document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1205419>Question 1205419</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The amount of time that people spend at Grover Hot Springs is normally distributed with a mean of 77 minutes and a standard deviation of 14 minutes. Suppose one person at the hot springs is randomly chosen. Let X = the amount of time that person spent at Grover Hot Springs . Round all answers to 4 decimal places where possible.\r<BR>\n" );
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document.write( "c. The park service is considering offering a discount for the 4% of their patrons who spend the least time at the hot springs. What is the longest amount of time a patron can spend at the hot springs and still receive the discount? <BR>\n" );
document.write( " minutes.\r<BR>\n" );
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document.write( "d. Find the Inter Quartile Range (IQR) for time spent at the hot springs.<BR>\n" );
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document.write( " minutes<BR>\n" );
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document.write( "IQR: <BR>\n" );
document.write( " minutes<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #842207 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=842207\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I'll focus on the 1st problem only. I'll mark it in blue.<BR>\n" );
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document.write( "The amount of time that people spend at Grover Hot Springs is normally distributed with a mean of 77 minutes and a standard deviation of 14 minutes. Suppose one person at the hot springs is randomly chosen. Let X = the amount of time that person spent at Grover Hot Springs . Round all answers to 4 decimal places where possible.\r<BR>\n" );
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document.write( "c. The park service is considering offering a discount for the 4% of their patrons who spend the least time at the hot springs. What is the longest amount of time a patron can spend at the hot springs and still receive the discount?<BR>\n" );
document.write( "minutes.<BR>\n" );
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document.write( "As mentioned by tutor Theo, you can use the online calculator at the link he posted. <BR>\n" );
document.write( "The calculator is very user friendly and provides a nice diagram. \r<BR>\n" );
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document.write( "For exam purposes, your teacher may not let you use that calculator (or similar online calculators). It's possible your teacher may only allow something like a TI83 or TI84.  \r<BR>\n" );
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document.write( "If so, then check out this resource<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.statology.org/inverse-normal-distribution/\">https://www.statology.org/inverse-normal-distribution/</a><BR>\n" );
document.write( "It explains how to reach the <font color=green>invNorm</font> function.\r<BR>\n" );
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document.write( "In this case you'll type in:<BR>\n" );
document.write( "<font color=green>invNorm(0.04)</font><BR>\n" );
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document.write( "The result of that command is approximately -1.750686\r<BR>\n" );
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document.write( "It will mean P(z < -1.750686) = 0.04 approximately.<BR>\n" );
document.write( "The area under the curve to the left of z = -1.750686 is roughly 0.04<BR>\n" );
document.write( "4% of the distribution is below the approximate cut off point of z = -1.750686\r<BR>\n" );
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document.write( "We'll use this to find the corresponding raw score x.<BR>\n" );
document.write( "z = (x - mu)/sigma<BR>\n" );
document.write( "z*sigma = x - mu<BR>\n" );
document.write( "x = z*sigma + mu<BR>\n" );
document.write( "x = -1.750686*14 + 77<BR>\n" );
document.write( "x = 52.490396<BR>\n" );
document.write( "x = <font color=red size=4>52.4904</font> when rounding to 4 decimal places.\r<BR>\n" );
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document.write( "The longest amount of time that can be spent is <font color=red size=4>approximately 52.4904 minutes</font>, and you will be in the \"lowest 4%\" category to get the discount. <BR>\n" );
document.write( "Any time higher than this will land you in the upper 96% of the distribution.\r<BR>\n" );
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document.write( "If you aren't allowed a calculator, then use of a Z table is the only other option.\r<BR>\n" );
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document.write( "Here's one such table<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.ztable.net/\">https://www.ztable.net/</a><BR>\n" );
document.write( "Tables like this should be found at the back of your stats textbook. <BR>\n" );
document.write( "Your teacher should likely hand them out as a reference sheet during exams.\r<BR>\n" );
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document.write( "The idea is to find the value of k such that P(Z < k) = 0.04<BR>\n" );
document.write( "Unfortunately the exact value 0.04 is NOT inside the table.<BR>\n" );
document.write( "The closest we can get is 0.04006<BR>\n" );
document.write( "This value can be found at the row starting with \"-1.7\" and in the column with \"0.05\" at the top.<BR>\n" );
document.write( "This means P(Z < -1.75) = 0.04006 approximately, which is roughly the same as P(Z < -1.75) = 0.04\r<BR>\n" );
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document.write( "Once you determine the proper z value, convert it to the raw score x as shown here:<BR>\n" );
document.write( "z = (x - mu)/sigma<BR>\n" );
document.write( "x = z*sigma + mu<BR>\n" );
document.write( "x = -1.75*14 + 77<BR>\n" );
document.write( "x = 52.5<BR>\n" );
document.write( "We get a slightly different answer because we're using a less accurate z value.<BR>\n" );
document.write( "Luckily 52.5 is close to 52.4904<BR>\n" );
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