document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1172983>Question 1172983</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For the year 2010, 33% of taxpayers with adjusted gross incomes between $30,000 and $60,000<BR>\n" );
document.write( "itemized deductions on their federal income tax return (The Wall Street Journal, October 25, 2012).<BR>\n" );
document.write( "The mean amount of deductions for this population of taxpayers was $16,642. Assume the standard<BR>\n" );
document.write( "deviation is 𝜎 = $2400.<BR>\n" );
document.write( "3.1. What is the probability that a sample of taxpayers from this income group who have itemized<BR>\n" );
document.write( "deductions will show a sample mean within $200 of the population mean for each of the<BR>\n" );
document.write( "following sample sizes: 30, 50, 100, and 400?<BR>\n" );
document.write( "3.2. What is the advantage of a larger sample size when attempting to estimate the population mean?<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #798895 by </B><A HREF=/tutors/aboutme.mpl?userid=Boreal><B>Boreal(15235)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Boreal><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=Boreal><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=798895\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> z < (x-mean)/sd/sqrt(n) and z> (-200*sqrt(30)/2400)<BR>\n" );
document.write( "z < 200*sqrt(30)/2400<BR>\n" );
document.write( " -0.46 < z < 0.46<BR>\n" );
document.write( "probability is 0.3545<BR>\n" );
document.write( "-<BR>\n" );
document.write( "with sample size of 50<BR>\n" );
document.write( "it is <abs (200*sqrt(50)/2400 or -0.59 < z < 0.59<BR>\n" );
document.write( "probability is 0.4448<BR>\n" );
document.write( "-<BR>\n" );
document.write( "with sample size of 100<BR>\n" );
document.write( "it is < abs (200*10/2400) or -0.83 < z < 0.83 <BR>\n" );
document.write( "probability is 0.5935<BR>\n" );
document.write( "-<BR>\n" );
document.write( "with n=400 <BR>\n" );
document.write( "-1.67 < z < 1.67<BR>\n" );
document.write( "probability is 0.9051<BR>\n" );
document.write( "-<BR>\n" );
document.write( "The larger the sample size, the more likely the estimate of the population mean will be within a fixed interval. <BR>\n" );
document.write( "Confidence intervals will be narrower.  The sampling distribution of the sample mean over different sample sizes will become sharper, less variable, the greater the sample size. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );