document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1131566>Question 1131566</A>: <I><SPAN STYLE=\"word-break: break-all;\"> . A population has variance &#963;^2=100.  A random sample of 25 from this population had a mean equal <BR>\n" );
document.write( "    17 and an estimated standard deviation equal 16.  Can we conclude that the population mean is less<BR>\n" );
document.write( "    than 25?  Let  &#945;=5%.  Assume x is normally distributed.  Hint, find &#963;_x .<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 #749092 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=749092\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> population variance is 100<BR>\n" );
document.write( "population standard deviation is sqrt(100) = 10<BR>\n" );
document.write( "sample size is 25.<BR>\n" );
document.write( "sample mean is 17<BR>\n" );
document.write( "sample standard deviation is 16.\r<BR>\n" );
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document.write( "if you know the population standard deviation, then you use that.<BR>\n" );
document.write( "otherwise you use the sample standard deviation and then the t-score rather than the z-score.\r<BR>\n" );
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document.write( "at 95% confidence level, the z-score will be plus or minus 1.96.\r<BR>\n" );
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document.write( "z-score formula is z = (x-m) / s\r<BR>\n" );
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document.write( "z is the z-score<BR>\n" );
document.write( "x is the raw score<BR>\n" );
document.write( "m is the mean<BR>\n" );
document.write( "s is the standard error.\r<BR>\n" );
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document.write( "standard error = population stnadard deviation / sqrt (sample size).\r<BR>\n" );
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document.write( "that becomes s - 10 / sqrt(25) = 10 / 5 = 2\r<BR>\n" );
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document.write( "on the high side for z, the formula becomes 1.96 = (x-17) / 2.\r<BR>\n" );
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document.write( "solve for x to get x = 2 * 1.96 + 17 = 3.92 + 17 = 20.92\r<BR>\n" );
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document.write( "on the low side for z, the formula becomes -1.96 = (1-17) / 2.\r<BR>\n" );
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document.write( "solve for x to get x = -2 * 1.96 + 17 = 13.08.\r<BR>\n" );
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document.write( "at 95% confident level, the true population mean is expected to be between 13.08 and 20.92.\r<BR>\n" );
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document.write( "based on this, you can conclude that the true population mean is less than 25.\r<BR>\n" );
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document.write( "that's my take.\r<BR>\n" );
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document.write( "here's some info on when to use t-score versus z-score.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"https://www.statisticshowto.datasciencecentral.com/probability-and-statistics/hypothesis-testing/t-score-vs-z-score/\" target = \"_blank\">https://www.statisticshowto.datasciencecentral.com/probability-and-statistics/hypothesis-testing/t-score-vs-z-score/</a>\r<BR>\n" );
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