document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1183652>Question 1183652</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 	 <BR>\n" );
document.write( "		Suppose the annual savings (in millions of euros) of a specific household is a random variable, which follows the normal distribution with mean μ=5 and standard deviation σ=1. If we take a random sample of 16 households from the population, what is the probability the sampling mean to be:\r<BR>\n" );
document.write( "\n" );
document.write( "	more than 5.5 million euros? <BR>\n" );
document.write( "	between 4.9 and 5.1 million euros? <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #814070 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=814070\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> when you're dealing with the mean of a sample, then  you have to use the standard error.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "standard error = standard deviation / square root of sample size.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "use of an online calculator helps.<BR>\n" );
document.write( "here's one for z-scores.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://davidmlane.com/hyperstat/z_table.html\" target = \"_blank\">https://davidmlane.com/hyperstat/z_table.html</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if you are using the population standard deviation, then use the z-score if the sample size is greater than 30, otherwise use the t-score.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "here's a reference.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<a  rel=nofollow HREF= \"https://math.stackexchange.com/questions/1817980/how-to-know-when-to-use-t-value-or-z-value\" target = \"_blank\">https://math.stackexchange.com/questions/1817980/how-to-know-when-to-use-t-value-or-z-value</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your problem is using the standard deviation from the population, but the sample size is less than 30, so use of the t-score is indicated.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "i'll do both so you can see the difference, but your official answer should be from the t-score unless the problem doesn't care whether you use the t-score or the z-score.<BR>\n" );
document.write( "that you need to get from your professor.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "using the z-score ********************************************\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "z = (x - m) / s\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "z is the z-score<BR>\n" );
document.write( "x is the raw score<BR>\n" );
document.write( "m is the raw mean<BR>\n" );
document.write( "s is the standard error.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "s = standard deviation from the population divided by square root of sample size = 1 / sqrt(16) = 1/4 = .25.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "z-score for 5.5 million euros becomes:<BR>\n" );
document.write( "z = (5.5 - 5) / .25 = 2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you will be looking for the area under the normal distribution curve to the right of a z-score of 2.<BR>\n" );
document.write( "that area will be .0228 rounded to 4 decimal places. *****\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the area to the right of the z-score is the probability that you will get a z-score greater than 2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "here is a graph of the results using the z-score.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2021/081701.jpg\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "z-score for 4.9 million euros becomes:<BR>\n" );
document.write( "z = 4.9 - 5) / .25 = -.4<BR>\n" );
document.write( "z-score for 5.1 million euros becomes:<BR>\n" );
document.write( "z = (5.1 - 5) / .25 = .4\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you will be looking for the area under the normal distribution curve between  z = -.4 and z = .4<BR>\n" );
document.write( "that area will be .31084 rounded to 5 decimal places.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "same standard error is used.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "here is a graph of the results using the z-score.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2021/081702.jpg\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "using the t-score **************************************\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when you are using the t-score, you need to use the degrees of freedom.<BR>\n" );
document.write( "for simple problem, such as this, the degrees of freedom are equal to the sample size minus of 1.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your degrees of freedom are 15.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your calculation of the t-score is the same, i.e......<BR>\n" );
document.write( "t = (x - m) / s\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x is the same.<BR>\n" );
document.write( "m is the same.<BR>\n" );
document.write( "s is the same.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "what will be different is the area to under the normal distribution curve.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "for area to the right of 5.5, the t-score will be 2.<BR>\n" );
document.write( "the area to the right of that t-score with 15 degrees of freedom will be equal to .031972502 = .032 rounded to 3 decimal places, as in the graphical display below.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2021/081703.jpg\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "for area between t-score of -.4 and.4, the area in between will be equal to .3052050642 = .3052 rounded to 4 decimal places, as in the graphical display below.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2021/081704.jpg\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the difference in the area between the z-score and the t-score are greater when the sample size is smaller and gets less as the sample size grows larger.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the z-score calculator can be found at <a  rel=nofollow HREF= \"https://davidmlane.com/hyperstat/z_table.html\" target = \"_blank\">https://davidmlane.com/hyperstat/z_table.html</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the t-score calculator can be found at <a  rel=nofollow HREF= \"https://mathcracker.com/t-distribution-graph-generator#results\" target = \"_blank\">https://mathcracker.com/t-distribution-graph-generator#results</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "for my own calculations, i used the TI-84 plus calculator.<BR>\n" );
document.write( "it provides results to a greater degree of accurac6y (more decimal places).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );