EXAMPLE 16 Calculating the Kruskal-Wallis test statistic

citybusiness

The U.S. Small Business Administration publishes the number of small businesses in medium-size cities. We are interested in testing whether the population median number of small businesses per city is the same in Florida, North Carolina, and Texas. For the following independent random samples given in the table below, calculate the test statistic for the Kruskal-Wallis test, using these steps:

  1. Temporarily combine the three samples and arrange them in increasing order. Then rank the data values from smallest to largest. Resolve ties using the mean rank, as we have done in the previous sections.
  2. Calculate the sum of the ranks for each sample, , , and .
  3. Finally, calculate .
Florida city Number
of small
businesses		 North
Carolina
city		 Number of
small
businesses		 Texas city Number
of small
businesses
Gainesville 3,718 Asheville 4,883 El Paso 8,150
Tallahassee 4,948 Wilmington 5,825 Lubbock 4,403
Daytona Beach 9,489 Greenville 2, 153 Killeen 3,274
Melbourne 8,771 Fayetteville 3,424 College Station 2,276
Sarasota 13,729 Rocky Mount 2,108 Laredo 3,070
Lakeland 6,865 Amarillo 3,855
Naples 7,184

Solution

  1. The combined data, and their ranks, are shown here.
    Combined data 2,108 2,153 2,276 3,070 3,274 3,424 3,718 3,855 4,403
    Rank 1 2 3 4 5 6 7 8 9
    Combined data 4,883 4,948 5,825 6,865 7,184 8,150 8,771 9,489 13,729
    Rank 10 11 12 13 14 15 16 17 18

    14-38

  2. The sum of the ranks for Florida is

    The sum of the ranks for North Carolina is

    The sum of the ranks for Texas is

    Also, there are 7 cities in the Florida sample, 5 cities in the North Carolina sample, and 6 cities in the Texas sample, so that , , and , and the total sample size is .

  3. Finally, the value of the test statistic is

    Later, we will find out if this value for the test statistic warrants rejection of the null hypothesis. But first, we need to learn the hypotheses for the Kruskal-Wallis test.

NOW YOU CAN DO

Exercises 7–14.