In this paper, we compare the powers of several discrete goodness-of-fit test statistics considered by Steele and Chaseling [10] under the null hypothesis of a 'zig-zag' distribution. The results suggest that the Discrete Kolmogorov-Smirnov test statistic is generally more powerful for the decreasing trend alternative. The Pearson Chi-Square statistic is generally more powerful for the increasing, unimodal, leptokurtic, platykurtic and bath-tub shaped alternatives. Finally, both the Nominal Kolmogorov- Smirnov and the Pearson Chi-Square test statistic are generally more powerful for the bimodal alternative. We also address the issue of the sensitivity of the test statistics to the alternatives under the 'zig-zag' null. In comparison to the uniform null of Steele and Chaseling [10], our investigation shows that the Discrete KS test statistic is most sensitive to the decreasing trend alternative; the Pearson Chi-Square statistic is most sensitive to both the leptokurtic and platykurtic trend alternatives. In particular, under the 'zig-zag' null we are able to clearly identify the most powerful test statistic for the platykurtic and leptokurtic alternatives, compared to the uniform null of Steele and Chaseling [10], which could not make such identification.