1/5/2016Slide 1 We will use a one-sample test of proportions to test whether or not our sample proportion supports the population proportion from which.

04/21/23 Slide 1

We will use a one-sample test of proportions to test whether or not our sample proportion supports the population proportion from which our sample was presumably drawn.

We are testing the research hypothesis that the sample proportion is different from (not equal to) the population proportion. The null hypothesis is that the sample proportion is equal to the population proportion (the difference between the two is zero).

We know that multiple samples drawn from the same population will have different proportions. The question we are answering is whether or not our sample proportion is so different from the population proportion that it is reasonable to conclude that our sample proportion is either not representative of the population, is likely to have come from a different population (other than the one we thought we were sampling), or the population proportion is different from what we thought it was.

04/21/23 Slide 2

There are two strategies for evaluating the null hypothesis:we can examine the probability of the z test statistic which compares the population proportion and the sample proportion

or we can examine the confidence interval around the sample proportion to see whether or not it includes the population proportion.

In these problems, we will use the confidence interval strategy, though either strategy will produce the same results.

If the population proportion falls in the confidence interval, we conclude that the sample proportion is “comparable” to the population proportion. The slight differences in the numeric for the sample proportion and the population proportion is attributable to sampling error.

04/21/23 Slide 3

If the population proportion falls outside the confidence interval, we conclude that the sample proportion is “different” from the population proportion. The case for comparing our results to those from previous research is weakened by the evidence that our sample is different from the population.

If we violate the assumptions for the test, the conclusion is “na” as the test should not be performed. For this test, the only assumption concerns sample size. The text states that the sample proportion should be less than 10% of the population and that there should be 10 subjects in each category of the dichotomous variable. We will assume that our sample is less than 10% of the population for all questions, but we will examine our results to assure ourselves that we have 10 or more subjects per category.

04/21/23 Slide 4

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

04/21/23 Slide 5

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

The notes list the variables to be compared (polhitok, polabuse, and polmurdr), the data set (GSS2000R), and the alpha level to use in testing the null hypothesis. An alpha of .05 is equivalent to using a confidence interval of 95%.

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

04/21/23 Slide 6

SPSS does not do tests of proportions and calculations of confidence intervals. We will use a script to do our calculations.

For the one-sample test, we select the variable from the Dependent variable column.

We must also enter the population proportion as a decimal fraction.

The script will produce a frequency table in the output and a table showing the results for the test of a proportion.

When a one-sample test is selected, the independent variable column is ignored.

04/21/23 Slide 7

The first paragraph identifies the test to use and the implied research question (comparing our proportion to that from previous research).

04/21/23 Slide 8

The second paragraph identifies population proportion from previous research for each variable that we will compare to our sample.

04/21/23 Slide 9

The first sentence in the third paragraph states the first comparison that we want to test.

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

04/21/23 Slide 10

Make sure the One-sample Test option is selected.

The test proportion is the proportion found in previous research, entered as a decimal fraction.

Select the variable to test from the list, polhitok.

Click on the Run button to produce the output.

04/21/23 Slide 11

The script requests SPSS produce a frequency distribution with counts and percentages in the rows.

Proportions are represented as decimal fractions in the in the one sample table rather than as percentage in the frequency distribution.

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

04/21/23 Slide 12

We transfer the count and valid percent for the “Yes” category from the frequency distribution to Table 1.

04/21/23 Slide 13

We transfer the count and valid percent for the “No” category from the frequency distribution to Table 1.

04/21/23 Slide 14

We transfer the percent for the “Yes” category from Table 1 to the narrative statement in the problem. “Yes” is equivalent to the narrative statement of saying “they would approve.”

04/21/23 Slide 15

If the population proportion falls in the confidence interval, we conclude that the sample proportion is “comparable” to the population proportion.

If the population proportion falls outside the confidence interval, we conclude that the sample proportion is “different” from the population proportion.

The next answer calls on us to determine whether the sample and population proportion should be characterized as different or the same (comparable within the margin of error).

04/21/23 Slide 16

The lower and upper bound for the 95% confidence interval are transferred from the table for the one sample test to Table 1, converting from decimal fractions to percentages.

04/21/23 Slide 17

Finally, we compare the population proportion from previous research (72%) to the upper and lower bound of the confidence interval (63.5% to 77.1%). Since 72% falls within the confidence interval, we interpret the proportions as comparable.

04/21/23 Slide 18

The second sentence in the third paragraph states the second comparison that we want to test.

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

04/21/23 Slide 19

Be sure to change the test proportion to 0.02 for this problem.

Select the variable to test from the list, polabuse.

Click on the Run button to produce the output.

04/21/23 Slide 20

The script requests SPSS produce a frequency distribution with counts and percentages in the rows.

Proportions are represented as decimal fractions in the in the one sample table rather than as percentage in the frequency distribution.

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

04/21/23 Slide 21

We transfer the count and valid percent for the “Yes” and “No” categories from the frequency distribution to Table 1. Only some of the transfers are shown on this slide.

04/21/23 Slide 22

We transfer the percent for the “Yes” category from Table 1 to the narrative statement in the problem. “Yes” is equivalent to the narrative statement of saying “they would approve.”

Next, we determine whether the sample and population proportion should be characterized as different or the same (comparable within the margin of error).

04/21/23 Slide 23

The lower and upper bound for the 95% confidence interval are transferred from the table for the one sample test to Table 1, converting from decimal fractions to percentages.

04/21/23 Slide 24

Finally, we compare the population proportion from previous research (2%) to the upper and lower bound of the confidence interval (3.4% to 10.8%). Since 2% falls below the confidence interval, we interpret the proportions as different.

04/21/23 Slide 25

The third sentence in the third paragraph states the third comparison that we want to test.

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

04/21/23 Slide 26

Be sure to change the test proportion to 0.06 for this problem.

Select the variable to test from the list, polmurdr.

Click on the Run button to produce the output.

04/21/23 Slide 27

The script requests SPSS produce a frequency distribution with counts and percentages in the rows.

Proportions are represented as decimal fractions in the in the one sample table rather than as percentage in the frequency distribution.

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

04/21/23 Slide 28

We transfer the count and valid percent for the “Yes” and “No” categories from the frequency distribution to Table 1. Only some of the transfers are shown on this slide.

04/21/23 Slide 29

We transfer the percent for the “Yes” category from Table 1 to the narrative statement in the problem. “Yes” is equivalent to the narrative statement of saying “they would approve.”

04/21/23 Slide 30

Because the number of cases in this cell was less than 10, we do not complete the confidence interval or the comparison of the sample proportion to the population proportion. The answer to each question is na.

04/21/23 Slide 31

When we submit the problem, we receive verification that our answers were correct.