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Running head: CHAPTER 6
Chapter 6: Chi-Square Tests
The introduction of this topic has given an example of a contingency table and how data
is arranged in the table. A contingency table is an array of data displayed in a matrix format of
rows and columns showing the frequency distribution of the respective variables. The purpose of
the contingency table is to determine whether there is a dependence between two variables.
Additionally, the contingency table can be used to calculate the probability of the individual
Motivating the Chi-square Test of Independence
In a statistical experiment, two sets are independent if they occur exclusively. Therefore,
one event does not affect the occurrence of another event. In a contingency table, the observed
frequencies indicate the actual number of the values while the expected frequencies indicate the
expected number assuming independence of the observed variables.
Making an Inference About a Population from a Sample
In this topic, the authors have revisited the hypothesis testing. In statistics, the null
hypothesis indicates the existence of no significance between the observed variables, while the
null hypothesis suggests the existence of significance between the observed variables. Therefore,
there are only two possible decisions for the test of independence. Rejecting the null hypothesis
indicates the existence of significance while failing to reject the null hypothesis suggests
significance between the observed variables, in this case, dependence. Furthermore, the topic has
restated that type I error is an error due to rejecting the null hypothesis when it is true while type
II error occurs due to failing to reject the null hypothesis when it is false. The null hypothesis is
symbolized as H0, while the alternative hypothesis is represented as H1 or Ha.
Pearson's Chi-Square Test Statistic
Pearson's Chi-square test statistic (X2) is the statistical test applied in a contingency table
to determine whether the observed values occurred by chance or if there exists variation in the
data. In computing the expected frequencies in the contingency table, the values must be at least
five. Where the values are less than five, two or more rows have to be combined with having a
value greater than five. The chi-square test statistic is given by [(O-E)2/E], where O is the
observed frequency and E the expected frequency.
Interpreting the Test Statistic: The Null Distribution
After computing the test statistic, it is essential to determine whether it warrants rejecting
the null hypothesis. This is checked against the degrees of freedom given by DF = (r-1) (c-1)
where r is the number of rows and c the number of columns. When making inferences using
Pearson's chi-square test statistics, we reject the null hypothesis if the value of the calculated test
statistic is greater than or equal to the critical value. The critical value can be obtained from
Chi-square test. A statistical test applied in a contingency table to determine whether the
observed values occurred by chance or if there exists variation in the data.
Contingency table. An array of data displayed in a matrix format of rows and columns
showing the frequency distribution of the respective variables.
Null hypothesis. A hypothesis stating no statistical significance between the observed
Alternative hypothesis. A hypothesis stating statistical significance between the
To memorize the vocabulary terms learned in this chapter, I composed a tune that
summarized the meaning of these terms as well as how they are applied. Furthermore, most of
the terms learned in this chapter were covered in the previous topics hence was able to master
within two days.
Multiple Choice Questions
1. A researcher wishes to perform some statistical procedures. These procedures only work
properly if the data is randomly selected from a normal population. Upon these data, she
performs a Shapiro-Wilk test. Her expensive statistical package reports a p-value of
At the 5% significance level, it is appropriate to use her data for the purposes she
At the 1% significance level, it is appropriate to use her data for the purposes she
a. I is true, and II is true.
b. I is true, and II is false....
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