document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=906208>Question 906208</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a random sample of 88 observations produced a mean x = 25.8 and standard deviation s=2.7. Find 95%, 90% and 99% confidence interval for u\r<BR>\n" );
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document.write( "I have been trying to figure out how to do this problem for hours but I keep getting stuck.  If someone can show me step by step it would help me alot.  Thank you so much. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #549684 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=549684\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> sample size = 88<BR>\n" );
document.write( "mean = 25.8<BR>\n" );
document.write( "standard deviation = 2.7\r<BR>\n" );
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document.write( "mu or what you show as u is presumably the mean of the population that this sample was taken from.\r<BR>\n" );
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document.write( "if my assumption is correct, then you would solve this as follows:\r<BR>\n" );
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document.write( "standard error, which is the standard deviation of the distribution of sample means, is equal to standard deviation divided by the square root of the sample size.\r<BR>\n" );
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document.write( "this makes se = 2.7 / sqrt(88) which makes se = .29 rounded to 2 decimal places which should be adequate for the accuracy that you need, and certainly enough to show you the concept of what you're trying to do.\r<BR>\n" );
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document.write( "se represents standard error.<BR>\n" );
document.write( "sqrt represents square root of.\r<BR>\n" );
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document.write( "you want to find the interval where the population mean is expected to be within 95% of the time, 90% of the time, and 99% of the time.\r<BR>\n" );
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document.write( "you will want to find the z-scores that correspond to those confidence limits.\r<BR>\n" );
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document.write( "x-scores and x-factors mean the same thing.  i sometimes use z-scores and i sometimes use z-factors but i'm talking about the same thing.\r<BR>\n" );
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document.write( "a two sided confidence interval is assumed.<BR>\n" );
document.write( "that means that the interval will have a tail at both ends that is outside the confidence interval.<BR>\n" );
document.write( "this is standard procedure unless indicated otherwise.\r<BR>\n" );
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document.write( "for a 95% confidence interval, the tail will be calculated as (100%-95%)/2 = 2.5% of the area at each end.\r<BR>\n" );
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document.write( "for a 90% confidence interval, you will have a tail of 5% at each end.\r<BR>\n" );
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document.write( "for a 99% confidence interval, you will have a tail of .5% at each end.\r<BR>\n" );
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document.write( "at 95%, you look for an area of 97.5% which is an area of .975 which will generate a z-factor of 1.96.\r<BR>\n" );
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document.write( "at 90%, you will look for an area of 95% which is an area of .95 which will generate a z-factor of 1.65.\r<BR>\n" );
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document.write( "at 99%, you will look for an area of 99.5% which is an area of .995 which will generate a z-factor of 2.58\r<BR>\n" );
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document.write( "all z factors have been rounded to 2 decimal places.\r<BR>\n" );
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document.write( "you now want to find the limits of your sample mean where you expect the population mean to be within 95% of the time, 90% of the time, or 99% of the time.\r<BR>\n" );
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document.write( "to find the margin of error, you multiply the limiting z factor * the standard error.\r<BR>\n" );
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document.write( "the limiting z-factor is also called the critical z-factor.\r<BR>\n" );
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document.write( "at 95% confidence interval, your margin of error will be plus or minus 1.96 * .29 which is equal to plus or minus .57 rounded to 2 decimal places.\r<BR>\n" );
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document.write( "since your sample mean is 25.8, then your 95% confidence interval based on that sample mean is 25.8 - .57 to 25.8 + .57.\r<BR>\n" );
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document.write( "this results in an interval from 25.23 to 26.37\r<BR>\n" );
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document.write( "you will be 95% confident that your population mean is within the interval of from 25.23 to 26.37 based on this sample.\r<BR>\n" );
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document.write( "you can do the same calculations using the critical z-factors for 90% confidence interval and 99% confidence interval.\r<BR>\n" );
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document.write( "what will change is the margin of error.\r<BR>\n" );
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document.write( "the higher the confidence level, the greater is the margin of error.\r<BR>\n" );
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document.write( "the formula for margin of error is standard error * critical z-factor.\r<BR>\n" );
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document.write( "at 90% confidence interval, the margin of error will be 25.8 plus or minus .29 * 1.65.\r<BR>\n" );
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document.write( "at 99% confidence interval, the margin of error will be 25.8 plus or minus .29 * 2.58\r<BR>\n" );
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document.write( "hopefully you can do the rest from here.<BR>\n" );
document.write( "if not, let me know and i'll guide you further depending on what your questions are.\r<BR>\n" );
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document.write( "a picture of what i just calculated for you will look like this from the z-score perspective:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/092603.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "a pictures of what i just calculated for you will look like this from the mean and standard error perspective:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/092604.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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