document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1194764>Question 1194764</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The following is sample information. Test the hypothesis that the treatment means are equal. Use the 0.05 significance level.<BR>\n" );
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document.write( "Treatment 1	Treatment 2	Treatment 3<BR>\n" );
document.write( "7           	7           	4           <BR>\n" );
document.write( "6           	10           	5           <BR>\n" );
document.write( "8           	4           	5           <BR>\n" );
document.write( "6           	7           	5           <BR>\n" );
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document.write( "a. State the null hypothesis and the alternate hypothesis.\r<BR>\n" );
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document.write( "H0:               <BR>\n" );
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document.write( "H1:               <BR>\n" );
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document.write( "b. What is the decision rule? (Round the final answer to 2 decimal places.)\r<BR>\n" );
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document.write( "Reject H0 if the test statistic is greater than                <BR>\n" );
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document.write( "c. Compute SST, SSE, and SS total. (Round the final answers to 3 decimal places.)\r<BR>\n" );
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document.write( "SST =                \r<BR>\n" );
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document.write( "SSE =                \r<BR>\n" );
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document.write( "SS total =                \r<BR>\n" );
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document.write( "d. Complete the ANOVA table. (Round the SS, MS, and F values to 3 decimal places.)\r<BR>\n" );
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document.write( "Source	SS	DF	MS	F<BR>\n" );
document.write( "Treatment	   <BR>\n" );
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document.write( "Error	<BR>\n" );
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document.write( "Total	<BR>\n" );
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document.write( "e. State your decision regarding the null hypothesis.\r<BR>\n" );
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document.write( " H0.\r<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 #848442 by </B><A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><B>ElectricPavlov(122)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848442\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a) State the null hypothesis and the alternate hypothesis.**\r<BR>\n" );
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document.write( "* **H0:** μ1 = μ2 = μ3 (The mean values for all three treatments are equal)<BR>\n" );
document.write( "* **H1:** At least one treatment mean is different from the others.\r<BR>\n" );
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document.write( "**b) What is the decision rule?**\r<BR>\n" );
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document.write( "* **Find the critical F-value:**<BR>\n" );
document.write( "    * Degrees of freedom between groups (df1): k - 1 = 3 - 1 = 2<BR>\n" );
document.write( "    * Degrees of freedom within groups (df2): N - k = 12 - 3 = 9 <BR>\n" );
document.write( "    * Significance level (α) = 0.05<BR>\n" );
document.write( "    * Using an F-distribution table or statistical software, find the critical F-value. <BR>\n" );
document.write( "        * **F-critical ≈ 4.26** \r<BR>\n" );
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document.write( "* **Decision Rule:** <BR>\n" );
document.write( "    * Reject H0 if the calculated F-statistic is greater than 4.26.\r<BR>\n" );
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document.write( "**c) Compute SST, SSE, and SS total.**\r<BR>\n" );
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document.write( "* **Calculate Treatment Means:**<BR>\n" );
document.write( "    * Treatment 1: (7 + 6 + 8 + 6) / 4 = 6.75<BR>\n" );
document.write( "    * Treatment 2: (7 + 10 + 4 + 7) / 4 = 7<BR>\n" );
document.write( "    * Treatment 3: (4 + 5 + 5 + 5) / 4 = 4.75<BR>\n" );
document.write( "    * Grand Mean: (Sum of all observations) / Total number of observations = 5.5\r<BR>\n" );
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document.write( "* **Calculate Sum of Squares (SS):**<BR>\n" );
document.write( "    * **SST (Total Sum of Squares):** Σ(x_ij - x̄)² <BR>\n" );
document.write( "        * x_ij: Individual observation<BR>\n" );
document.write( "        * x̄: Grand mean<BR>\n" );
document.write( "        * SST = (7 - 5.5)² + (6 - 5.5)² + ... + (5 - 5.5)² = 26.75\r<BR>\n" );
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document.write( "    * **SSB (Sum of Squares Between Groups):** Σ(n_i * (x̄_i - x̄)²) <BR>\n" );
document.write( "        * n_i: Number of observations in each group (4)<BR>\n" );
document.write( "        * x̄_i: Mean of each group<BR>\n" );
document.write( "        * SSB = 4 * (6.75 - 5.5)² + 4 * (7 - 5.5)² + 4 * (4.75 - 5.5)² = 12.5\r<BR>\n" );
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document.write( "    * **SSE (Sum of Squares Error):** SST - SSB = 26.75 - 12.5 = 14.25\r<BR>\n" );
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document.write( "**d) Complete the ANOVA Table**\r<BR>\n" );
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document.write( "| Source | SS | DF | MS | F |<BR>\n" );
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document.write( "| Treatment | 12.500 | 2 | 6.250 | 4.167 |<BR>\n" );
document.write( "| Error | 14.250 | 9 | 1.583 |  |<BR>\n" );
document.write( "| Total | 26.750 | 11 |  |  |\r<BR>\n" );
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document.write( "**e) State your decision regarding the null hypothesis.**\r<BR>\n" );
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document.write( "* Since the calculated F-statistic (4.167) is less than the critical F-value (4.26), we **fail to reject the null hypothesis (H0)**. \r<BR>\n" );
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document.write( "**Conclusion:**\r<BR>\n" );
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document.write( "There is not sufficient evidence at the 0.05 significance level to conclude that there are significant differences among the treatment means.<BR>\n" );
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