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# The Sign Test

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For comparing characteristics of two populations

For comparing characteristics of two populations

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• 1. The sign test is used to determine if there is a significant difference between the mean characteristics of two populations (say A and B). Responses on each pair of A and B are compared. The number of times A exceeded B is used as the test statistic. It is denoted as the letter X. This is called the sign test because X is the number of positive (or negative) signs associated with the difference between the pairs in population A and B. In such cases, the implied null hypothesis is that the two population distributions are identical or there is no significant difference between the two population distributions A and B. For any given pair, the probability that A exceeds B is p=0.5 when the null hypothesis is true.
• 2. The following are the steps in doing the sign test: a. State the null hypothesis. Ho : There is no significant difference between the two populations being compared. b. State the alternative hypothesis. HA : There is a significant difference between the two populations being compared. c. Test the null hypothesis using the following procedure. 1. Examine each pair (A and B) of observations. If A>B, assign a plus (+) sign, if A<B, assign a minus (-) sign, if A=B, the pair must be discard. 2. Count the number of pairs remaining and denote it by n. 3. Count the number of times the less frequent sign occurs and denote this by r. d. To test the null hypothesis, compare r with the critical values tabulated n the table for critical values of r for the sign test in the Appendix. e. Make your decision. If r is less than or equal to the tabulated value for the chosen significance level, reject the null hypothesis.
• 3. Example: The response time, in seconds, for two different stimuli (S1 and S2) were recorded for 10 subjects who participated in a psychological experiment. The following data were obtained. Subject S1 S2 1 2 3 6.4 9.4 7.8 7.8 10.3 8.9 4 7.7 5.2 5 8.8 11.3 6 7 8 5.6 12.1 6.9 4.1 14.7 8.7 9 4.2 7.1 10 5.6 8.1 Use the sign test to test the null hypothesis that state that there is no significant difference between stimulus 1 and stimulus 2 in terms of mean response to them by the subjects.
• 4. Solution: Step 1. Ho: There is no significant difference between stimulus 1 and stimulus 2 in terms of mean response to them by the two samples or the two stimuli are identical. Step 2. Ha: Stimulus 1 and stimulus 2 are not identical. Step 3. Get the sign of the difference between stimulus 1 and stimulus 2, then find r or the number of times the less frequent sign occurs. Subject D 1 2 3 6.4 9.4 7.8 7.8 10.3 8.9 - 4 5 6 7 8 7.7 8.8 5.6 12.1 6.9 5.2 11.3 4.1 14.7 8.7 + + - 9 4.2 7.1 - 10 5.6 8.1 -
• 5. Step 4. Compare r=2 to the critical value of r. Step 5. Make the decision. Since r=2 is greater than rcritical =1, Ho is not rejected. There is no significant difference between the two stimuli in terms of the reaction time of the subjects on them.
• 6. Thank you for listening! 