document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199714>Question 1199714</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The scores archived by the entired population of students follow a normal distribution.<BR>\n" );
document.write( "The average score for the entire population was 21.<BR>\n" );
document.write( "47.72% of the students scored between 15 and 21.\r<BR>\n" );
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document.write( "Full scholarship is offered to all students with a z-score of 3.2 or more.\r<BR>\n" );
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document.write( "a. Find the standard deviation. Show your work.\r<BR>\n" );
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document.write( "b. What is the minimum score a student must achieve in order to earn the scholarship? \r<BR>\n" );
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document.write( "c. What percentage of students were offered this scholarship? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848164 by </B><A HREF=/tutors/aboutme.mpl?userid=textot><B>textot(100)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=textot><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=848164\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a) Find the standard deviation**\r<BR>\n" );
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document.write( "1. **Find the z-score corresponding to the 22.28th percentile:**\r<BR>\n" );
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document.write( "   * Since 47.72% of students scored between 15 and 21, and the distribution is symmetric, 22.28% of students scored below 15. <BR>\n" );
document.write( "   * Using a standard normal distribution table or calculator, find the z-score corresponding to the 22.28th percentile. <BR>\n" );
document.write( "     * This z-score is approximately **-0.77**.\r<BR>\n" );
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document.write( "2. **Calculate the standard deviation:**\r<BR>\n" );
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document.write( "   * Use the z-score formula: <BR>\n" );
document.write( "      * z = (X - μ) / σ <BR>\n" );
document.write( "      * where: <BR>\n" );
document.write( "          * z = z-score (-0.77)<BR>\n" );
document.write( "          * X = score (15)<BR>\n" );
document.write( "          * μ = mean (21)<BR>\n" );
document.write( "          * σ = standard deviation\r<BR>\n" );
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document.write( "   * Rearrange the formula to solve for σ:<BR>\n" );
document.write( "      * σ = (X - μ) / z<BR>\n" );
document.write( "      * σ = (15 - 21) / -0.77 <BR>\n" );
document.write( "      * σ = -6 / -0.77 <BR>\n" );
document.write( "      * σ ≈ 7.79\r<BR>\n" );
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document.write( "**Therefore, the standard deviation of the scores is approximately 7.79.**\r<BR>\n" );
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document.write( "**b) Find the minimum score for a scholarship**\r<BR>\n" );
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document.write( "1. **Find the raw score corresponding to a z-score of 3.2:**\r<BR>\n" );
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document.write( "   * Use the z-score formula:<BR>\n" );
document.write( "      * X = μ + z * σ<BR>\n" );
document.write( "      * X = 21 + 3.2 * 7.79<BR>\n" );
document.write( "      * X = 21 + 24.928<BR>\n" );
document.write( "      * X = 45.928\r<BR>\n" );
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document.write( "**Therefore, the minimum score a student must achieve to earn the scholarship is approximately 45.93.**\r<BR>\n" );
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document.write( "**c) Percentage of students offered the scholarship**\r<BR>\n" );
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document.write( "1. **Find the area to the right of z = 3.2:**\r<BR>\n" );
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document.write( "   * Using a standard normal distribution table or calculator, find the area to the right of z = 3.2. <BR>\n" );
document.write( "     * This area represents the proportion of students with a z-score of 3.2 or more. <BR>\n" );
document.write( "     * This area is approximately 0.0007 or 0.07%.\r<BR>\n" );
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document.write( "**Therefore, approximately 0.07% of students were offered the scholarship.**<BR>\n" );
document.write( " </SPAN>\n" );
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