document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177272>Question 1177272</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Grades assigned by an economics lecturer for his postgraduate coursework students at a oxford have historically followed a symmetrical distribution: 5% HDs, 25% Ds, 40% Cs, 25% Ps and 5% Fs. This year, a sample of grades revealed 11 HDs, 32 Ds, 62 Cs, 29 Ps and 16 Fs. Can you conclude, at the 1% level of significance, that this year’s marks are distributed differently from marks in the past? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850494 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=850494\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We will use a chi-square goodness-of-fit test to determine if this year's grades are distributed differently from the historical distribution.\r<BR>\n" );
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document.write( "**1. Define Hypotheses:**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H0):** The distribution of grades this year is the same as the historical distribution.<BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** The distribution of grades this year is different from the historical distribution.\r<BR>\n" );
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document.write( "**2. Set Significance Level:**\r<BR>\n" );
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document.write( "* α = 0.01\r<BR>\n" );
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document.write( "**3. Observed and Expected Frequencies:**\r<BR>\n" );
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document.write( "* Total number of students: 11 + 32 + 62 + 29 + 16 = 150<BR>\n" );
document.write( "* Historical percentages: 5% HD, 25% D, 40% C, 25% P, 5% F<BR>\n" );
document.write( "* Expected frequencies:<BR>\n" );
document.write( "    * HD: 150 * 0.05 = 7.5<BR>\n" );
document.write( "    * D: 150 * 0.25 = 37.5<BR>\n" );
document.write( "    * C: 150 * 0.40 = 60<BR>\n" );
document.write( "    * P: 150 * 0.25 = 37.5<BR>\n" );
document.write( "    * F: 150 * 0.05 = 7.5<BR>\n" );
document.write( "* Observed frequencies:<BR>\n" );
document.write( "    * HD: 11<BR>\n" );
document.write( "    * D: 32<BR>\n" );
document.write( "    * C: 62<BR>\n" );
document.write( "    * P: 29<BR>\n" );
document.write( "    * F: 16\r<BR>\n" );
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document.write( "**4. Calculate the Chi-Square Statistic:**\r<BR>\n" );
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document.write( "* χ² = Σ [(Observed - Expected)² / Expected]<BR>\n" );
document.write( "* χ² = [(11 - 7.5)² / 7.5] + [(32 - 37.5)² / 37.5] + [(62 - 60)² / 60] + [(29 - 37.5)² / 37.5] + [(16 - 7.5)² / 7.5]<BR>\n" );
document.write( "* χ² = (3.5² / 7.5) + (-5.5² / 37.5) + (2² / 60) + (-8.5² / 37.5) + (8.5² / 7.5)<BR>\n" );
document.write( "* χ² = (12.25 / 7.5) + (30.25 / 37.5) + (4 / 60) + (72.25 / 37.5) + (72.25 / 7.5)<BR>\n" );
document.write( "* χ² = 1.6333 + 0.8067 + 0.0667 + 1.9267 + 9.6333<BR>\n" );
document.write( "* χ² ≈ 14.0667\r<BR>\n" );
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document.write( "**5. Determine Degrees of Freedom:**\r<BR>\n" );
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document.write( "* Degrees of freedom (df) = number of categories - 1<BR>\n" );
document.write( "* df = 5 - 1 = 4\r<BR>\n" );
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document.write( "**6. Find the Critical Chi-Square Value:**\r<BR>\n" );
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document.write( "* Using a chi-square distribution table or calculator, with df = 4 and α = 0.01, the critical chi-square value is approximately 13.277.\r<BR>\n" );
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document.write( "**7. Compare the Calculated Chi-Square and Critical Value:**\r<BR>\n" );
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document.write( "* Calculated χ² (14.0667) > Critical χ² (13.277)\r<BR>\n" );
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document.write( "**8. Make a Decision:**\r<BR>\n" );
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document.write( "* Since the calculated chi-square value is greater than the critical chi-square value, we reject the null hypothesis.\r<BR>\n" );
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document.write( "**9. Conclusion:**\r<BR>\n" );
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document.write( "* At the 1% level of significance, we can conclude that this year's marks are distributed differently from marks in the past.\r<BR>\n" );
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document.write( "Final Answer: Yes, we can conclude at the 1% level of significance that this year’s marks are distributed differently from marks in the past.<BR>\n" );
document.write( " </SPAN>\n" );
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