document.write( "Question 1199714: The scores archived by the entired population of students follow a normal distribution.
\n" ); document.write( "The average score for the entire population was 21.
\n" ); document.write( "47.72% of the students scored between 15 and 21.\r
\n" ); document.write( "\n" ); document.write( "Full scholarship is offered to all students with a z-score of 3.2 or more.\r
\n" ); document.write( "\n" ); document.write( "a. Find the standard deviation. Show your work.\r
\n" ); document.write( "\n" ); document.write( "b. What is the minimum score a student must achieve in order to earn the scholarship? \r
\n" ); document.write( "\n" ); document.write( "c. What percentage of students were offered this scholarship? \n" ); document.write( "
Algebra.Com's Answer #848190 by ikleyn(52855)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "The scores archived by the \"highlight%28cross%28entired%29%29\" entire population of students follow a normal distribution.
\n" ); document.write( "The average score for the entire population was 21.
\n" ); document.write( "47.72% of the students scored between 15 and 21.\r
\n" ); document.write( "\n" ); document.write( "Full scholarship is offered to all students with a z-score of 3.2 or more.\r
\n" ); document.write( "\n" ); document.write( "a. Find the standard deviation. Show your work.\r
\n" ); document.write( "\n" ); document.write( "b. What is the minimum score a student must achieve in order to earn the scholarship? \r
\n" ); document.write( "\n" ); document.write( "c. What percentage of students were offered this scholarship?
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\n" ); document.write( "\n" ); document.write( "        The solution and the answers in the post by the other tutor are incorrect in parts  (a)  and  (b).\r
\n" ); document.write( "\n" ); document.write( "        I came to bring correct solution to all parts  (a),  (b),  (c).\r
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document.write( "                 Part (a)\r\n" );
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document.write( "     - Since 47.72% of the students scored between 15 and 21,\r\n" );
document.write( "     - since 21 is the average score, and\r\n" );
document.write( "     - since a normal distribution is symmetric,\r\n" );
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document.write( "we conclude that twice more percentage of the students, 2*47.72% = 95.44%  are scored between\r\n" );
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document.write( "     15  and  21 + (21-15) = 21 + 6 = 27.\r\n" );
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document.write( "Now recall the \"empirical rule\" of the normal distribution: 95.45% of values are within 2 standard deviations.\r\n" );
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document.write( "From the data above, we conclude that  21-15 = 6 = 27 - 21  scores is two standard deviations.\r\n" );
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document.write( "Hence, one standard deviation of this normal distribution is 6/2 = 3 scores.\r\n" );
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document.write( "It is the ANSWER to question (a): the standard deviation of this normal distribution is 3 scores.\r\n" );
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document.write( "Part (a) is completed.\r\n" );
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document.write( "                 Part (b)\r\n" );
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document.write( "As we completed part (a), the ANSWER to question (b) is immediate: this minimum score\r\n" );
document.write( "is 3.2 standard deviations above the average. i.e.\r\n" );
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document.write( "    21 + 3.2*3 = 30.6.\r\n" );
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document.write( "It is the ANSWER to question (b):  this minimum score is  30.6. \r\n" );
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document.write( "                 Part (c)\r\n" );
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document.write( "Question (c) asks to find the area under this normal curve on the right of the score 30.6.\r\n" );
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document.write( "The easiest way to do it is to use online free of charge calculator \r\n" );
document.write( "    https://onlinestatbook.com/2/calculators/normal_dist.html\r\n" );
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document.write( "It has very convenient and intuitively clear interface, so it, actually, does not require\r\n" );
document.write( "any explanations.  The calculator gives the value (the ANSWER)\r\n" );
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document.write( "    the area = 0.0007.\r\n" );
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document.write( "It means that 0.07% of students will be offered this scholarship.\r\n" );
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document.write( "Alternatively, instead of the online calculator, you may use your regular calculator like TI-83/84\r\n" );
document.write( "with the standard function normcdf.  Then the formula to get the desired area is\r\n" );
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document.write( "                    z1    z2   mean,  SD    <<<---===  formatting pattern\r\n" );
document.write( "    area = normcdf(31.6. 9999, 21,    3).\r\n" );
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document.write( "It will give you the same result.\r\n" );
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