document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1153359>Question 1153359</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A research firm conducted a survey to determine the mean amount Americans spend on coffee during a week. \r<BR>\n" );
document.write( "\n" );
document.write( "They found the distribution of weekly spending followed the normal distribution with a population standard deviation of $5. <BR>\n" );
document.write( "A sample of 49 Americans revealed that Xbar=$20. \r<BR>\n" );
document.write( "\n" );
document.write( " <BR>\n" );
document.write( "What is the point estimate of the population mean?\r<BR>\n" );
document.write( "\n" );
document.write( "Using the 95% level of confidence, determine the confidence interval for μ. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #775567 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=775567\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3>\r<BR>\n" );
document.write( "\n" );
document.write( "Question: What is the point estimate of the population mean?<BR>\n" );
document.write( "Answer: <font color=red>xbar = 20</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Question: What is the 95% confidence interval for mu?<BR>\n" );
document.write( "Answer: <font color=red>approximately (18.6, 21.4)</font>, in which we can alternatively write as 18.6 < mu < 21.4 or as <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P \pm M = 20 \pm 1.4\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "======================================================<BR>\n" );
document.write( "Work Shown:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "n = 49 is the sample size<BR>\n" );
document.write( "mu = population mean<BR>\n" );
document.write( "xbar = 20 is the sample mean<BR>\n" );
document.write( "sigma = 5 is the population standard deviation\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "xbar is the point estimate of the population mean mu. It is our best guess at the population mean based on the sample information we gathered/computed. Of course we aren't likely to have xbar and mu match up perfectly. This is where the confidence interval comes in. The confidence interval is like a net we cast out and we say we are 95% confident that the population mean is in this range. For any confidence interval involving mu, the point estimate xbar is always at the center of the confidence interval.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "At 95% confidence, the critical z value is roughly z = 1.96; use a table or calculator to determine this. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "L = lower bound of confidence interval<BR>\n" );
document.write( "L = xbar - z*(sigma/sqrt(n))<BR>\n" );
document.write( "L = 20 - 1.96*(5/sqrt(49))<BR>\n" );
document.write( "L = 20 - 1.96*(5/7)<BR>\n" );
document.write( "L = 20 - 1.96*(0.714285714285714)<BR>\n" );
document.write( "L = 20 - 1.4<BR>\n" );
document.write( "L = 18.6\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "U = upper bound of confidence interval<BR>\n" );
document.write( "U = xbar + z*(sigma/sqrt(n))<BR>\n" );
document.write( "U = 20 + 1.96*(5/sqrt(49))<BR>\n" );
document.write( "U = 20 + 1.96*(5/7)<BR>\n" );
document.write( "U = 20 + 1.96*(0.714285714285714)<BR>\n" );
document.write( "U = 20 + 1.4<BR>\n" );
document.write( "U = 21.4\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(L, U) = (18.6, 21.4) is the approximate 95% confidence interval.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If you want to write the confidence interval in the form L < mu < U, then you would write 18.6 < mu < 21.4\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Another format is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P \pm M\">, which has P as the point estimate and M as the margin of error. So, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P \pm M = 20 \pm 1.4\"> meaning we have a point estimate of 20 and a margin of error of 1.4; this is reflected in L = 20-1.4 and U = 20+1.4<BR>\n" );
document.write( " <BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );