document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1118054>Question 1118054</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You wish to test the following claim (Ha) at a significance level of <BR>\n" );
document.write( "&#945;=0.01<BR>\n" );
document.write( "Ho:&#956;=59.5<BR>\n" );
document.write( "Ha:&#956;>59.5\r<BR>\n" );
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document.write( "You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size <BR>\n" );
document.write( "n=16 with a mean of <BR>\n" );
document.write( "¯x=68.1<BR>\n" );
document.write( " and a standard deviation of <BR>\n" );
document.write( "s=13.2\r<BR>\n" );
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document.write( "What is the critical value for this test?\r<BR>\n" );
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document.write( "Please show work and explain how to find critical value, I can't figure it out for any question!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #733286 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=733286\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> if you don't know the standard devition for the population, then you use the T-test rather than the Z-test.\r<BR>\n" );
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document.write( "your alpha = .01.\r<BR>\n" );
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document.write( "it's all on the right side of the normal distribution because your test is greater than rather than not equal.\r<BR>\n" );
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document.write( "your null hypothesis is that the mean is 59.5\r<BR>\n" );
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document.write( "your alternate hypothesis is that the mean is greater than 59.5.\r<BR>\n" );
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document.write( "your sample size is 16 with a mean of 68.1 and a standard deviation of 13.2.\r<BR>\n" );
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document.write( "you want to know what the critical value for this test is.\r<BR>\n" );
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document.write( "you need to look that up in the critical T-score Table.\r<BR>\n" );
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document.write( "your degrees of freedom are the sample size minus 1 = 15.\r<BR>\n" );
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document.write( "you are looking for a one-tail critical T-score with an alpha of .01.\r<BR>\n" );
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document.write( "the T-score Table i looked at is at <a  rel=nofollow HREF= \"http://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf\" target = \"_blank\">http://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf</a>\r<BR>\n" );
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document.write( "here's a picture of what i found.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/060201.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "i looked up a one tail probability of .01.<BR>\n" );
document.write( "that was in the 8th column to the right of the first column which contained the title of one-tail.\r<BR>\n" );
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document.write( "i then looked down in the df column (degrees of freedom) until i found 15.\r<BR>\n" );
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document.write( "i then went to the 8th column to the right of that on that row and found a critical T-score of 2.602.\r<BR>\n" );
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document.write( "i then confirmed with my TI-84 Plus Calculator that the critical T-score was 2.602.\r<BR>\n" );
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document.write( "in that, the calculator gave me the answer to more decimal places (2.60248029).\r<BR>\n" );
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document.write( "my critical T-score is what i am testing against.\r<BR>\n" );
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document.write( "if my test T-score is greater than tht, then the results are significant and i would say that it is likely that the T-score of my test was not due to random variations in samples of size 16, but possibly something more significant, like a real difference dues to something other than chance.\r<BR>\n" );
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document.write( "the critical alpha of = .01 means that i can be fairly confident that, if i did this test an infinite number of times., that the test T-score would be within the confidence limits 99% of the time, meaning that only 1 out of 100 samples would give me the results i saw.\r<BR>\n" );
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document.write( "to do the test, i need to calculate the standard error.\r<BR>\n" );
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document.write( "the standard error is the standard deviation of the distribution of sample means.\r<BR>\n" );
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document.write( "the standard error is affected by the sample size.\r<BR>\n" );
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document.write( "the larger the sample size, the smaller the standard error.\r<BR>\n" );
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document.write( "if i can find a reference to explain this to you, i'll include it down below.\r<BR>\n" );
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document.write( "for now, just accept that it is so.\r<BR>\n" );
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document.write( "the formula for sample error is s = sqrt(sd/n)\r<BR>\n" );
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document.write( "s is the standard error.<BR>\n" );
document.write( "sd is the standard deviation of the population or the sample.<BR>\n" );
document.write( "n is the sample size.\r<BR>\n" );
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document.write( "if you know the standard deviation of the population, you use that.\r<BR>\n" );
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document.write( "if not, you use the standard deviation of the sample itself.\r<BR>\n" );
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document.write( "using this formula, you get s = sqrt(13.2/16) = .9082951062.\r<BR>\n" );
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document.write( "i also need to calculate the T-score for the test, using that standard error.\r<BR>\n" );
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document.write( "the T-score formula is T(15) = (x-m)/s.\r<BR>\n" );
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document.write( "x is the mean of the sample.<BR>\n" );
document.write( "m is the mean of the population, or the assumed mean of your null hypothesis.<BR>\n" );
document.write( "s is the sample size.\r<BR>\n" );
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document.write( "it's the same as the Z-score formula, except you are dealing with a T-score and not a Z-score, and you have to take into consideration the degrees of freedom.\r<BR>\n" );
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document.write( "using the information i know have, i get:\r<BR>\n" );
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document.write( "T(15) = (68.1 - 59.5) / .9082951062 = 9.46828838.\r<BR>\n" );
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document.write( "my critical T-score is 2.602 for samples of size 16.\r<BR>\n" );
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document.write( "this is much greater than that, so i can safely assume that the difference is not due to random variation in samples of size 16, but a real difference due to something other than chance.\r<BR>\n" );
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document.write( "here's a reference on sample error.\r<BR>\n" );
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document.write( "check it out and then go to the next section on the central limit theorem.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://davidmlane.com/hyperstat/A103397.html\" target = \"_blank\">http://davidmlane.com/hyperstat/A103397.html</a>\r<BR>\n" );
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document.write( "if you go a bit further, there is even a simulation that you might be able to perform that demonstrates the central limit theorem in action.\r<BR>\n" );
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document.write( "the rules for when you would use a T-score rather than a Z-score are discussed in the following reference.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.statisticshowto.com/probability-and-statistics/hypothesis-testing/t-score-vs-z-score/\" target = \"_blank\">http://www.statisticshowto.com/probability-and-statistics/hypothesis-testing/t-score-vs-z-score/</a>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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