document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1188071>Question 1188071</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Samples of 6 members, each representing three unions, were tested with their respect to their union loyalty. The test had a potential range of scores from 1 to 100, with 100 reflecting the greatest loyalty. Do these data indicate any significant differences in union loyalty between the three unions? Perform the appropriate analysis. Test at the .05 level of significance. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849309 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849309\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You're describing a situation where you need to perform a Kruskal-Wallis test. This is a non-parametric test used to compare three or more groups when the data is not assumed to be normally distributed.  Here's how to perform the test:\r<BR>\n" );
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document.write( "**1. State the Hypotheses:**\r<BR>\n" );
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document.write( "*   **Null Hypothesis (H0):** There is no significant difference in union loyalty scores among the three unions.<BR>\n" );
document.write( "*   **Alternative Hypothesis (H1):** There is a significant difference in union loyalty scores among the three unions.\r<BR>\n" );
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document.write( "**2. Organize the Data and Rank:**\r<BR>\n" );
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document.write( "You'll need the actual data values for each of the 6 members from the three unions.  Let's assume you have the following data:\r<BR>\n" );
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document.write( "| Union 1 | Union 2 | Union 3 |<BR>\n" );
document.write( "|---|---|---|<BR>\n" );
document.write( "| 85 | 72 | 92 |<BR>\n" );
document.write( "| 90 | 68 | 88 |<BR>\n" );
document.write( "| 78 | 75 | 95 |<BR>\n" );
document.write( "| 82 | 65 | 89 |<BR>\n" );
document.write( "| 87 | 70 | 91 |<BR>\n" );
document.write( "| 80 | 73 | 93 |\r<BR>\n" );
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document.write( "Now, combine all the scores and rank them from lowest to highest.  Give tied scores the average of the ranks they would have received.\r<BR>\n" );
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document.write( "| Score | Rank |<BR>\n" );
document.write( "|---|---|<BR>\n" );
document.write( "| 65 | 1 |<BR>\n" );
document.write( "| 68 | 2 |<BR>\n" );
document.write( "| 70 | 3 |<BR>\n" );
document.write( "| 72 | 4 |<BR>\n" );
document.write( "| 73 | 5 |<BR>\n" );
document.write( "| 75 | 6 |<BR>\n" );
document.write( "| 78 | 7 |<BR>\n" );
document.write( "| 80 | 8 |<BR>\n" );
document.write( "| 82 | 9 |<BR>\n" );
document.write( "| 85 | 10 |<BR>\n" );
document.write( "| 87 | 11 |<BR>\n" );
document.write( "| 88 | 12 |<BR>\n" );
document.write( "| 89 | 13 |<BR>\n" );
document.write( "| 90 | 14 |<BR>\n" );
document.write( "| 91 | 15 |<BR>\n" );
document.write( "| 92 | 16 |<BR>\n" );
document.write( "| 93 | 17 |<BR>\n" );
document.write( "| 95 | 18 |\r<BR>\n" );
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document.write( "**3. Calculate the Rank Sums:**\r<BR>\n" );
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document.write( "Add up the ranks for each union:\r<BR>\n" );
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document.write( "*   Union 1: 10 + 14 + 7 + 9 + 11 + 8 = 59<BR>\n" );
document.write( "*   Union 2: 4 + 2 + 3 + 1 + 5 + 6 = 21<BR>\n" );
document.write( "*   Union 3: 16 + 12 + 18 + 13 + 15 + 17 = 91\r<BR>\n" );
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document.write( "**4. Calculate the Kruskal-Wallis H Statistic:**\r<BR>\n" );
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document.write( "H = (12 / (N(N+1))) * Σ(Ri² / ni) - 3(N+1)\r<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "*   N = Total number of observations (6 + 6 + 6 = 18)<BR>\n" );
document.write( "*   ni = Number of observations in each group (6)<BR>\n" );
document.write( "*   Ri = Sum of ranks for each group\r<BR>\n" );
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document.write( "H = (12 / (18 * 19)) * [(59² / 6) + (21² / 6) + (91² / 6)] - 3(18 + 1)<BR>\n" );
document.write( "H = (12 / 342) * [580.17 + 73.5 + 1380.17] - 57<BR>\n" );
document.write( "H ≈ 0.0351 * 2033.84 - 57<BR>\n" );
document.write( "H ≈ 71.49 - 57<BR>\n" );
document.write( "H ≈ 14.49\r<BR>\n" );
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document.write( "**5. Determine the Degrees of Freedom:**\r<BR>\n" );
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document.write( "df = Number of groups - 1 = 3 - 1 = 2\r<BR>\n" );
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document.write( "**6. Find the Critical Value:**\r<BR>\n" );
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document.write( "Consult a chi-square distribution table or use a calculator. For α = 0.05 and df = 2, the critical value is approximately 5.991.\r<BR>\n" );
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document.write( "**7. Make a Decision:**\r<BR>\n" );
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document.write( "Our calculated H statistic (14.49) is *greater* than the critical value (5.991).  Therefore, we *reject* the null hypothesis.\r<BR>\n" );
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document.write( "**8. Conclusion:**\r<BR>\n" );
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document.write( "There is sufficient evidence at the 0.05 significance level to conclude that there is a significant difference in union loyalty scores among the three unions.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );