document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1060679>Question 1060679</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can someone show me how to solve these problem below using the Central Limit Theorem?\r<BR>\n" );
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document.write( "A specific study found that the average number of doctor visits per year for people over 55 is 8 with a standard deviation of 2. Assume that the variable is normally distributed.\r<BR>\n" );
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document.write( "1.Identify the population mean.\r<BR>\n" );
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document.write( "2.Identify the population standard deviation.\r<BR>\n" );
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document.write( "3.Suppose a random sample of 15 people over 55 is selected.  What is the probability that the sample mean is above 9?\r<BR>\n" );
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document.write( "4.Suppose a random sample of 100 people over 55 is selected.  What is the probability that the sample mean will be below 7?\r<BR>\n" );
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document.write( "A study found that citizens spend on average $1950 per year on groceries with a standard deviation of $100.  Assume that the variable is normally distributed.\r<BR>\n" );
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document.write( "5.Identify the population mean.\r<BR>\n" );
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document.write( "6.Identify the population standard deviation.\r<BR>\n" );
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document.write( "7.Find the probability that a sample of 20 citizens will have a mean less than $1800.\r<BR>\n" );
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document.write( "8.Find the probability that a sample of 500 citizens will have a mean greater than $2000.\r<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 #675563 by </B><A HREF=/tutors/aboutme.mpl?userid=rothauserc><B>rothauserc(4718)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rothauserc><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=rothauserc><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=675563\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> When we have a population that is normally distributed, we make decision/s based on the sample size<BR>\n" );
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document.write( "1) population mean(u) is 8<BR>\n" );
document.write( "2) population standard deviation(std.dev.) is 2<BR>\n" );
document.write( "3) sample size(n) is 15 < 30 ( that is, the sample size is small per the Central Limit Theorem(CLT), therefore we will use the t-distribution.<BR>\n" );
document.write( "Note that the sample mean is the same as the population mean per CLT<BR>\n" );
document.write( "t-value = (9 - 8) / (2/square root(15) = 1.9365<BR>\n" );
document.write( "degrees of freedom = 15 - 1 = 14<BR>\n" );
document.write( "we use the table of t-values to get probability<BR>\n" );
document.write( "Probability(Pr) ( X > 9 ) = 1 - Pr ( X < 9 )<BR>\n" );
document.write( "Pr ( X > 9 ) = 1 - 0.9634 = 0.0366 approximately 0.04<BR>\n" );
document.write( "4) n = 100 > 30, we use normal distribution tables and compute z-score<BR>\n" );
document.write( "z-score = (7 - 8) / 2 = -0.50<BR>\n" );
document.write( "Pr ( X < 7 ) = 0.3085 approximately 0.31<BR>\n" );
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document.write( "5) population mean(u) is 1950<BR>\n" );
document.write( "6) population std.dev. is 100<BR>\n" );
document.write( "7) n = 20 < 30, we use the t-statistic<BR>\n" );
document.write( "t-value = (1800 - 1950) / (100/square root(20) = -6.7082<BR>\n" );
document.write( "Note we use the absolute value of the t-value with the t-tables, then subtract from 1<BR>\n" );
document.write( "degrees of freedom = 20 - 1 = 19<BR>\n" );
document.write( "Pr ( X < 1800 ) = 1.00 - 1.00 = 0<BR>\n" );
document.write( "8) n = 500 > 30, we use the z-score and the z-tables<BR>\n" );
document.write( "z-value = ( 2000 - 1950 ) / 100 = 0.50<BR>\n" );
document.write( "Pr ( X > 2000 ) = 1 - Pr ( X < 2000) = 1 - 0.6915 = 0.3085 approximately 0.31<BR>\n" );
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