The document provides an overview of goodness-of-fit tests for multinomial experiments and contingency tables, which are used to test if observed frequency distributions fit expected distributions. It defines multinomial experiments, goodness-of-fit tests, and contingency tables, and explains how to perform tests of independence and homogeneity using chi-square tests on contingency tables. Sample problems are provided to test claims about categories of outcomes and the independence of variables in contingency tables.

1. 1
Statistics, Practice (Sample) Test
Chapters 11: Goodness-of-Fit (Multinomial Experiments) and Contingency Tables)
Name: ______________________________
Instructions: Read this Mini Lecture or your text, or study the tutorials online
thoroughly to be able to handle this last Sample Test.
Mini Lecture:
Chapter 11: Multinomial Experiments & Contingency Tables
Definition:
A Multinomial Experiment is an experiment that meets the following conditions:
1. The number of trials is fixed.
2. The trials are independent.
3. All outcomes of each trial must be classified into exactly one of several
different categories.
4. The probabilities for the different categories remain constant for each trial.
11.1: Goodness-of-fit Test in Multinomial Experiments:
Definition
A goodness-of-fit test is used to test the hypothesis that an observed frequency
distribution fits (or conforms to) some claimed distribution.
0 represents the observed frequency of an outcome
E represents the expected frequency of an outcome
k represents the number of different categories or outcomes
n represents the total number of trials
If all expected frequencies are equal, then
n
E
k
 , the sum of all observed frequencies divided by the number of categories.
If the expected frequencies are not all equal:
E np
 , each expected frequency is found by multiplying the sum of all observed
frequencies by the probability for the category.

2. 2
Test Statistic:
 
2
2 O E
E


 
Critical Values
 Found in Table A-4 using k – 1 degrees of freedom where k = number of
categories
 Goodness-of-fit hypothesis tests are always right-tailed.
11.2: Contingency Tables: Independence and Homogeneity
Contingency Table (or two-way frequency table)
Definition
A contingency table is a table in which frequencies correspond to two variables.
(One variable is used to categorize rows, and a second variable is used to categorize
columns.) Contingency tables have at least two rows and at least two columns.
Test of Independence
This method tests the null hypothesis that the row variable and column variable in a
contingency table are not related. (The null hypothesis is the statement that the row and
column variables are independent.)
Assumptions
 The sample data are randomly selected.
 The null hypothesis H0 is the statement that the row and column variables are
independent; the alternative hypothesis H1 is the statement that the row and
column variables are dependent.
 For every cell in the contingency table, the expected frequency E is at least 5.
(There is no requirement that every observed frequency must be at least 5.)
Test of Independence
Test Statistic:
 
2
2 O E
E


 
Critical Values
1. Found in Table A-4 using: degrees of freedom = (r – 1)(c – 1)
r is the number of rows and c is the number of columns
2. Tests of Independence are always right-tailed.

3. 3
Expected Frequency for Contingency Tables
 
 
 
 
Rowtotal Columntotal
E np Grandtotal
Grandtotal Grandtotal
   
  
 
rowtotal columntotal
E
Grandtotal

Grand Total = Total number of all observed frequencies in the table
Test of Homogeneity
In a test of homogeneity, we test the claim that different populations have the
same proportions of some characteristics.
How to distinguish between a test of homogeneity and a test for independence:
Were predetermined sample sizes used for different populations (are they similar in
nature?) (test of homogeneity,), or was one big sample drawn so both row and column
totals were determined randomly (test of independence)?
Statistics, Sample Test
Chapters 11: Multinomial Experiments & Contingency Tables
Name: ______________________________ Student ID: _______________________
Instructions: Print out the following pages that contain the Sample Test, solve all
problems, show your work completely, and bring it on the day of the Final on
campus exam. Show all your work for full credit.
1) (11-1: Goodness-of-fit Test in Multinomial Experiments :)
Here are the observed frequencies from four categories: 5, 6, 8, and 13. At 0.05
significance level, test the claim that the four categories are all equally likely.
A. State the null and alternative hypothesis.
B. What is the expected frequency for each of the four categories?
C. What is the value of the test statistic?
D. Find the critical value(s).
E. Make a decision

4. 4
2) (11.1: Goodness-of-fit Test in Multinomial Experiments: ) A
professor asked 40 of his students to identify the tire they would select as a flat tire of
a car carrying 4 students who misses a test (an excuse). The following table
summarizes the result, Use a 0.05 significance level to test the claim that all 4 tires
have equal proportions of being claimed as flat.
Tire Left Front Right Front Left Rear Right Rear
Number selected 11 15 8 6
3) (11.2: Test of Independence) Using a 0.05 significance level, test the claim
that when the Titanic sank, whether someone survived or died is independent of
whether that person is a man, woman, boy, or girl.
4) (11.2: Test of Homogeneity) Using the following table, with a 0.05
significance level, test the effect of pollster gender on survey responses by men.