1. Assess whether or not a data set is symmetric.
2. Perform the Wilcoxon signed rank test for matched-pair data from two dependent samples.
3. Perform the Wilcoxon signed rank test for a single population median.

In the sign test, the data are converted into plus signs or minus signs. The magnitude of the data values is lost, which contributes to the low efficiency of the sign test. In 1945, Frank Wilcoxon developed the Wilcoxon signed rank test for a single population median, which takes the magnitude of the data into account by ranking the data values.

1. Calculate the signed rank for each community college.
2. Find the sums of the positive signed ranks and the negative signed ranks.

1. 1. For each data value, find the difference between the data values for each matched pair. That is, for each community college, we find d = the number of enrolled students in 2014 minus the number of enrolled students in 2013. Omit observations where . These differences are shown in Column 4 of Table 7.
   2. Find the absolute values of the differences. The absolute values of the differences are shown in Column 5 of Table 7.
   3. Rank the absolute values of the differences from smallest to largest. If two or more data values are tied with the same rank, assign to each the mean value of their ranks had they not been tied. (There are no ties in this data set. See Example 11 to see how ties are handled.) The ranks of the absolute differences are shown in Column 6 of Table 7.
   4. Attach to each rank the sign of its corresponding value of . This is its signed rank. For example, the rank of for Los Angeles Community College is 3, but the sign of for Los Angeles is negative (–). We attach this negative sign to the rank to give us Los Angeles's signed rank of −3. Replace each original data value with its corresponding signed rank. The signed ranks are shown in the last column of Table 7.
      Table 14.26: Table 7 Students enrolled at California community colleges

      Community college	Enrollment Spring 2013	Enrollment Spring 2014	Difference (2014 – 2013)		Rank of	Signed rank
      Los Angeles CC	148,754	148,362	–392	392	3	–3
      Santa Monica CC	31,719	31,437	–282	282	2	–2
      El Camino CC	22,657	22,791	134	134	1	1
      North Orange CC	51,780	53,993	2,213	2,213	5	5
      Foothill CC	34,415	33,574	–841	841	4	–4
      Los Rios CC	75,230	71,911	–3,319	3,319	6	–6

2. The sum of the positive sign ranks is

   The sum of the negative signed ranks is

• Step 1 State the hypotheses.

  Choose one of the forms in Table 8.

  Table 14.27: Table 8 Hypotheses for the Wilcoxon signed rank test for matched-pair data

  Null hypothesis	Alternative hypothesis	Type of test
  		Right-tailed test
  		Left-tailed test
  		Two-tailed test

• Step 2 Find the critical value and state the rejection rule.
  • Small-Sample Case : Use Appendix Table J. Choose the column with the appropriate level of significance () and the applicable one-tailed or two-tailed test. Then select the row with the appropriate sample size , where is the number of data values for which does not equal zero. The number in that row and column is your critical value . The rejection rule is to reject if
  • Large-Sample Case : Use Appendix Table C, the standard normal table. The critical value for this sign test is always found in the left tail of the standard normal distribution, so that is always less than 0. For a left-tailed test or a right-tailed test, the critical value is the value of with area to the left of it. For a two-tailed test, the critical value is the value of with area to the left of it. Table 4 in Chapter 9 (page 500) contains values of for some common values of . The rejection rule is to reject if

• Step 3 Find the value of the test statistic.

  First find the signed ranks using the following steps:

  1. For each data value, find the difference between each data value and the hypothesized median . Omit data values for which .
  2. Find the absolute values of the differences.
  3. Rank the absolute values of the differences from smallest to largest. If two or more data values have the same rank, assign to each the mean value of their ranks had they not been tied.
  4. Attach to each rank the sign of its corresponding value of . This is its signed rank. Replace each original data value with its corresponding signed rank.

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  • Small-Sample Case : Use Table 9 to find , where is the sum of the positive signed ranks, and is the absolute value of the sum of the negative signed ranks.
    Table 14.28: Table 9 Finding

    Type of test	Test statistic
    Right-tailed test
    Left-tailed test
    Two-tailed test	, whichever is smaller

  • Large-Sample Case : Use Table 9 to find , and then calculate the test statistic :
    

• Step 4 State the conclusion and the interpretation. Compare the test statistic with the critical value, using the rejection rule.

Figure 14.11: FIGURE 11 TI-83/84 boxplot of the differences.

