document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1119447>Question 1119447</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A distribution of values is normal with a mean of 210 and a standard deviation of 23. From this distribution, you are drawing samples of size 33.\r<BR>\n" );
document.write( "\n" );
document.write( "Find the interval containing the middle-most 94% of sample means: </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #735012 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=735012\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> mean is 210<BR>\n" );
document.write( "standard deviation is 23<BR>\n" );
document.write( "sample size is 33.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "standard error = standard deviation divided by square root of sample size = sqrt(23/33) = .8348471099.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you want the middle 94% of the distribution of sample means.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that would be .94 of the area under the normal distribution curve.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1 - .94 = .6 area that is split between the lower end and the upper end.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that makes .03 area on the left of the confidence interval and .03 area on the right of the confidence interval.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "an area of .03 to the left of the confidence interval is associated with a z-score of -1.88079361.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since the normal distribution curve is symmetric about the mean, then your confidence interval is between a z-score of plus or minus 1.88079361.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "to find the raw score associated with that, use the z-score formula of 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( "you get 1.88079361 = (x-210)/.8348471099 and you get -1.88079361 = (x-210)/.8348471099.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "solve for x to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x = .8348471099 * 1.88079361 + 210 and you get x = .8348471099 * -1.88079361 + 210.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this results in a confidence interval that is 94% of the area under the normal distribution curve that goes between the raw score of 208.4298249 to 211.5701751.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this can be seen in the following z-score calculator output.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the first tells you the critical z-score for an area of .94 in the middle of the normal distribution curve.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the second tells you the critical raw score for an area of.94 in the middle of the normal distribution curve.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the first has a mean of 0 and a standard deviation of 1 which is the mean and standard deviation for the z-score.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the second has a mean of 210 and a standard deviation (actually a standard error) of .8348471099.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "there is some rounding of the inputs when using this calculator.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this is more than enough for most applications, especially when the final answer is normally rounded to 3 decimal digits.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "here's the display of the calculations.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/070503.jpg\" alt=\"$$$\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/070504.jpg\" alt=\"$$$\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<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" );