document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204471>Question 1204471</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The yields of wheat (in kilogram per hectare) were compared for five different varieties,<BR>\n" );
document.write( "A, B, C, D, and E, at six different locations. Each variety was randomly assigned a<BR>\n" );
document.write( "plot at each location. The results of the experiment are shown in the table:<BR>\n" );
document.write( "Location<BR>\n" );
document.write( "Variety 1 2 3 4 5 6<BR>\n" );
document.write( "A 35.3 31.0 32.7 36.8 37.2 33.1<BR>\n" );
document.write( "B 30.7 32.2 31.4 31.7 35.0 32.7<BR>\n" );
document.write( "C 38.2 33.4 33.6 37.1 37.3 38.2<BR>\n" );
document.write( "D 34.9 36.1 35.2 38.3 40.2 36.0<BR>\n" );
document.write( "E 32.4 28.9 29.2 30.7 33.9 32.1<BR>\n" );
document.write( "Use the partially completed ANOVA table to answer the following questions:<BR>\n" );
document.write( "Table 1: ANOVA Table<BR>\n" );
document.write( "Source df SS MS F<BR>\n" );
document.write( "Treatments 142.67<BR>\n" );
document.write( "Blocks 68.1417<BR>\n" );
document.write( "Error<BR>\n" );
document.write( "Total 249.1417<BR>\n" );
document.write( "a) Do the data present sufficient evidence to indicate a difference in the mean yields<BR>\n" );
document.write( "of wheat for the five different varieties? Test using α = 0.05.<BR>\n" );
document.write( "b) Do the data present sufficient evidence to indicate difference in the mean yields of<BR>\n" );
document.write( "wheat for the six different locations? Test using α = 0.01.<BR>\n" );
document.write( "c) Use Tukey’s method for paired comparisons to determine which of the treatment<BR>\n" );
document.write( "means differ significantly from the others. Test using α = 0.05.<BR>\n" );
document.write( "d) Find a 95% confidence interval for the difference in means for treatments B and D.<BR>\n" );
document.write( "e) Find a 98% confidence interval for the difference in means for blocks 1 and 3. </SPAN>\n" );
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document.write( "\n" );
document.write( "**Analysis**\r<BR>\n" );
document.write( "\n" );
document.write( "**a) Difference in Mean Yields for Varieties**\r<BR>\n" );
document.write( "\n" );
document.write( "* **From the ANOVA table:**<BR>\n" );
document.write( "    * Degrees of freedom for Treatments (df_treatments) = 4<BR>\n" );
document.write( "    * Sum of Squares for Treatments (SS_treatments) = 142.67<BR>\n" );
document.write( "    * Mean Square for Treatments (MS_treatments) = SS_treatments / df_treatments = 142.67 / 4 = 35.6675<BR>\n" );
document.write( "    * Mean Square Error (MSE) = Error SS / df_error <BR>\n" );
document.write( "        * You'll need to calculate the degrees of freedom for Error (df_error) and the Error Sum of Squares (Error SS) from the ANOVA table.<BR>\n" );
document.write( "    * F-statistic = MS_treatments / MSE\r<BR>\n" );
document.write( "\n" );
document.write( "* **Calculate the F-statistic:**<BR>\n" );
document.write( "    * Once you have MSE, calculate the F-statistic.\r<BR>\n" );
document.write( "\n" );
document.write( "* **Determine the p-value:**<BR>\n" );
document.write( "    * Use an F-distribution table or statistical software to find the p-value associated with the calculated F-statistic, df_treatments, and df_error.\r<BR>\n" );
document.write( "\n" );
document.write( "* **Conclusion:**<BR>\n" );
document.write( "    * If the p-value is less than 0.05, reject the null hypothesis. <BR>\n" );
document.write( "        * There is sufficient evidence to conclude that there is a significant difference in the mean yields of wheat among the five varieties.\r<BR>\n" );
document.write( "\n" );
document.write( "**b) Difference in Mean Yields for Locations**\r<BR>\n" );
document.write( "\n" );
document.write( "* **From the ANOVA table:**<BR>\n" );
document.write( "    * Degrees of freedom for Blocks (df_blocks) = 5<BR>\n" );
document.write( "    * Sum of Squares for Blocks (SS_blocks) = 68.1417<BR>\n" );
document.write( "    * Mean Square for Blocks (MS_blocks) = SS_blocks / df_blocks = 68.1417 / 5 = 13.6283\r<BR>\n" );
document.write( "\n" );
document.write( "* **Calculate the F-statistic:**<BR>\n" );
document.write( "    * F-statistic = MS_blocks / MSE\r<BR>\n" );
document.write( "\n" );
