document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177142>Question 1177142</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Application of the analysis of variance to a particular problem has resulted in the following ANOVA table. Complete the test at a significance level of 5%.<BR>\n" );
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document.write( "Source  |    DF   |       SS     |        MS    |      F  |<BR>\n" );
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document.write( "Between |     2   |    457.1241  |    228.5621  |   52.74 |<BR>\n" );
document.write( "within  |    17   |    117.0165  |      4.3339  |         |<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850514 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.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=850514\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Hypotheses**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H0):** The means of all the groups are equal.<BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** At least one group mean is different from the others.\r<BR>\n" );
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document.write( "**2. Significance Level**\r<BR>\n" );
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document.write( "* α = 0.05\r<BR>\n" );
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document.write( "**3. Degrees of Freedom**\r<BR>\n" );
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document.write( "* df(between) = 2 (from the ANOVA table)<BR>\n" );
document.write( "* df(within) = 17 (from the ANOVA table)\r<BR>\n" );
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document.write( "**4. F-statistic**\r<BR>\n" );
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document.write( "* F = 52.74 (from the ANOVA table)\r<BR>\n" );
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document.write( "**5. Critical F-value**\r<BR>\n" );
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document.write( "* Using an F-distribution table or calculator with α = 0.05, df1 = 2, and df2 = 17, we find the critical F-value to be approximately 3.59.\r<BR>\n" );
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document.write( "**6. Decision**\r<BR>\n" );
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document.write( "* Since the calculated F-statistic (52.74) is greater than the critical F-value (3.59), we reject the null hypothesis.\r<BR>\n" );
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document.write( "**7. Conclusion**\r<BR>\n" );
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document.write( "At a 5% significance level, there is enough evidence to conclude that there is a statistically significant difference between the means of at least two of the groups.\r<BR>\n" );
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document.write( "**In simpler terms:** The ANOVA test shows that it's very unlikely that the observed differences between the group means are just due to random chance. Therefore, we can confidently say that there's a real difference in the average values of those groups.\r<BR>\n" );
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document.write( "**Further Analysis**\r<BR>\n" );
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document.write( "While ANOVA tells us that there's a difference, it doesn't tell us *which* groups are different. To find that out, you would need to perform post-hoc tests (like Tukey's HSD or Bonferroni correction) to compare specific pairs of group means. </SPAN>\n" );
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