document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1184639>Question 1184639</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help, I've been trying to find out how to do these problems online for hours.\r<BR>\n" );
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document.write( "For the following distribution of scores:<BR>\n" );
document.write( " x      f<BR>\n" );
document.write( "10     1<BR>\n" );
document.write( "9      2<BR>\n" );
document.write( "8      4<BR>\n" );
document.write( "7      5<BR>\n" );
document.write( "6      3<BR>\n" );
document.write( "5      1<BR>\n" );
document.write( "4      1<BR>\n" );
document.write( "3      1\r<BR>\n" );
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document.write( "a) Find the middle 30% of the scores.<BR>\n" );
document.write( "b) Find the bottom 20% of the scores.<BR>\n" );
document.write( "c) Find the top 10% of the scores.<BR>\n" );
document.write( "d) Find the bottom 70% of the scores.<BR>\n" );
document.write( "e) Find the top 60% of the scores. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849817 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=849817\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You're right, these percentile problems can be tricky! Let's break down each one step by step, and I'll explain the logic clearly.\r<BR>\n" );
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document.write( "**1. Calculate Cumulative Frequencies:**\r<BR>\n" );
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document.write( "This is the crucial first step.  Cumulative frequency tells you how many scores are *at or below* a given value.\r<BR>\n" );
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document.write( "| x   | f   | Cumulative f |<BR>\n" );
document.write( "| --- | --- | ------------- |<BR>\n" );
document.write( "| 10  | 1   | 1             |<BR>\n" );
document.write( "| 9   | 2   | 3             |<BR>\n" );
document.write( "| 8   | 4   | 7             |<BR>\n" );
document.write( "| 7   | 5   | 12            |<BR>\n" );
document.write( "| 6   | 3   | 15            |<BR>\n" );
document.write( "| 5   | 1   | 16            |<BR>\n" );
document.write( "| 4   | 1   | 17            |<BR>\n" );
document.write( "| 3   | 1   | 18            |\r<BR>\n" );
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document.write( "The total number of scores (N) is 18.\r<BR>\n" );
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document.write( "**2. Understanding Percentiles:**\r<BR>\n" );
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document.write( "A percentile tells you the score at or below which a certain percentage of the data falls.  For example, the 20th percentile is the score at or below which 20% of the scores lie.\r<BR>\n" );
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document.write( "**3. The Calculations:**\r<BR>\n" );
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document.write( "**a) Middle 30%:**\r<BR>\n" );
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document.write( "*   **Logic:** The middle 30% is *between* the 35th and 65th percentiles.<BR>\n" );
document.write( "*   **35th Percentile:** (0.35) * 18 = 6.3. Since this isn't a whole number, we round *up* to the nearest whole number, which is 7.  The 7th score in the cumulative frequency table is 8.<BR>\n" );
document.write( "*   **65th Percentile:** (0.65) * 18 = 11.7. We round *up* to 12. The 12th score is 7.<BR>\n" );
document.write( "*   **Answer:** The middle 30% of scores are 7 and 8.\r<BR>\n" );
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document.write( "**b) Bottom 20%:**\r<BR>\n" );
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document.write( "*   **Logic:** We're looking for the score at or below which 20% of the data falls.<BR>\n" );
document.write( "*   **(0.20) * 18 = 3.6. Round up to 4.** The 4th score in the cumulative frequency table is 8.<BR>\n" );
document.write( "*   **Answer:** The bottom 20% of the scores are 8 and below.  (It's a little counterintuitive because the *value* is 8, but it represents the *cutoff* for the bottom 20% of the *scores*.)\r<BR>\n" );
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document.write( "**c) Top 10%:**\r<BR>\n" );
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document.write( "*   **Logic:** This is the score at or above which 10% of the data falls. This is the same as the 90th percentile (100% - 10% = 90%).<BR>\n" );
document.write( "*   **(0.90) * 18 = 16.2. Round up to 17.** The 17th score is 4.<BR>\n" );
document.write( "*   **Answer:** The top 10% of the scores are 4 and above.\r<BR>\n" );
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document.write( "**d) Bottom 70%:**\r<BR>\n" );
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document.write( "*   **(0.70) * 18 = 12.6. Round up to 13.** The 13th score is 7.<BR>\n" );
document.write( "*   **Answer:** The bottom 70% of the scores are 7 and below.\r<BR>\n" );
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document.write( "**e) Top 60%:**\r<BR>\n" );
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document.write( "*   **Logic:**  This is the same as the 40th percentile (100% - 60% = 40%).<BR>\n" );
document.write( "*   **(0.40) * 18 = 7.2. Round up to 8.** The 8th score is 8.<BR>\n" );
document.write( "*   **Answer:** The top 60% of the scores are 8 and above.\r<BR>\n" );
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document.write( "**Key Points to Remember:**\r<BR>\n" );
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document.write( "*   **Round Up:** When the percentile calculation results in a decimal, *always* round up to the next whole number.  This is because we're looking for the *position* of the score in the ordered list.<BR>\n" );
document.write( "*   **Cumulative Frequency:** The cumulative frequency table is essential for these problems.<BR>\n" );
document.write( "*   **\"At or Below/Above\":**  Percentiles represent the value *at or below* a certain percentage of the data.  So, the bottom 20% includes all scores up to and including the score at the 20th percentile.\r<BR>\n" );
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document.write( "I hope this detailed explanation helps! Let me know if you have any more questions.<BR>\n" );
document.write( " </SPAN>\n" );
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