document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200796>Question 1200796</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A normal population has a mean of $62 and standard deviation of $14. You select random samples of nine.<BR>\n" );
document.write( "a.  What is the probability that a sample mean is greater than $65? (Round z-value to 2 decimal places and final answer to 4 decimal places.)<BR>\n" );
document.write( " What is the probability that a sample mean is less than $58? (Round z-value to 2 decimal places and final answer to 4 decimal places.)<BR>\n" );
document.write( " <BR>\n" );
document.write( " What is the probability that a sample mean is between $58 and $65? (Round z-value to 2 decimal places and final answer to 4 decimal places.)<BR>\n" );
document.write( "What is the probability that the sampling error (  x¯<BR>\n" );
document.write( "  − μ) would be $9 or more? That is, what is the probability that the estimate of the population mean is less than $53 or more than $71? (Round z-value to 2 decimal places and final answer to 4 decimal places.) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #838626 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=838626\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> population mean is 62.<BR>\n" );
document.write( "population standard deviation is 14.<BR>\n" );
document.write( "sample size is 9.<BR>\n" );
document.write( "standard error is standard deviation / sqrt(sample size) = 14 / sqrt(9) = 4.66667.<BR>\n" );
document.write( "round z-score to 2 decimal places.<BR>\n" );
document.write( "round probability to 4 decimal places.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "formula to use is z = (x-m)/s<BR>\n" );
document.write( "z is the z-score<BR>\n" );
document.write( "x is the sample mean<BR>\n" );
document.write( "m is the population mean<BR>\n" );
document.write( "s is the standard error that was derived from the population standard deviation divided by the square root of the sample size = 4.66667 as shown above.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "a. What is the probability that a sample mean is greater than $65? (Round z-value to 2 decimal places and final answer to 4 decimal places.)<BR>\n" );
document.write( "What is the probability that a sample mean is less than $58? (Round z-value to 2 decimal places and final answer to 4 decimal places.)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "formula becomes z = (58 - 62) / 4.66667 = -4 / 4.66667 = -.86.<BR>\n" );
document.write( "area to the left of that = .1949 *****<BR>\n" );
document.write( "that's the probbility of getting a z-score less than -.86, which is the same as getting a sample mean less than 58.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "What is the probability that a sample mean is between $58 and $65? (Round z-value to 2 decimal places and final answer to 4 decimal places.)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "lower limit z = (58 - 62) / 4.66667 = -.86<BR>\n" );
document.write( "high limit z = (65 - 62) / 4.66667 = .64<BR>\n" );
document.write( "area to the left of z = -.86 = .1949<BR>\n" );
document.write( "area to the left of z = .64 = .7389<BR>\n" );
document.write( "area in between = .7389 minus .1949 = .5440 *****<BR>\n" );
document.write( "that's the probability of getting a z-score between -.86 and .64, whiich is the same as getting a sample mean between 58 and 65.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "What is the probability that the sampling error ( x¯<BR>\n" );
document.write( "− μ) would be $9 or more? That is, what is the probability that the estimate of the population mean is less than $53 or more than $71? (Round z-value to 2 decimal places and final answer to 4 decimal places.)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "lower z-score = (53 - 62) / 4.66667 = -1.93<BR>\n" );
document.write( "higher z-score = (71 - 62) / 4.66667 = 1.93\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "area to the left of z = -1.93 = .0268<BR>\n" );
document.write( "area to the right of z = 1.93 = .0268<BR>\n" );
document.write( "the probability of either getting a z-score less than -1.93 or greater than 1.93 = .0268 + .0268 = .0536. *****<BR>\n" );
document.write( "that's the same as getting a sample mean less than 53 or greater than 71.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "i confirmd these results through the use of the calculator at <a  rel=nofollow HREF= \"https://www.hackmath.net/en/calculator/normal-distribution?mean=0&sd=1&above=&below=&ll=-0.86&ul=0.64&area=outside&outsideLL=-1.93&outsideUL=1.93&draw=Calculate\" target = \"_blank\">https://www.hackmath.net/en/calculator/normal-distribution?mean=0&sd=1&above=&below=&ll=-0.86&ul=0.64&area=outside&outsideLL=-1.93&outsideUL=1.93&draw=Calculate</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "here are the results from using the clculator.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2023/080804.jpg\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2023/080805.jpg\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2023/080806.jpg\">\r<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" );