document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1194707>Question 1194707</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A lecturer in a university wants to investigate the differences in students’ marks with the <BR>\n" );
document.write( "different majors from School A, School B, School C and School D for MSG 162 subject. <BR>\n" );
document.write( "It was thought that students with four different majors might result in different grades for <BR>\n" );
document.write( "MSG 162. The lecturer examined exam scores for a sample of four students each from the <BR>\n" );
document.write( "school that participated in the MSG 162 class<BR>\n" );
document.write( "The following are the MSG 162 exam results for the students:<BR>\n" );
document.write( "A B C D<BR>\n" );
document.write( "85 67 88 77<BR>\n" );
document.write( "79 75 79 56<BR>\n" );
document.write( "76 72 85 60<BR>\n" );
document.write( "83 80 82 70<BR>\n" );
document.write( "Assume that all assumptions to perform an ANOVA test are satisfied.<BR>\n" );
document.write( "a) Test the hypothesis that there is no difference between exam scores of students from <BR>\n" );
document.write( "four schools at the significance level, α = 0.05. Interpret the results of the hypothesis <BR>\n" );
document.write( "testing.<BR>\n" );
document.write( "b) Determine the level of significance (p-value) of the test results. <BR>\n" );
document.write( "c) Determine a 95% interval estimate for the students marks from school B. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848504 by </B><A HREF=/tutors/aboutme.mpl?userid=proyaop><B>proyaop(69)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=proyaop><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=848504\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a) Hypothesis Testing**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H0):** There is no significant difference in the mean exam scores of students from the four schools (μA = μB = μC = μD).<BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** At least one school's mean exam score is significantly different from the others.\r<BR>\n" );
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document.write( "**1. Calculate the necessary statistics:**\r<BR>\n" );
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document.write( "* **Calculate the mean and variance for each group:**<BR>\n" );
document.write( "    * School A: Mean = 80.75, Variance = 8.6875<BR>\n" );
document.write( "    * School B: Mean = 73.50, Variance = 10.1250<BR>\n" );
document.write( "    * School C: Mean = 83.50, Variance = 4.6875<BR>\n" );
document.write( "    * School D: Mean = 65.75, Variance = 66.1875\r<BR>\n" );
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document.write( "* **Calculate the overall mean:** <BR>\n" );
document.write( "    * Overall Mean = (Sum of all scores) / Total number of scores = 75.875\r<BR>\n" );
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document.write( "* **Calculate the between-groups sum of squares (SSB):**<BR>\n" );
document.write( "    * SSB = Σ(n_i * (mean_i - overall_mean)^2) <BR>\n" );
document.write( "    * SSB = 4 * ((80.75 - 75.875)^2 + (73.50 - 75.875)^2 + (83.50 - 75.875)^2 + (65.75 - 75.875)^2) <BR>\n" );
document.write( "    * SSB = 4 * (23.890625 + 5.421875 + 58.007813 + 100.15625) <BR>\n" );
document.write( "    * SSB = 752.5\r<BR>\n" );
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document.write( "* **Calculate the within-groups sum of squares (SSW):**<BR>\n" );
document.write( "    * SSW = Σ(n_i - 1) * variance_i<BR>\n" );
document.write( "    * SSW = (4 - 1) * (8.6875 + 10.1250 + 4.6875 + 66.1875) <BR>\n" );
document.write( "    * SSW = 246\r<BR>\n" );
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document.write( "* **Calculate the degrees of freedom:**<BR>\n" );
document.write( "    * Between-groups degrees of freedom (dfB) = k - 1 = 4 - 1 = 3<BR>\n" );
document.write( "    * Within-groups degrees of freedom (dfW) = N - k = 16 - 4 = 12\r<BR>\n" );
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document.write( "* **Calculate the mean squares:**<BR>\n" );
document.write( "    * Mean Square Between (MSB) = SSB / dfB = 752.5 / 3 = 250.83<BR>\n" );
document.write( "    * Mean Square Within (MSW) = SSW / dfW = 246 / 12 = 20.5\r<BR>\n" );
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document.write( "* **Calculate the F-statistic:**<BR>\n" );
document.write( "    * F = MSB / MSW = 250.83 / 20.5 = 12.23\r<BR>\n" );
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document.write( "**2. Determine the critical value:**\r<BR>\n" );
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document.write( "* Using an F-distribution table with dfB = 3 and dfW = 12, and α = 0.05, the critical value is approximately 3.49.\r<BR>\n" );
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document.write( "**3. Make a decision:**\r<BR>\n" );
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document.write( "* Since the calculated F-statistic (12.23) is greater than the critical value (3.49), we reject the null hypothesis.\r<BR>\n" );
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document.write( "**Interpretation:**\r<BR>\n" );
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document.write( "* There is sufficient evidence at the 0.05 significance level to conclude that there is a significant difference in the mean exam scores of students from the four schools.\r<BR>\n" );
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document.write( "**b) Determine the p-value:**\r<BR>\n" );
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document.write( "* Using statistical software (like R or Python), we can find the exact p-value associated with the calculated F-statistic (12.23) and the degrees of freedom (dfB = 3, dfW = 12). <BR>\n" );
document.write( "* The p-value will be very small (likely less than 0.001).\r<BR>\n" );
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document.write( "**c) 95% Confidence Interval for the Mean of School B:**\r<BR>\n" );
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document.write( "* **Calculate the standard error of the mean for School B:**<BR>\n" );
document.write( "    * Standard Error (SE) = √(variance_B / n_B) = √(10.1250 / 4) = 1.586\r<BR>\n" );
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document.write( "* **Find the critical t-value:**<BR>\n" );
document.write( "    * For a 95% confidence interval and df = n_B - 1 = 3, the critical t-value (from a t-distribution table) is approximately 3.182.\r<BR>\n" );
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document.write( "* **Calculate the margin of error:**<BR>\n" );
document.write( "    * Margin of Error = t_critical * SE = 3.182 * 1.586 = 5.05\r<BR>\n" );
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document.write( "* **Calculate the confidence interval:**<BR>\n" );
document.write( "    * Lower limit = Mean_B - Margin of Error = 73.50 - 5.05 = 68.45<BR>\n" );
document.write( "    * Upper limit = Mean_B + Margin of Error = 73.50 + 5.05 = 78.55\r<BR>\n" );
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document.write( "* **95% Confidence Interval for School B:** (68.45, 78.55)\r<BR>\n" );
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document.write( "**Interpretation:**\r<BR>\n" );
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document.write( "We are 95% confident that the true mean exam score for students from School B lies between 68.45 and 78.55.\r<BR>\n" );
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document.write( "**Note:** This analysis assumes that the assumptions for ANOVA are met, including normality of the data within each group and homogeneity of variances. \r<BR>\n" );
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document.write( "I hope this comprehensive analysis is helpful!<BR>\n" );
document.write( " </SPAN>\n" );
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