document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1182694>Question 1182694</A>: <I><SPAN STYLE=\"word-break: break-all;\"> c)	A manufacturer of machine parts is considering buying one of the three machines currently in the market. The following is their daily output for five different days; <BR>\n" );
document.write( "machine<BR>\n" );
document.write( "A	B	C<BR>\n" );
document.write( "25	31	24<BR>\n" );
document.write( "30	39	30<BR>\n" );
document.write( "36	38	28<BR>\n" );
document.write( "38	42	25<BR>\n" );
document.write( "31	35	28\r<BR>\n" );
document.write( "\n" );
document.write( "Do the machines have equal output rate? Test at α=0.05     (10 marks)\r<BR>\n" );
document.write( "\n" );
document.write( " <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849909 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</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=849909\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to determine if the machines have equal output rates using an ANOVA test:\r<BR>\n" );
document.write( "\n" );
document.write( "**1. State the Hypotheses:**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Null Hypothesis (H0):** The machines have equal output rates (μA = μB = μC).<BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** At least one machine has a different output rate.\r<BR>\n" );
document.write( "\n" );
document.write( "**2. Significance Level:** α = 0.05\r<BR>\n" );
document.write( "\n" );
document.write( "**3. Calculate the Sum of Squares:**\r<BR>\n" );
document.write( "\n" );
document.write( "We need to calculate the Sum of Squares Between Groups (SSB) and the Sum of Squares Within Groups (SSW).\r<BR>\n" );
document.write( "\n" );
document.write( "* **Calculate the Grand Mean:**<BR>\n" );
document.write( "   Grand Mean = (Sum of all observations) / (Total number of observations)<BR>\n" );
document.write( "   Grand Mean = (25+30+36+38+31 + 31+39+38+42+35 + 24+30+28+25+28) / 15<BR>\n" );
document.write( "   Grand Mean = 480 / 15 = 32\r<BR>\n" );
document.write( "\n" );
document.write( "* **Calculate SSB:**<BR>\n" );
document.write( "   SSB = 5 * (Mean_A - Grand_Mean)² + 5 * (Mean_B - Grand_Mean)² + 5 * (Mean_C - Grand_Mean)²\r<BR>\n" );
document.write( "\n" );
document.write( "   * Mean_A = (25+30+36+38+31)/5 = 32<BR>\n" );
document.write( "   * Mean_B = (31+39+38+42+35)/5 = 37<BR>\n" );
document.write( "   * Mean_C = (24+30+28+25+28)/5 = 27\r<BR>\n" );
document.write( "\n" );
document.write( "   SSB = 5 * (32 - 32)² + 5 * (37 - 32)² + 5 * (27 - 32)²<BR>\n" );
document.write( "   SSB = 0 + 5 * 25 + 5 * 25<BR>\n" );
document.write( "   SSB = 250\r<BR>\n" );
document.write( "\n" );
document.write( "* **Calculate SSW:**\r<BR>\n" );
document.write( "\n" );
document.write( "   SSW = Σ(each observation - its group mean)²<BR>\n" );
document.write( "   SSW = (25-32)² + (30-32)² + ... + (28-27)²  (add up the squared differences for all 15 observations)<BR>\n" );
document.write( "   SSW = 49 + 4 + 16 + 36 + 1 + 1 + 49 + 36 + 100 + 9 + 4 + 4 + 1 + 9 + 1<BR>\n" );
document.write( "   SSW = 310\r<BR>\n" );
document.write( "\n" );
document.write( "* **Calculate SST (Total Sum of Squares):** SST = SSW + SSB = 310+250 = 560\r<BR>\n" );
document.write( "\n" );
document.write( "**4. Degrees of Freedom:**\r<BR>\n" );
document.write( "\n" );
document.write( "* df_between (degrees of freedom between groups) = Number of groups - 1 = 3 - 1 = 2<BR>\n" );
document.write( "* df_within (degrees of freedom within groups) = Total number of observations - Number of groups = 15 - 3 = 12<BR>\n" );
document.write( "* df_total = n-1 = 15-1 = 14\r<BR>\n" );
document.write( "\n" );
document.write( "**5. Calculate Mean Square:**\r<BR>\n" );
document.write( "\n" );
document.write( "* MSB (Mean Square Between) = SSB / df_between = 250 / 2 = 125<BR>\n" );
document.write( "* MSW (Mean Square Within) = SSW / df_within = 310 / 12 ≈ 25.83\r<BR>\n" );
document.write( "\n" );
document.write( "**6. Calculate the F-statistic:**\r<BR>\n" );
document.write( "\n" );
document.write( "F = MSB / MSW = 125 / 25.83 ≈ 4.84\r<BR>\n" );
document.write( "\n" );
document.write( "**7. Determine the Critical Value:**\r<BR>\n" );
document.write( "\n" );
document.write( "Using an F-distribution table with α = 0.05, df_between = 2, and df_within = 12, the critical F-value is approximately 3.89.\r<BR>\n" );
document.write( "\n" );
document.write( "**8. Decision:**\r<BR>\n" );
document.write( "\n" );
document.write( "Our calculated F-statistic (4.84) is *greater than* the critical value (3.89). Therefore, we *reject* the null hypothesis.\r<BR>\n" );
document.write( "\n" );
document.write( "**9. Conclusion:**\r<BR>\n" );
document.write( "\n" );
document.write( "At the 5% level of significance, there *is* sufficient evidence to conclude that the machines do *not* have equal output rates. At least one machine has a different output rate.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );