
1980 Population in Percents
Expected Counts (1996)
In order to determine whether the distribution has changed since 1980, we need a way to measure
how well the observed counts (O) from 1996 fit the expected counts (E) under H o . The procedure
is to calculate the quantity ____________.
Calculate the goodness of fit
Age group
Observed (O)
O  E 
Expected (E)
2
E
0 to 24
25 to 44
45 to 64
65 and older
2 =
In Table E, for a P-value of 0.05 and degrees of freedom = 3, we find that the critical value is
7.81. Since our  2 = ______________ is _______________ than the critical value, we say that
the probability of observing a result ________________ as the one we actually observed, by
chance alone, is less that ________. Therefore ________________________________________
_____________________________________________________________________________.
The Chi-Square Distributions
The ____________________ are a family of distributions that take only _____________ and
are __________________________. A specific chi-square distribution is specified by one
parameter, called the _________________>
The chi-square density curves have the following properties:
1.
2.
3.
One of the most common applications of the chi-square goodness of fit test is in the field of
genetics. Scientists want to investigate the genetic characteristics of offspring that result from
mating (also called “crossing”) parents with known genetic makeups. They use rules about
dominant and recessive genes to predict that ratio of offspring that will fall in each possible
genetic category. Then, the scientists mate the parents and classify the resulting offspring. The
chi-square goodness of fit test helps the scientists assess the validity of their hypothesized ratios.
Red-Eyed Fruit Flies
Biologists wish to mate fruit flies having genetic makeup RrCc, indicating that it has one
dominant gene (R) and one recessive gene (r) for eye color, along with one dominant (C) and one
recessive (c) gene for wing type. Each offspring will receive one gene for each of the two traits
from both parents. The following table, often called a Punnet square, shows the possible
combinations of genes received by the offspring.
RC
Rc
rC
rc
RC
Rc
rC
rc
Any offspring receiving an R gene will have red eyes, and any offspring receiving a C gene will
have straight wings. So based on this Punnet square, the biologists predict a ration of
______________, ______________, (x): ____________, _____________, (y) ______________,
______________, (z): _____________, ______________(w) offspring. In order to test their
hypothesis about the distribution of offspring, the biologists mate the fruit flies. Of 200
offspring, 101 had red eyes and straight wings, 42 had red eyes and curly wings, 49 had white
eyes and straight wings, and 10 had white eyes and curly wings. Do these data differ
significantly from what the biologists have predicted?