document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1195794>Question 1195794</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 9.A random sample of 24 management students in each batch was taken. The total (100%) scores for those students were recorded as follows. <BR>\n" );
document.write( "1st year students	2nd year students	3rd year students<BR>\n" );
document.write( "82	83	38<BR>\n" );
document.write( "83	73	59<BR>\n" );
document.write( "97	68	55<BR>\n" );
document.write( "93	61	66<BR>\n" );
document.write( "55	77	45<BR>\n" );
document.write( "67	54	52<BR>\n" );
document.write( "63	69	52<BR>\n" );
document.write( "63	51	61<BR>\n" );
document.write( "Required: At the 0.01 α, do the three batch students lead to different levels of score? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848422 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=848422\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Set up Hypotheses**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H0):** The mean scores of the three batches of students are equal. (μ1 = μ2 = μ3)<BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** At least one of the batch means is different from the others.\r<BR>\n" );
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document.write( "**2. Calculate Necessary Statistics**\r<BR>\n" );
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document.write( "* **Calculate the mean score for each batch:**<BR>\n" );
document.write( "    * 1st Year: (82 + 83 + 97 + 93 + 55 + 67 + 63 + 63) / 8 = 73.125<BR>\n" );
document.write( "    * 2nd Year: (83 + 73 + 68 + 61 + 77 + 54 + 69 + 51) / 8 = 65.75<BR>\n" );
document.write( "    * 3rd Year: (38 + 59 + 55 + 66 + 45 + 52 + 52 + 61) / 8 = 53.5\r<BR>\n" );
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document.write( "* **Calculate the overall mean:** <BR>\n" );
document.write( "    * (Sum of all scores) / (Total number of students) <BR>\n" );
document.write( "    * (Sum of all scores for 1st Year + Sum of all scores for 2nd Year + Sum of all scores for 3rd Year) / (24 students)\r<BR>\n" );
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document.write( "* **Calculate Sum of Squares Between Groups (SSB):**<BR>\n" );
document.write( "    * SSB = Σ(n_i * (x̄_i - x̄_grand)²), where: <BR>\n" );
document.write( "        * n_i is the sample size of each group (8 in this case)<BR>\n" );
document.write( "        * x̄_i is the mean of each group<BR>\n" );
document.write( "        * x̄_grand is the overall mean\r<BR>\n" );
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document.write( "* **Calculate Sum of Squares Within Groups (SSW):**<BR>\n" );
document.write( "    * SSW = Σ(Σ(x_ij - x̄_i)²), where: <BR>\n" );
document.write( "        * x_ij is the score of each individual student<BR>\n" );
document.write( "        * x̄_i is the mean of the group that student belongs to\r<BR>\n" );
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document.write( "* **Calculate Degrees of Freedom:**<BR>\n" );
document.write( "    * Between groups: df_between = k - 1 = 3 groups - 1 = 2<BR>\n" );
document.write( "    * Within groups: df_within = N - k = 72 - 3 = 69 (where N is the total number of students)\r<BR>\n" );
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document.write( "* **Calculate Mean Squares:**<BR>\n" );
document.write( "    * MSB = SSB / df_between<BR>\n" );
document.write( "    * MSW = SSW / df_within\r<BR>\n" );
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document.write( "* **Calculate F-statistic:**<BR>\n" );
document.write( "    * F = MSB / MSW\r<BR>\n" );
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document.write( "**3. Determine Critical Value**\r<BR>\n" );
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document.write( "* **Find the critical F-value** from the F-distribution table using:<BR>\n" );
document.write( "    * α = 0.01<BR>\n" );
document.write( "    * df_numerator = df_between = 2<BR>\n" );
document.write( "    * df_denominator = df_within = 69\r<BR>\n" );
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document.write( "**4. Decision**\r<BR>\n" );
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document.write( "* **Compare the calculated F-statistic to the critical F-value:**<BR>\n" );
document.write( "    * If F-statistic > F-critical, reject the null hypothesis (H0).<BR>\n" );
document.write( "    * If F-statistic ≤ F-critical, fail to reject the null hypothesis.\r<BR>\n" );
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document.write( "**Interpretation**\r<BR>\n" );
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document.write( "* If you reject the null hypothesis, it means there is sufficient evidence at the 0.01 significance level to conclude that there is a significant difference in the mean scores between at least two of the three student batches. \r<BR>\n" );
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document.write( "**Note:**\r<BR>\n" );
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document.write( "* This analysis requires the use of statistical software (like R, Python, or statistical packages like SPSS or Excel) to perform the calculations efficiently and accurately. <BR>\n" );
document.write( "* The ANOVA test only tells you that there is a significant difference between groups. To determine which specific groups differ, you would need to perform post-hoc tests (such as Tukey's HSD).\r<BR>\n" );
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document.write( "**Let me know if you'd like help with the actual calculations using statistical software.** <BR>\n" );
document.write( " </SPAN>\n" );
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