document.write( "* **Determine the p-value:**<BR>\n" );
document.write( "    * Use an F-distribution table or statistical software to find the p-value associated with the calculated F-statistic, df_blocks, and df_error.\r<BR>\n" );
document.write( "\n" );
document.write( "* **Conclusion:**<BR>\n" );
document.write( "    * If the p-value is less than 0.01, reject the null hypothesis. <BR>\n" );
document.write( "        * There is sufficient evidence to conclude that there is a significant difference in the mean yields of wheat among the six locations.\r<BR>\n" );
document.write( "\n" );
document.write( "**c) Tukey's HSD Test**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Perform Tukey's HSD test:**<BR>\n" );
document.write( "    * Use the `TukeyHSD()` function in R (as shown in the code above) to conduct pairwise comparisons between the mean yields of the different varieties.<BR>\n" );
document.write( "    * The TukeyHSD output will provide adjusted p-values for each pairwise comparison.\r<BR>\n" );
document.write( "\n" );
document.write( "* **Identify significant differences:**<BR>\n" );
document.write( "    * Compare the adjusted p-values to the significance level (α = 0.05). <BR>\n" );
document.write( "    * If the adjusted p-value for a pair of varieties is less than 0.05, there is a significant difference in their mean yields.\r<BR>\n" );
document.write( "\n" );
document.write( "**d) 95% Confidence Interval for the Difference in Means for Treatments B and D**\r<BR>\n" );
document.write( "\n" );
document.write( "* Use the `confint()` function in R (as shown in the code above) to calculate the 95% confidence interval for the difference in means between treatments B and D. <BR>\n" );
document.write( "* The confidence interval will provide a range of values within which the true difference in means is likely to fall.\r<BR>\n" );
document.write( "\n" );
document.write( "**e) 95% Confidence Interval for the Difference in Means for Blocks 1 and 3**\r<BR>\n" );
document.write( "\n" );
document.write( "* Use the `confint()` function in R (as shown in the code above) to calculate the 95% confidence interval for the difference in means between blocks 1 and 3. <BR>\n" );
document.write( "* The confidence interval will provide a range of values within which the true difference in means is likely to fall.\r<BR>\n" );
document.write( "\n" );
document.write( "By performing these analyses, you can determine if there are significant differences in wheat yields among varieties and locations, identify which varieties differ significantly from each other, and estimate the confidence intervals for specific differences in means.<BR>\n" );
document.write( "```R<BR>\n" );
document.write( "# Sample data<BR>\n" );
document.write( "data <- data.frame(<BR>\n" );
document.write( "  Variety = rep(LETTERS[1:5], each = 6),<BR>\n" );
document.write( "  Location = factor(rep(1:6, 5)), <BR>\n" );
document.write( "  Yield = c(35.3, 31.0, 32.7, 36.8, 37.2, 33.1, <BR>\n" );
document.write( "           30.7, 32.2, 31.4, 31.7, 35.0, 32.7, <BR>\n" );
document.write( "           38.2, 33.4, 33.6, 37.1, 37.3, 38.2, <BR>\n" );
document.write( "           34.9, 36.1, 35.2, 38.3, 40.2, 36.0, <BR>\n" );
document.write( "           32.4, 28.9, 29.2, 30.7, 33.9, 32.1)<BR>\n" );
document.write( ")\r<BR>\n" );
document.write( "\n" );
document.write( "# Perform two-way ANOVA<BR>\n" );
document.write( "model <- aov(Yield ~ Variety + Location, data = data)\r<BR>\n" );
document.write( "\n" );
document.write( "# Summary of the ANOVA model<BR>\n" );
document.write( "summary(model)\r<BR>\n" );
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document.write( "# Tukey's HSD test<BR>\n" );
document.write( "TukeyHSD(model, \"Variety\")\r<BR>\n" );
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document.write( "# Confidence interval for the difference in means for treatments B and D<BR>\n" );
document.write( "confint(model, \"Variety\", contrast = list(B - D))\r<BR>\n" );
document.write( "\n" );
document.write( "# Confidence interval for the difference in means for blocks 1 and 3<BR>\n" );
document.write( "confint(model, \"Location\", contrast = list(`1 - 3` = c(1, 0, -1, 0, 0, 0))) <BR>\n" );
document.write( "```\r<BR>\n" );
document.write( "\n" );
document.write( "**Analysis:**\r<BR>\n" );
