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AP Statistics
Chapter 26: Comparing Counts
Test for Goodness of Fit:
To analyze categorical data, we construct two-way tables and examine the counts or percents of the explanatory and response variables.
Hershey Activity:Kind Milk
ChocolateMr.
Goodbar Krackel Special Dark
Claimed % 40% 21% 21% 18%
Proportion of Hershey Miniatures
KIND ObservedCount, O
ExpectedCount, E
Milk Chocolate
Mr. Goodbar
Krackel
Special Dark
2O E
E
We want to compare the observed counts to the expected counts.
The null hypothesis is that there is no difference between the observed and expected counts.
The alternative hypothesis is that there is a difference between the observed and expected counts.
is called the
chi-square test statistic . It measures how well the observed counts fit the expected counts, assuming that the null hypothesis is true.
22 O E
E
The distribution of the chi-square statistic is called the chi-square distribution, 2. This distribution is a density curve.
The total area under the curve is 1 .
The curve begins at zero on the horizontal axis and is skewed right .
As the degrees of freedom increase, the shape of the curve becomes more symmetric .
Using the Hershey’s Miniatures chi-square statistic, find the probability of obtaining a 2 value at least this extreme assuming the null hypothesis is true.
This is known as the “Goodness of Fit
Test.” Graph the chi-square distribution with (k
– 1) = 3 degrees of freedom, where k = # of categories:
TI-83+: 2pdf (X, 5)
We would expect to obtain a 2 value at least this extreme in about _____ out of every _____ samples, assuming the null hypothesis is true.
CONDITIONS: The Goodness of Fit Test may be used when:SRSwe are working with counts .all counts are at least 1 and most (at least 80%) of the counts are at least 5 .
*Following the Goodness of Fit Test,
check to see which component made the greatest contribution to the chi-square statistic to see where the biggest changes occurred.
Inference for Two Way Tables:
To compare two proportions, we use a 2-Proportion Z Test. If we want to compare three or more proportions, we need a new procedure.
The first step in the overall test for comparing
several proportions is to arrange the data in a 2-way table.
Think of the counts as elements of a matrix with r
rows and c columns. This is called an r x c table with (r)(c) cells.
Our null hypothesis is that there is no difference
among the proportions. The alternative hypothesis is that there are some difference among the proportions.
We will use the chi-square test to measure how far the observed values are from the expected values.
To calculate the expected counts, multiply the row total by the column total, and divide by the table total : row total column total
expected counttable total
The chi-square statistic is the sum over all r x c cells in the table:
The degrees of freedom is (r – 1)
(c – 1) . The P-value is the area to the
right of the statistic under the chi-square density curve.
In Summary:Use a chi-square goodness of fit
test when you want to show that the distribution of counts in one categorical variable matches the distribution predicted by a model.
( one categorical variable, one -way table)H0: The distribution of Hershey Miniatures is the same as the advertised distribution.
Use a chi-square test for homogeneity when you want to show that the distribution of counts for two or more groups is the same.
“Are the groups homogeneous?”; more than one population; split, then sample
( one categorical variable, two -way table)
H0: The distribution of X is the same as the distribution of Y.
Use a chi-square test for independence when you want to show that two categorical variables are independent for one group of individuals.
“Are the variables independent?”; one population; sample, then split
( two categorical variables, two -way table)
H0: X and Y are independent.
Slide 26 - 16
What Can Go Wrong?
Don’t use chi-square methods unless you have counts.
Just because numbers are in a two-way table doesn’t make them suitable for chi-square analysis.
Beware large samples.With a sufficiently large sample size, a chi-square test can always reject the null hypothesis.
Don’t say that one variable “depends” on the other just because they’re not independent.
Association is not causation.
Slide 26 - 17
What have we learned?We’ve learned how to test hypotheses about categorical variables.All three methods we examined look at counts of data in categories and rely on chi-square models.
Goodness-of-fit tests compare the observed distribution of a single categorical variable to an expected distribution based on theory or model.Tests of homogeneity compare the distribution of several groups for the same categorical variable.Tests of independence examine counts from a single group for evidence of an association between two categorical variables.
Slide 26 - 18
What have we learned? (cont.)
Mechanically, these tests are almost identical.While the tests appear to be one-sided, conceptually they are many-sided, because there are many ways that the data can deviate significantly from what we hypothesize.When we reject the null hypothesis, we know to examine standardized residuals to better understand the patterns in the data.
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