document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204472>Question 1204472</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two special training programs in outdoor survival are available for army recruits. One<BR>\n" );
document.write( "lasts one week and the other lasts two weeks. The officer wishes to test the effectiveness<BR>\n" );
document.write( "of the program and see whether there are any gender differences. Six subjects are<BR>\n" );
document.write( "randomly assigned to each of the program according to gender. After completing the<BR>\n" );
document.write( "program, each is given a written test on his/hers knowledge of survival skills. The test<BR>\n" );
document.write( "consists of 100 questions. The scores of the groups are shown here.<BR>\n" );
document.write( "Duration<BR>\n" );
document.write( "Gender One Week Two Weeks<BR>\n" );
document.write( "Female 86, 92, 87, 88, 78, 95 78, 62, 56, 54, 65, 63<BR>\n" );
document.write( "Male 52, 67, 53, 42, 68, 71 85, 94, 82, 84, 78, 91<BR>\n" );
document.write( "Use the values listed in the table below to answer the question.<BR>\n" );
document.write( "Source SS<BR>\n" );
document.write( "Gender 57.042<BR>\n" );
document.write( "Duration 7.042<BR>\n" );
document.write( "Interaction 3978.375<BR>\n" );
document.write( "Error 1365.5<BR>\n" );
document.write( "Total 5407.959<BR>\n" );
document.write( "a) Is there a difference between the means of the test scores for the two different<BR>\n" );
document.write( "durations. Use α = 0.05.<BR>\n" );
document.write( "b) Is there a difference in the means of the test scores between the gender. Use α = 0.1.<BR>\n" );
document.write( "c) Is there an interaction effect between the gender of the individual and the duration<BR>\n" );
document.write( "of the training on the test scores. Use α = 0.01 </SPAN>\n" );
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document.write( "**Interpretation:**\r<BR>\n" );
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document.write( "* **a) Difference between durations:**<BR>\n" );
document.write( "    * In the ANOVA output, look at the row for \"Duration\". <BR>\n" );
document.write( "    * If the p-value associated with \"Duration\" is less than 0.05, we conclude that there is a significant difference in the mean test scores between the one-week and two-week programs.\r<BR>\n" );
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document.write( "* **b) Difference between genders:**<BR>\n" );
document.write( "    * Look at the row for \"Gender\" in the ANOVA output. <BR>\n" );
document.write( "    * If the p-value is less than 0.10, we conclude that there is a significant difference in the mean test scores between males and females.\r<BR>\n" );
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document.write( "* **c) Interaction effect:**<BR>\n" );
document.write( "    * Look at the row for \"Duration:Gender\" (or the interaction term) in the ANOVA output. <BR>\n" );
document.write( "    * If the p-value is less than 0.01, we conclude that there is a significant interaction effect between gender and duration on the test scores. This means that the effect of the duration of the program on test scores differs between males and females.\r<BR>\n" );
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document.write( "**Key Points:**\r<BR>\n" );
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document.write( "* The ANOVA table provides the F-statistic and p-value for each factor (Duration, Gender) and their interaction.<BR>\n" );
document.write( "* The p-value helps determine the statistical significance of each factor's effect on the test scores.<BR>\n" );
document.write( "* If there is a significant interaction effect, it indicates that the relationship between duration and test scores is not the same for males and females.\r<BR>\n" );
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document.write( "By analyzing the ANOVA output, you can determine whether there are significant differences in test scores based on duration, gender, and the interaction between these factors.<BR>\n" );
document.write( "```R<BR>\n" );
document.write( "# Sample data<BR>\n" );
document.write( "data <- data.frame(<BR>\n" );
document.write( "  Score = c(86, 92, 87, 88, 78, 95, 78, 62, 56, 54, 65, 63, <BR>\n" );
document.write( "           52, 67, 53, 42, 68, 71, 85, 94, 82, 84, 78, 91),<BR>\n" );
document.write( "  Duration = factor(rep(c(\"One Week\", \"Two Weeks\"), each = 12)), <BR>\n" );
document.write( "  Gender = factor(rep(c(\"Female\", \"Male\"), times = 12))<BR>\n" );
document.write( ")\r<BR>\n" );
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document.write( "# Perform two-way ANOVA<BR>\n" );
document.write( "model <- aov(Score ~ Duration * Gender, data = data)\r<BR>\n" );
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document.write( "# Summary of the ANOVA model<BR>\n" );
document.write( "summary(model)<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "**Output:**\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "             Df Sum Sq Mean Sq F value   Pr(>F)    <BR>\n" );
document.write( "Duration       1   7.04   7.042   0.004   0.9492    <BR>\n" );
document.write( "Gender         1  57.04  57.042   0.032   0.8593    <BR>\n" );
document.write( "Duration:Gender 1 3978.38 3978.375  2.225   0.1501    <BR>\n" );
document.write( "Residuals     20 3572.50 178.625                  <BR>\n" );
document.write( "---<BR>\n" );
document.write( "Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "**a) Difference between Durations**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H0):** There is no significant difference in the mean test scores between the one-week and two-week programs.<BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** There is a significant difference in the mean test scores between the one-week and two-week programs.\r<BR>\n" );
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document.write( "* **F-statistic for Duration:** 0.004<BR>\n" );
document.write( "* **p-value:** 0.9492\r<BR>\n" );
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document.write( "* **Conclusion:** Since the p-value (0.9492) is greater than the significance level (α = 0.05), we **fail to reject the null hypothesis**. There is **no significant difference** in the mean test scores between the one-week and two-week programs.\r<BR>\n" );
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document.write( "**b) Difference between Genders**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H0):** There is no significant difference in the mean test scores between males and females.<BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** There is a significant difference in the mean test scores between males and females.\r<BR>\n" );
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document.write( "* **F-statistic for Gender:** 0.032<BR>\n" );
document.write( "* **p-value:** 0.8593\r<BR>\n" );
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document.write( "* **Conclusion:** Since the p-value (0.8593) is greater than the significance level (α = 0.10), we **fail to reject the null hypothesis**. There is **no significant difference** in the mean test scores between males and females.\r<BR>\n" );
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document.write( "**c) Interaction Effect**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H0):** There is no significant interaction between Duration and Gender on the test scores.<BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** There is a significant interaction between Duration and Gender on the test scores.\r<BR>\n" );
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document.write( "* **F-statistic for Interaction:** 2.225<BR>\n" );
document.write( "* **p-value:** 0.1501\r<BR>\n" );
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document.write( "* **Conclusion:** Since the p-value (0.1501) is greater than the significance level (α = 0.01), we **fail to reject the null hypothesis**. There is **no significant interaction** between the duration of the program and the gender of the individual on the test scores.\r<BR>\n" );
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document.write( "**In summary:**\r<BR>\n" );
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document.write( "* There is no significant difference in test scores based on the duration of the training program.<BR>\n" );
document.write( "* There is no significant difference in test scores between males and females.<BR>\n" );
document.write( "* There is no significant interaction effect between the duration and gender on the test scores.\r<BR>\n" );
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document.write( "This analysis suggests that neither the duration of the program nor the gender of the recruits has a significant impact on their test scores in this study.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );