document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197052>Question 1197052</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Last year's freshman class at Big State University totaled 5,327 students. Of those, 1,265 received a merit scholarship to help offset tuition costs their freshman year (although the amount varied per student). The amount received was N($3,468, $489). If the cost of full tuition was $4,100 last year, what percentage of students who received a merit scholarship did not received enough to cover full tuition? (Round your answer to the nearest whole percent.) </SPAN>\n" );
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document.write( "The notation N(3468, 489) tells us we have a normal distribution with these parameters<BR>\n" );
document.write( "mu = 3468 = mean<BR>\n" );
document.write( "sigma = 489 = standard deviation\r<BR>\n" );
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document.write( "If a student got a scholarship, then they landed somewhere on this normal distribution. <BR>\n" );
document.write( "The majority of the students will cluster around the mean of $3468\r<BR>\n" );
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document.write( "Let's convert the raw score x = 4100 to its corresponding z score<BR>\n" );
document.write( "z = (x - mu)/sigma<BR>\n" );
document.write( "z = (4100 - 3468)/489<BR>\n" );
document.write( "z = 1.29 approximately\r<BR>\n" );
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document.write( "Then we'll use a Z table such as this one<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.ztable.net/\">https://www.ztable.net/</a><BR>\n" );
document.write( "or one found in the back of your book.\r<BR>\n" );
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document.write( "That table says<BR>\n" );
document.write( "P(Z < 1.29) = 0.90147<BR>\n" );
document.write( "So about 90.147% of the students who got a scholarship received less than $4,100\r<BR>\n" );
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document.write( "If you were to use a calculator such as this one<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://davidmlane.com/normal.html\">https://davidmlane.com/normal.html</a><BR>\n" );
document.write( "then you'll find that P(Z < 1.29) = 0.9015<BR>\n" );
document.write( "The area under the curve to the left of z = 1.29 is roughly 0.9015\r<BR>\n" );
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document.write( "Alternatively you can use a TI83 or TI84 calculator to type in <font color=blue>normalcdf(-99,1.29)</font> and it will display 0.9014746057 approximately. The normalcdf function can be found by hitting the key labeled \"2nd\" then pressing the VARS key.\r<BR>\n" );
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document.write( "If you wish to use a spreadsheet, then type in <font color=blue>=NORM.DIST(1.29,0,1,true)</font><BR>\n" );
document.write( "The 1.29 refers to the z score we calculated<BR>\n" );
document.write( "the 0 and 1 are the mean and standard deviation respectively<BR>\n" );
document.write( "The \"true\" tells the spreadsheet to compute the cdf rather than pdf<BR>\n" );
document.write( "You could also type in <font color=blue>=NORM.DIST(4100,3468,489,true)</font> if you wanted to skip converting to a z score. This avoids the slight error that happens when we rounded the z score to 1.29\r<BR>\n" );
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document.write( "As you can see, there are various methods to calculating the area under the standard normal curve to the left of z = 1.29\r<BR>\n" );
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document.write( "Answer: <font color=red>Approximately 90%</font> (when rounding to the nearest whole percent)<BR>\n" );
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