Chapter 10
Chi-Square Tests and
the F-Distribution
,§ 10.1
Goodness of Fit
, Multinomial Experiments
A multinomial experiment is a probability experiment
consisting of a fixed number of trials in which there are
more than two possible outcomes for each independent
trial. (Unlike the binomial experiment in which there were
only two possible outcomes.)
Example:
A researcher claims that the distribution of favorite pizza
toppings among teenagers is as shown below.
Topping Frequency, f
Each outcome is Cheese 41% The probability
classified into Pepperoni 25% for each possible
categories. Sausage 15% outcome is fixed.
Mushrooms 10%
Onions 9%
Larson & Farber, Elementary Statistics: Picturing the World, 3e 3
, Chi-Square Goodness-of-Fit Test
A Chi-Square Goodness-of-Fit Test is used to test whether a
frequency distribution fits an expected distribution.
To calculate the test statistic for the chi-square goodness-of-fit test,
the observed frequencies and the expected frequencies are used.
The observed frequency O of a category is the frequency for the
category observed in the sample data.
The expected frequency E of a category is the calculated frequency
for the category. Expected frequencies are obtained assuming the
specified (or hypothesized) distribution. The expected frequency
for the ith category is
Ei = npi
where n is the number of trials (the sample size) and pi is the
assumed probability of the ith category.
Larson & Farber, Elementary Statistics: Picturing the World, 3e 4
Chi-Square Tests and
the F-Distribution
,§ 10.1
Goodness of Fit
, Multinomial Experiments
A multinomial experiment is a probability experiment
consisting of a fixed number of trials in which there are
more than two possible outcomes for each independent
trial. (Unlike the binomial experiment in which there were
only two possible outcomes.)
Example:
A researcher claims that the distribution of favorite pizza
toppings among teenagers is as shown below.
Topping Frequency, f
Each outcome is Cheese 41% The probability
classified into Pepperoni 25% for each possible
categories. Sausage 15% outcome is fixed.
Mushrooms 10%
Onions 9%
Larson & Farber, Elementary Statistics: Picturing the World, 3e 3
, Chi-Square Goodness-of-Fit Test
A Chi-Square Goodness-of-Fit Test is used to test whether a
frequency distribution fits an expected distribution.
To calculate the test statistic for the chi-square goodness-of-fit test,
the observed frequencies and the expected frequencies are used.
The observed frequency O of a category is the frequency for the
category observed in the sample data.
The expected frequency E of a category is the calculated frequency
for the category. Expected frequencies are obtained assuming the
specified (or hypothesized) distribution. The expected frequency
for the ith category is
Ei = npi
where n is the number of trials (the sample size) and pi is the
assumed probability of the ith category.
Larson & Farber, Elementary Statistics: Picturing the World, 3e 4
