document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1182170>Question 1182170</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The final grades of students in Foreign Language are normally distributed. records show that the average grade of students is 85 with the standard deviation of 2.3; suppose the professor wants to convert the grades of the students into an alpha-grade; A as the highest and D as the lowest grades, and DECIDES to assign: 10% A's, 20% B's, 30% C's, and 40% D's. What shall be the cut-off grades of these grade ranges? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849943 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</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=849943\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to determine the cut-off grades for the Foreign Language class:\r<BR>\n" );
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document.write( "**Understanding the Problem**\r<BR>\n" );
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document.write( "The professor wants to assign letter grades based on a normal distribution. This means we need to find the scores that correspond to the given percentages of each grade.\r<BR>\n" );
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document.write( "**Steps**\r<BR>\n" );
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document.write( "1. **Percentiles:** We'll use the given percentages to find the corresponding percentiles in the normal distribution.<BR>\n" );
document.write( "   * 10% A's: This means the top 10% of scores are A's. So, the cutoff for a B is at the 90th percentile.<BR>\n" );
document.write( "   * 20% B's: The cutoff for a C is at the 70th percentile (90th percentile - 20%).<BR>\n" );
document.write( "   * 30% C's: The cutoff for a D is at the 40th percentile (70th percentile - 30%).\r<BR>\n" );
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document.write( "2. **Z-scores:** We need to find the z-scores that correspond to these percentiles. A z-score tells us how many standard deviations a score is away from the mean. You can use a z-table or a calculator with statistical functions to find these z-scores.<BR>\n" );
document.write( "   * 90th percentile: z ≈ 1.28<BR>\n" );
document.write( "   * 70th percentile: z ≈ 0.52<BR>\n" );
document.write( "   * 40th percentile: z ≈ -0.25\r<BR>\n" );
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document.write( "3. **Cut-off Scores:** Now, use the z-scores, the mean (85), and the standard deviation (2.3) to calculate the cut-off scores:<BR>\n" );
document.write( "   * B cutoff (90th percentile): Score = Mean + (z-score * Standard Deviation) = 85 + (1.28 * 2.3) ≈ 87.94<BR>\n" );
document.write( "   * C cutoff (70th percentile): Score = 85 + (0.52 * 2.3) ≈ 86.20<BR>\n" );
document.write( "   * D cutoff (40th percentile): Score = 85 + (-0.25 * 2.3) ≈ 84.43\r<BR>\n" );
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document.write( "**Results**\r<BR>\n" );
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document.write( "* **A:** 87.94 and above<BR>\n" );
document.write( "* **B:** Between 86.20 and 87.94<BR>\n" );
document.write( "* **C:** Between 84.43 and 86.20<BR>\n" );
document.write( "* **D:** Below 84.43 </SPAN>\n" );
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