EXAMPLE 5 Sign test for matched-pair data from two dependent samples

The National Center for Educational Statistics publishes the results from the Trends in International Math and Science Study (TIMSS). The following table contains the 2007 and 2011 average eighth-grade mathematics scores for a random sample of 12 countries. Test whether the population median math score has decreased from 2007 to 2011, using .

14-11

Country 2007 2011 Difference
(2011 – 2007)		 Sign
Korea 597 613 +16 +
Singapore 593 611 +18 +
United States 508 509 +1 +
Lithuania 506 502 −4 −
Hungary 517 505 −12 −
Romania 461 458 −3 −
Russia 512 539 +27 +
Australia 496 505 +9 +
Indonesia 397 386 −11 −
Norway 469 475 +6 +
Sweden 491 484 −7 −
Malaysia 474 440 −34 −

Solution

The countries represent a random sample of matched-pair data, so the condition for performing the sign test for the population median of the differences is met.

  • Step 1 State the hypotheses. We have a left-tailed test:

    where represents the population median of the differences in eighth-grade math scores from 2007 to 2011.

  • Step 2 Find the critical value and state the rejection rule. The sample size is the sum of the number of plus signs and minus signs: . Because , we use the small-sample case. To find the critical value, we use Appendix Table I. We have a one-tailed test, with and , which gives us . The rejection rule is to reject if .
  • Step 3 Find the value of the test statistic. From Table 3, we have .
  • Step 4 State the conclusion and the interpretation. Because is not ≤2, we do not reject . There is insufficient evidence that the population median eighth-grade math score has decreased from 2007 to 2011.

NOW YOU CAN DO

Exercises 21–24.