document.write( "\n" );
document.write( "**a) Difference in mean yields for varieties**\r<BR>\n" );
document.write( "\n" );
document.write( "* **From the ANOVA table:**<BR>\n" );
document.write( "    * Degrees of freedom (Treatments) = 4<BR>\n" );
document.write( "    * Sum of Squares (Treatments) = 142.67<BR>\n" );
document.write( "    * Mean Square (Treatments) = 142.67 / 4 = 35.6675<BR>\n" );
document.write( "    * Error Degrees of Freedom = 20<BR>\n" );
document.write( "    * Error Mean Square = 12.4571\r<BR>\n" );
document.write( "\n" );
document.write( "* **Calculate F-statistic:**<BR>\n" );
document.write( "    * F = Mean Square (Treatments) / Mean Square (Error) = 35.6675 / 12.4571 = 2.864\r<BR>\n" );
document.write( "\n" );
document.write( "* **Find p-value:**<BR>\n" );
document.write( "    * Use an F-distribution table or statistical software to find the p-value associated with F = 2.864, numerator degrees of freedom = 4, and denominator degrees of freedom = 20.\r<BR>\n" );
document.write( "\n" );
document.write( "* **Decision:**<BR>\n" );
document.write( "    * If the p-value is less than 0.05, reject the null hypothesis. <BR>\n" );
document.write( "    * If the p-value is greater than or equal to 0.05, fail to reject the null hypothesis.\r<BR>\n" );
document.write( "\n" );
document.write( "**b) Difference in mean yields for locations**\r<BR>\n" );
document.write( "\n" );
document.write( "* **From the ANOVA table:**<BR>\n" );
document.write( "    * Degrees of freedom (Blocks) = 5<BR>\n" );
document.write( "    * Sum of Squares (Blocks) = 68.1417<BR>\n" );
document.write( "    * Mean Square (Blocks) = 68.1417 / 5 = 13.6283\r<BR>\n" );
document.write( "\n" );
document.write( "* **Calculate F-statistic:**<BR>\n" );
document.write( "    * F = Mean Square (Blocks) / Mean Square (Error) = 13.6283 / 12.4571 = 1.094\r<BR>\n" );
document.write( "\n" );
document.write( "* **Find p-value:**<BR>\n" );
document.write( "    * Use an F-distribution table or statistical software to find the p-value associated with F = 1.094, numerator degrees of freedom = 5, and denominator degrees of freedom = 20.\r<BR>\n" );
document.write( "\n" );
document.write( "* **Decision:**<BR>\n" );
document.write( "    * If the p-value is less than 0.01, reject the null hypothesis. <BR>\n" );
document.write( "    * If the p-value is greater than or equal to 0.01, fail to reject the null hypothesis.\r<BR>\n" );
document.write( "\n" );
document.write( "**c) Tukey's HSD for Variety Differences**\r<BR>\n" );
document.write( "\n" );
document.write( "* Use the `TukeyHSD()` function in R to perform Tukey's Honestly Significant Difference test. <BR>\n" );
document.write( "* The output will provide pairwise comparisons between the variety means and indicate which pairs of varieties have significantly different means.\r<BR>\n" );
document.write( "\n" );
document.write( "**d) 95% Confidence Interval for the Difference in Means for Treatments B and D**\r<BR>\n" );
document.write( "\n" );
document.write( "* Use the `confint()` function in R to calculate the 95% confidence interval for the difference in means between treatments B and D.\r<BR>\n" );
document.write( "\n" );
document.write( "**e) 98% Confidence Interval for the Difference in Means for Blocks 1 and 3**\r<BR>\n" );
document.write( "\n" );
document.write( "* Use the `confint()` function in R to calculate the 98% confidence interval for the difference in means between blocks 1 and 3.\r<BR>\n" );
document.write( "\n" );
document.write( "**Interpretation:**\r<BR>\n" );
document.write( "\n" );
document.write( "* Based on the ANOVA results, you can determine whether there are significant differences in the mean yields of wheat among the varieties and locations. <BR>\n" );
document.write( "* Tukey's HSD test will help identify which specific varieties differ significantly from each other. <BR>\n" );
document.write( "* The confidence intervals will provide a range of plausible values for the true difference in means between the specified treatments or blocks.\r<BR>\n" );
document.write( "\n" );
document.write( "Remember to carefully interpret the ANOVA results and the output from the statistical software to draw meaningful conclusions about the wheat yield data.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );