04/20/23 Slide 1

We will use a two-sample test of proportions to test whether or not there are group differences in the proportions of cases that have a particular characteristic, such that the two groups are likely to represent populations (or sub-populations) that have different proportions of a particular response on a variable of interest.

For example, we might test whether or not there is a gender difference in political party affiliation. We would compare the political party affiliations of a sample of women drawn from the population of women to the political party affiliations of a sample of men drawn from the population of men.

While we could draw a sample of men and a sample of women, as a practical matter, we would probably draw men and women in the same sample, and record variables for both gender and political party. As long as we have a reasonable chance of obtaining a sample that adequately represents men and women, the way we conceptualize the sample strategy does not alter the results.

04/20/23 Slide 2

We are testing the research hypothesis that the proportion in the population represented by one group (e.g. males) is different from (not equal to) the proportion in the population represented by the second group (e.g. females). The null hypothesis is that the proportions are equal in the two groups, i.e. the difference between the proportions is zero.

We know that multiple samples drawn from the same population will have different proportions and differences among groups. The question we are answering is whether or not the two groups are sufficiently different that it is reasonable to conclude that the group differences hold true for the groups in the population.

In two-sample problems, we are testing the numeric difference between groups to see if the difference is substantially distant from zero, i.e., beyond the margin of error for sampling differences.

04/20/23 Slide 3

Similar to evaluation hypotheses for one-sample tests, there are two strategies for evaluating the null hypothesis for two samples:

we can examine the probability of the z test statistic which compares the difference between the group proportions to no difference (zero difference)

or we can examine the confidence interval around the difference in group proportions to see whether or not it includes no difference (zero difference).

In these problems, we will use the probability of the z test statistic, though either strategy will produce the same results.

If the probability of the z-statistic is less than or equal to the probability that we are willing to make an incorrect statement of difference (alpha), we reject the null hypothesis and support the finding stated in the research hypothesis.

04/20/23 Slide 4

Like the one-sample test, there is a minimum sample size requirement for each cell in the contingency table formed by the two variables, i.e. 10 cases per cell.

If we violate the assumptions for the test, we report “na” for the results that would be based on the statistical test.

04/20/23 Slide 5

This is a prototype of the problems, with the correct answers completed.

04/20/23 Slide 6

Because the number of males who agreed (7) was less than 10, we do not do the two-sample test. na is used for the confidence interval, and for the values for z, p, and the comparison symbol (< or =).

An example of when “na” is the correct response.

04/20/23 Slide 7

The first paragraph identifies the test to use and the groups that will be compared.

The notes list the variables to be compared (wrkslf and wrkgovt), the variable defining groups (sex), the data set (GSS2000R), and the alpha level to use in testing the null hypothesis.

The second note lists the sample size requirement we will use for the problem.

04/20/23 Slide 8

The two-sample test of proportion is tested separately for each variable (wrkslf and wrkgovt). The second paragraph in the problem contains the test for the first variable.

To answer this question, we complete the first block of the table.

04/20/23 Slide 9

Third, make sure the Two-sample Test option is selected.

The test proportion is ignored when a two-sample test is requested.

First, select the variable to test from the list, wrkslf, from the list of dependent variables.

Fourth, click on the Run button to produce the output.

Second, select the independent variable, sex, which defines the groups being compared.

Since SPSS does not compute tests of proportions, we will use the script for our calculations.

04/20/23 Slide 10

The script requests SPSS produce a contingency table with counts and column percentages in the cells.

Proportions are represented as decimal fractions in the in the two-sample table rather than percentages.

The script creates a table for the two-sample test, both as the z test statistic and its probability, and as a 95% confidence interval.

04/20/23 Slide 11

We first complete the entries in the table.

The counts and percents are transferred from the contingency table to Table 1 for the problem.

Note: only a few counts and percents are shown, but all cells in the Self-Employment block should be completed.

04/20/23 Slide 12

To obtain the percentage difference, we can either subtract the percent for females from the percent for males:

or we can convert the difference in proportions for the two-sample test (-.040) to a percentage (-4%) and transfer it to table 1.

