### Math 221 – Principles of Statistics

Comparing Two or More Unknown Populations Reference Page

Purpose: To test whether two or more unknown populations have the same distribution.

Requirements: Let  be a random variable having the first unknown distribution,  a random variable having the second unknown distribution, and so on, up to . Set up a two-way table of observed counts. One way is for the factor “population,” with the populations as its levels. The other way is for values of  , , and so on, up to . Use the table to compute the expected counts for each cell in the table, thus:. The conditions for the test are

(1) The data for each population are from a (simple) random sample taken from that population

(2) The samples are all independent of each other.

(3) All expected counts are greater than 1.

(4) No more than 20% of the expected counts are less than 5.

Hypotheses:

The populations all have the same distribution. The alternative hypothesis is  At least one of the populations is different from at least one of the others. (Hopefully, this reminds you of the ANOVA hypotheses.)

Test Statistic: We recommend using software such as SPSS to calculate the  test statistic, the degrees of freedom, and the p-value. However, you can use the formula , with degrees of freedom equal to , and get the (one-tailed) p-value from a  table such as Table F in Moore & McCabe’s The Practice of Business Statistics.

WARNING: The  test statistic is very sensitive to round-off errors made during calculation. We recommend using the above formula instead of the one usually found in textbooks, as it suffers less from this problem than they do. Also, we recommend entering the entire calculation in your calculator all at once, rather than computing each term separately, to help minimize round-off error.

Examples of testing claims about several proportions:

Related topics:

Chi-squared tests master reference page

Tests of independence

Tests of homogeneity (several proportions)

Goodness-of-fit (comparing populations when one population is “known”)