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

  where represents the population median of the differences in number of enrolled students at California community colleges from 2013 to 2014.

• Step 2 Find the critical value and state the rejection rule. The sample size is the number of data values for which the difference does not equal zero. Because none of the differences equals zero, our sample size is . Because , we use the small-sample case. To find the critical value, we use Appendix Table J. We have a one-tailed test, with level of significance and , which gives us , as shown in Figure 12. The rejection rule is to reject if .

• Step 3 Find the value of the test statistic. The signed ranks are given in Table 7. We have a left-tailed test, so from Table 9, we have

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  Figure 14.12: FIGURE 12 Finding the critical value .
• Step 4 State the conclusion and the interpretation. Because is not ≤2, we do not reject . The evidence is insufficient to conclude that the population median number of students enrolled at California community colleges has decreased from 2013 to 2014.

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

  where represents the population median age of the missing children. Thus, the hypothesized value for the median is .

• Step 2 Find the critical value and state the rejection rule. To find the critical value, we use Appendix Table J, excerpted here in Figure 13. We have a two-tailed test, with level of significance and , which gives us . The rejection rule is to reject if .
  

  Figure 14.13: FIGURE 13 Using Appendix Table J to find the critical value .
• Step 3 Find the value of the test statistic. The calculations to find the signed ranks are shown in Table 11.
  Table 14.31: Table 11 Finding the signed ranks for the child age data

  Child	Age			Rank of	Signed rank
  Adam	4		2	3	−3
  Juan	9		3	4.5	4.5
  Benjamin	5		1	1.5	−1.5
  Samantha	7		1	1.5	1.5
  Kayleen	6		—	—	—
  Aiko	3		3	4.5	−4.5

  1. Find for each child. Note that the value of for Kayleen is zero, so we omit Kayleen's age from further calculations.
  2. The absolute values of the differences are shown in the fourth column of Table 11.
  3. We rank the absolute differences. Notice that the absolute values for Benjamin and Samantha are . Had they not been tied, their ranks would have been 1 and 2. The mean of 1 and 2 is . Thus, each child's age is assigned the rank of 1.5. There is also a tie between Juan and Aiko, with . Had they not been tied, their ranks would have been 4 and 5, so each child's age is assigned the mean rank of 4.5. The ranks of the absolute differences are shown in the fifth column of Table 11.
  4. Attach to each rank the sign of its corresponding value of . This is its signed rank. For example, the rank of for Adam is 3, but the sign of for Adam is negative (–). We attach this negative sign to the rank for Adam to give us Adam's signed rank of –3. Replace each original data value with its corresponding signed rank, shown in the last column of Table 11.

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  Next, we need to sum the positive ranks and the negative ranks. There are two positive signed ranks: Juan's 4.5 and Samantha's 1.5. Thus, . There are three negative signed ranks, which we add to get . Taking the absolute value gives us . Table 9 tells us that Thus, .

• Step 4 State the conclusion and the interpretation. The rejection rule is to reject if . Because is not ≤1, we do not reject . There is insufficient evidence that the population median age of missing children differs from 6 years old.

Boxplot of childern's ages.

• Step 1 State the hypotheses.

  where represents the population median age of the missing children.

• Step 2 Find the critical value and state the rejection rule. There are 50 children. Ten of these children are 6 years old, so that . These 10 children are therefore omitted from this hypothesis test. This leaves us with 40 children, which is greater than 30, so we use the large-sample case. From Table 4 in Chapter 9 (page 500), the two-tailed test with level of significance gives us . We will reject if .

• Step 3 Find the value of the test statistic. We use the instructions provided in the Step-by-Step Technology Guide at the end of this section. Figure 14 shows the Minitab results, and Figure 15 shows the SPSS results, from the Wilcoxon signed rank test for the population median. Note that the original sample size (“N”) is 50, but that “N for Test” is , because 10 data values have been omitted. The “Wilcoxon Statistic” is the value of , which represents the smaller of and . We use this value to find the test statistic:

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  Figure 14.14: FIGURE 14 Minitab output for the Wilcoxon signed rank test for a population median.
  

  Figure 14.15: FIGURE 15 SPSS output for the Wilcoxon signed rank test for a population median.
• Step 4 State the conclusion and the interpretation. Because , we reject . There is evidence that the population median age of the missing children differs from 6 years old. Acquiring more data has changed our conclusion.