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One Sample Hypothesis Testing Paper

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Since the amount of money that a bank can loan out is based on a percentage of the deposits that they have, it is important for a bank considering opening a branch in a new city to have some estimate of the amount of money that will be deposited by customers into their bank.   Century National Bank is considering opening up a branch bank in one of four cities is the surrounding area.   In making the decision about where to open up their bank, they decide they was to test to determine if the mean account balance in each city is the same or if the 4 cities are potentially different from one another.   To that end, they hire a consultant to analyze the following sample of banking customers which they have obtained from the 4 different cities:

City 1 City 2 City 3 City 4
748 1756 1831 1622
1501 2125 740 1169
1886 1995 1554 2215
1593 1526 137 167
1474 1746 2276 2557
1913 1616 2144 634
1218 1958 1053 789
1006 1675 1120 2051
343 1885 1838 765
1494 2204 1735 1645
580 2409 1326 1266
1320 1338 1790 2138
1784 2076 32 1487
1044 2375 1455
890 1125
1708 1989
Number of observations 16 17 14 13
Median 1397 1958 1504.5 1487

Since the bankers are unsure what sort of distribution the current balances of customers might follow, and since the sample sizes are not particularly large, a nonparametric test is most appropriate.   The nonparametric alternative to ANOVA is the Kruskall Wallis Test.
First, however, we need to formulate the hypotheses.   The null hypothesis is that the medians current balance for each city is the same.   The alternative hypothesis is that at least one of the cities has a different median account balance than the other cities.   Stated symbolically, the hypotheses are:

We choose   .
To calculate the test statistic, we need to calculate the average ranks of each city as they related to the combined sample.   Consider the following table with the ranks and average ranks:


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