84.9 – 88.9 = -4.0

04/20/23 Slide 13

The lower and upper bound for the 95% confidence interval are transferred from the table for the two sample test to Table 1.

04/20/23 Slide 14

Having completed the table for self-employment, we can enter the results in the question narrative.

The first part of the sentence compares the percentage of males and the percentage of females working for someone else.

04/20/23 Slide 15

The percentages are transferred from Table 1 to the question narrative.

Since the percentage for males (84.9%) was less than the percentage for females (88.9%), males were less likely to be employed by someone else.

04/20/23 Slide 16

The second part of the sentence asks about the statistical significance of the difference of -4.0%.

04/20/23 Slide 17

The next drop down list selects the symbol that appears between the p for probability and the probability value. See the discussion below.

We transfer the value of the z-statistic (-0.93) from the two-sample table to the problem narrative.

Probabilities can become quite small, e.g. p = 0.00001, which SPSS will display as 0.000. Since the statement that p = 0.000 would incorrectly imply that there was no chance that the statistical value could occur, we need a way to represent very small probabilities.

The usual remedy for this is to write the probability as p < .001 rather than p = .00001. Since p = .001 can also occur, we need a mechanism to select the correct symbol.

04/20/23 Slide 18

We transfer the significance value (.352) from the two-sample table to the problem narrative.

Since the sig value really equals .352, we choose “=“ as the symbol.

04/20/23 Slide 19

Now that we have entered the values for z and p, we can answer the question of statistical significance.

We compare the value for p to the value for alpha. If p is less than or equal to alpha, we reject the null hypothesis and conclude that the difference is statistically significant and not equal to zero. Since p (.352) is greater than alpha of .05, the difference is not statistically significant. We do not reject the null hypothesis and cannot conclude that the difference in group proportions is greater than 0.

04/20/23 Slide 20

If we were using the confidence interval strategy, we examine the lower and upper bounds of the interval. Since the value for the difference under the null hypothesis (zero) falls within the bound of the interval (-12.4 to 4.4), the difference is not statistically significant.

04/20/23 Slide 21

The third paragraph contains the second comparison that we will test.

To answer this question, we complete the second block of the table.

04/20/23 Slide 22

Third, make sure the Two-sample Test option is selected.

First, select the variable to test from the list, wrkgovt, from the list of dependent variables.

Fourth, click on the Run button to produce the output.

Second, select the independent variable, sex, which defines the groups being compared.

04/20/23 Slide 23

The crosstabs table and the two-sample test table are produced in the output.

04/20/23 Slide 24

The counts and percents are transferred from the contingency table to Table 1 for the problem.

Note: only a few counts and percents are shown, but all cells in the Government Employment block should be completed.

04/20/23 Slide 25

To obtain the percentage difference, we can either subtract the percent for females from the percent for males:

or we can convert the difference in proportions for the two-sample test (-.106) to a percentage (-10.6%) and transfer it to table 1.

04/20/23 Slide 26

The lower and upper bound for the 95% confidence interval are transferred from the table for the two sample test to Table 1.

04/20/23 Slide 27

Having completed the table for works for government, we can enter the results in the question narrative.

The first part of the sentence compares the percentage of males and the percentage of females working for a private employer.

04/20/23 Slide 28

Since the percentage for males (79.1%) was less than the percentage for females (89.7%), males were less likely to be employed by privately.

04/20/23 Slide 29

The second part of the sentence asks about the statistical significance of the difference of -10.6%.

04/20/23 Slide 30

We transfer the value of the z-statistic (-2.37) and the p-value (.018) from the two-sample table to the problem narrative.

04/20/23 Slide 31

We compare the value for p to the value for alpha. If p is less than or equal to alpha, we reject the null hypothesis and conclude that the difference is statistically significant and not equal to zero. Since p (.018) is less than alpha of .05, the difference is statistically significant. We reject the null hypothesis and conclude that the difference in group proportions is greater than 0.

If we were using the confidence interval strategy, we examine the lower and upper bounds of the interval. Since the value for the difference under the null hypothesis (zero) falls outside the bound of the interval, the difference is statistically significant.

04/20/23 Slide 32