document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170416>Question 1170416</A>: <I><SPAN STYLE=\"word-break: break-all;\"> (i) When do you apply the analysis of variance technique?<BR>\n" );
document.write( "(ii) The three samples given below have been obtained from three normal populations with equal variances. Test the hypothesis that the population means are equal at 5% level of significance.\r<BR>\n" );
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document.write( "Sample 1\r<BR>\n" );
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document.write( "6\r<BR>\n" );
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document.write( "8\r<BR>\n" );
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document.write( "5\r<BR>\n" );
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document.write( "12\r<BR>\n" );
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document.write( "9\r<BR>\n" );
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document.write( "Sample 2\r<BR>\n" );
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document.write( "5\r<BR>\n" );
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document.write( "3\r<BR>\n" );
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document.write( "8\r<BR>\n" );
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document.write( "7\r<BR>\n" );
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document.write( "7\r<BR>\n" );
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document.write( "Sample 3\r<BR>\n" );
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document.write( "10\r<BR>\n" );
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document.write( "7\r<BR>\n" );
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document.write( "11\r<BR>\n" );
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document.write( "10\r<BR>\n" );
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document.write( "12 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851190 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=851190\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's address each part of the question.\r<BR>\n" );
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document.write( "**(i) When do you apply the analysis of variance (ANOVA) technique?**\r<BR>\n" );
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document.write( "The analysis of variance (ANOVA) technique is used to test the hypothesis that the means of two or more populations are equal. Specifically, you would apply ANOVA when:\r<BR>\n" );
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document.write( "* **You have more than two groups (or populations) to compare:** If you only have two groups, a t-test is appropriate. ANOVA is designed for situations where you need to compare three or more groups.<BR>\n" );
document.write( "* **You want to determine if there are statistically significant differences between the group means:** ANOVA helps you determine if the observed differences between the sample means are likely due to real differences in the population means or simply due to random chance.<BR>\n" );
document.write( "* **The dependent variable is continuous:** The variable you are measuring should be a continuous variable (e.g., height, weight, test scores).<BR>\n" );
document.write( "* **The independent variable is categorical:** The independent variable represents the groups or categories you are comparing (e.g., different treatment groups, different teaching methods).<BR>\n" );
document.write( "* **The assumptions of ANOVA are met:** These assumptions include:<BR>\n" );
document.write( "    * Normality: The data within each group should be approximately normally distributed.<BR>\n" );
document.write( "    * Homogeneity of variances: The variances of the populations should be equal.<BR>\n" );
document.write( "    * Independence: The observations should be independent of each other.\r<BR>\n" );
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document.write( "**(ii) Testing the hypothesis of equal population means**\r<BR>\n" );
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document.write( "We will perform a one-way ANOVA test.\r<BR>\n" );
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document.write( "**1. Set up the hypotheses:**\r<BR>\n" );
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document.write( "* Null hypothesis (H₀): μ₁ = μ₂ = μ₃ (The population means are equal).<BR>\n" );
document.write( "* Alternative hypothesis (H₁): At least one mean is different.\r<BR>\n" );
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document.write( "**2. Calculate the sample means and variances:**\r<BR>\n" );
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document.write( "* Sample 1 (x₁): 6, 8, 5, 12, 9<BR>\n" );
document.write( "    * n₁ = 5<BR>\n" );
document.write( "    * Mean (x̄₁) = (6 + 8 + 5 + 12 + 9) / 5 = 40 / 5 = 8<BR>\n" );
document.write( "    * Variance (s₁²) = Σ(x₁ᵢ - x̄₁)² / (n₁ - 1) = (4 + 0 + 9 + 16 + 1) / 4 = 30 / 4 = 7.5<BR>\n" );
document.write( "* Sample 2 (x₂): 5, 3, 8, 7, 7<BR>\n" );
document.write( "    * n₂ = 5<BR>\n" );
document.write( "    * Mean (x̄₂) = (5 + 3 + 8 + 7 + 7) / 5 = 30 / 5 = 6<BR>\n" );
document.write( "    * Variance (s₂²) = Σ(x₂ᵢ - x̄₂)² / (n₂ - 1) = (1 + 9 + 4 + 1 + 1) / 4 = 16 / 4 = 4<BR>\n" );
document.write( "* Sample 3 (x₃): 10, 7, 11, 10, 12<BR>\n" );
document.write( "    * n₃ = 5<BR>\n" );
document.write( "    * Mean (x̄₃) = (10 + 7 + 11 + 10 + 12) / 5 = 50 / 5 = 10<BR>\n" );
document.write( "    * Variance (s₃²) = Σ(x₃ᵢ - x̄₃)² / (n₃ - 1) = (0 + 9 + 1 + 0 + 4) / 4 = 14 / 4 = 3.5\r<BR>\n" );
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document.write( "**3. Calculate the overall mean (x̄):**\r<BR>\n" );
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document.write( "* x̄ = Σ(Σxᵢⱼ) / N = (40 + 30 + 50) / 15 = 120 / 15 = 8\r<BR>\n" );
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document.write( "**4. Calculate the sum of squares between groups (SSB):**\r<BR>\n" );
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document.write( "* SSB = Σnᵢ(x̄ᵢ - x̄)² = 5(8 - 8)² + 5(6 - 8)² + 5(10 - 8)² = 0 + 20 + 20 = 40\r<BR>\n" );
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document.write( "**5. Calculate the sum of squares within groups (SSW):**\r<BR>\n" );
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document.write( "* SSW = Σ(nᵢ - 1)sᵢ² = 4(7.5) + 4(4) + 4(3.5) = 30 + 16 + 14 = 60\r<BR>\n" );
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document.write( "**6. Calculate the degrees of freedom:**\r<BR>\n" );
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document.write( "* Degrees of freedom between groups (dfB) = k - 1 = 3 - 1 = 2 (where k is the number of groups)<BR>\n" );
document.write( "* Degrees of freedom within groups (dfW) = N - k = 15 - 3 = 12 (where N is the total number of observations)\r<BR>\n" );
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document.write( "**7. Calculate the mean squares:**\r<BR>\n" );
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document.write( "* Mean square between groups (MSB) = SSB / dfB = 40 / 2 = 20<BR>\n" );
document.write( "* Mean square within groups (MSW) = SSW / dfW = 60 / 12 = 5\r<BR>\n" );
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document.write( "**8. Calculate the F-statistic:**\r<BR>\n" );
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document.write( "* F = MSB / MSW = 20 / 5 = 4\r<BR>\n" );
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document.write( "**9. Find the critical F-value:**\r<BR>\n" );
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document.write( "* Using an F-distribution table or calculator, with dfB = 2 and dfW = 12, and α = 0.05, the critical F-value is approximately 3.89.\r<BR>\n" );
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document.write( "**10. Make a decision:**\r<BR>\n" );
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document.write( "* Since the calculated F-statistic (4) is greater than the critical F-value (3.89), we reject the null hypothesis.\r<BR>\n" );
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document.write( "**11. Conclusion:**\r<BR>\n" );
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document.write( "* There is sufficient evidence to conclude that the population means are not all equal at the 5% level of significance.\r<BR>\n" );
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document.write( "Therefore, we reject the null hypothesis.<BR>\n" );
document.write( " </SPAN>\n" );
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