document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1139503>Question 1139503</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello, i am having a bit of a problem that i am struggling with. I was wondering can you please break the problem down? A survey was conducted that asked<BR>\n" );
document.write( "1004 people how many books they had read in the past year. Results indicated that <BR>\n" );
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document.write( "12.5<BR>\n" );
document.write( "12.5 books and s<BR>\n" );
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document.write( "16.6<BR>\n" );
document.write( "16.6 books. Construct a <BR>\n" );
document.write( "99<BR>\n" );
document.write( "99​% confidence interval for the mean number of books people read. Interpret the interval. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #760010 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=760010\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"times\" color=\"black\" size=\"3\"><BR>\n" );
document.write( "Given info:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n = 1004\"> is the sample size<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \overline{x} = 12.5\"> is the sample mean<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large s = 16.6\"> is the sample standard deviation\r<BR>\n" );
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document.write( "At 99% confidence, the z critical value is approximately z = 2.576<BR>\n" );
document.write( "This can be found either through a table or a calculator. I used <a  rel=nofollow HREF= \"http://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf\">this table</a> to get the value (Go to the bottom of the table, look at the row that starts with Z and the column that has 99% at the bottom; you should see 2.576 just above the \"99%\".)\r<BR>\n" );
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document.write( "note: we don't know the population standard deviation (sigma) so we should use the T distribution; however, since n = 1004 is so large, this means we can use an approximation of the normal Z distribution instead. For more info, check out the <a  rel=nofollow HREF= \"https://stattrek.com/statistics/dictionary.aspx?definition=central_limit_theorem\">Central Limit Theorem</a>. So this is why I'm using Z instead of T. For large values of n, computing the critical T value is often not possible with a table (though a calculator can handle it just fine). Often anything over n = 30 is considered \"large\".  \r<BR>\n" );
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document.write( "Using the values mentioned, the lower boundary of the confidence interval is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large L = \overline{x} - z*\frac{s}{\sqrt{n}}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large L \approx 12.5 - 2.576*\frac{16.6}{\sqrt{1004}}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large L \approx 12.5 - 2.576*\frac{16.6}{31.685959}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large L \approx 12.5 - 2.576*0.523891\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large L \approx 12.5 - 1.349 543\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large L \approx 11.150457\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large L \approx 11.2\">\r<BR>\n" );
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document.write( "And the upper boundary is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large U = \overline{x} + z*\frac{s}{\sqrt{n}}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large U \approx 12.5 + 2.576*\frac{16.6}{\sqrt{1004}}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large U \approx 12.5 + 2.576*\frac{16.6}{31.685959}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large U \approx 12.5 + 2.576*0.523891\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large U \approx 12.5 + 1.349 543\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large U \approx 13.849543\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large U \approx 13.8\">\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "The 99% confidence interval is <font color=red size=4>(11.2, 13.8)</font> which can be written as <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large {\color{red}{11.2 < \mu < 13.8}}\">\r<BR>\n" );
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document.write( "Interpretation: We are 99% confident that the population parameter is in this interval. So we are 99% confident that the population mean of the number of books read is somewhere between 11.2 and 13.8 books. If you took 100 repeated samples and created confidence intervals from them, then about 99 of them should contain the true parameter mu.\r<BR>\n" );
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document.write( "Be careful not to say that \"there is a 99% chance that the parameter mu is in the interval (11.2, 13.8)\" as this is misleading and often a trick/trap that your teacher may set up to make sure the student is paying attention. <BR>\n" );
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