document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204966>Question 1204966</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Business Weekly conducted a survey of graduates from 30 top MBA programs. On the basis of the survey, assume the mean annual salary for graduates 10 years after graduation is 169000 dollars. Assume the standard deviation is 36000 dollars. Suppose you take a simple random sample of 91 graduates. Assume the population is normally distributed.\r<BR>\n" );
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document.write( "Note: You should carefully round any intermediate values you calculate to 4 decimal places to match wamap's approach and calculations.\r<BR>\n" );
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document.write( "Find the probability that a single randomly selected salary exceeds 166000 dollars. <BR>\n" );
document.write( "P(X > 166000) = 0.2023<BR>\n" );
document.write( " Enter your answer as a number accurate to 4 decimal places.\r<BR>\n" );
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document.write( "Find the probability that a sample of size 91 is randomly selected with a mean that exceeds 166000 dollars.<BR>\n" );
document.write( "P(x > 166000) = <BR>\n" );
document.write( " Enter your answer as a number accurate to 4 decimal places.\r<BR>\n" );
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document.write( "I did the first part but can't find P(xbar > 166000) with sample of 91<BR>\n" );
document.write( " part but can not figure out the bottom because when I do it I get a number that does not / can not exist for this problem. please help<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 #841528 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=841528\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part 1\r<BR>\n" );
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document.write( "mu = 169000 = mean<BR>\n" );
document.write( "sigma = 36000 = population standard deviation.<BR>\n" );
document.write( "x = 166000\r<BR>\n" );
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document.write( "z = (x - mu)/sigma<BR>\n" );
document.write( "z = (166000 - 169000)/36000<BR>\n" );
document.write( "z =  -0.08333333333333 approximately<BR>\n" );
document.write( "z = -0.08\r<BR>\n" );
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document.write( "Then use <a  rel=nofollow HREF=\"https://www.ztable.net/\">Z table</a> to find that P(z < -0.08) = 0.46812 approximately.\r<BR>\n" );
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document.write( "Such tables should be found at the back of your stats textbook. <BR>\n" );
document.write( "For exams, the teacher will likely hand out the table.\r<BR>\n" );
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document.write( "Then,<BR>\n" );
document.write( "P(Z > -0.08) = 1-P(Z < -0.08)<BR>\n" );
document.write( "P(Z > -0.08) = 1-0.46812<BR>\n" );
document.write( "P(Z > -0.08) = 0.53188\r<BR>\n" );
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document.write( "This leads back to P(x > 166000) = 0.53188 approximately<BR>\n" );
document.write( "Rounding to four decimal places gets us 0.5319\r<BR>\n" );
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document.write( "I'm not familiar with wamap, so I don't have access to that particular stats calculator.\r<BR>\n" );
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document.write( "However, there are many other available calculators out there as alternatives.\r<BR>\n" );
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document.write( "For example, <a  rel=nofollow HREF=\"https://www.statology.org/normal-probabilities-ti-84-calculator/\">using the TI83 or TI84</a> gives<BR>\n" );
document.write( "<font color=red>normalCDF(-0.08, 99, 0, 1)</font> = 0.531 881 440 4<BR>\n" );
document.write( "I put a space between each group of 3 decimal digits to make the number a bit more readable. <BR>\n" );
document.write( "That value rounds to 0.5319 when rounding to four decimal places.\r<BR>\n" );
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document.write( "Or another TI83 calculation we could do is<BR>\n" );
document.write( "<font color=red>normalCDF(-0.0833, 99, 0, 1)</font> = 0.533 205 493 9<BR>\n" );
document.write( "That value rounds to 0.5332 when rounding to four decimal places.\r<BR>\n" );
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document.write( "Or another TI83 calculation we could do is<BR>\n" );
document.write( "<font color=red>normalCDF(-0.083333, 99, 0, 1)</font> = 0.533 206 686 5<BR>\n" );
document.write( "That value rounds to 0.5332 when rounding to four decimal places.\r<BR>\n" );
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document.write( "It appears that the results steadily approach the value 0.5332 as the z score gets closer to -0.0833333 where the '3's go on forever.\r<BR>\n" );
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document.write( "I'm not sure how you came up with 0.2023<BR>\n" );
document.write( "Please let me know your scratch work and thought process. Thank you.\r<BR>\n" );
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document.write( "Here are some alternative calculators (for future students reading this question who do not have a TI83/TI84 and also don't have access to wamap)<ul><li>A <a  rel=nofollow HREF=\"https://davidmlane.com/hyperstat/z_table.html\">very user friendly calculator</a> by professor David M Lane. The calculator also displays the shaded diagram which is a nice bonus.</li><li>Use <a  rel=nofollow HREF=\"https://www.wolframalpha.com/\">WolframAlpha</a>. Type in <font color=red>P(Z > -0.0833)</font> for instance. Make sure the \"referring to statistics\" option is selected.</li><li>Use <a  rel=nofollow HREF=\"https://www.geogebra.org/?lang=en\">GeoGebra</a>. It can be accessed through either the normal input bar, the CAS mode, or the probability distribution mode. The reference page can be found <a  rel=nofollow HREF=\"https://wiki.geogebra.org/en/Normal_Command\">here</a></li><li>Use the <font color=red>normDist</font> function on a spreadsheet. See this <a  rel=nofollow HREF=\"https://support.microsoft.com/en-us/office/normdist-function-126db625-c53e-4591-9a22-c9ff422d6d58\">reference page</a> for more info.</li></ul>There are many other alternatives that I haven't listed. You can do an internet search to find your favorite one.<BR>\n" );
document.write( "Of all of the choices, the 1st option (the David M Lane one) is probably the best for new students. <BR>\n" );
document.write( "The only drawback is there doesn't appear to be a way to change the precision.\r<BR>\n" );
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document.write( "Part 2\r<BR>\n" );
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document.write( "mu = 169000 = mean<BR>\n" );
document.write( "sigma = 36000 = population standard deviation.<BR>\n" );
document.write( "xbar = 166000 = sample mean<BR>\n" );
document.write( "n = 91 = sample size\r<BR>\n" );
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document.write( "xbar means we have x with a horizontal line over top\r<BR>\n" );
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document.write( "When working in the xbar distribution, aka sample mean distribution, we have this z score conversion formula:<BR>\n" );
document.write( "z = (xbar - mu)/( sigma/sqrt(n) )\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "z = (xbar - mu)/( sigma/sqrt(n) )<BR>\n" );
document.write( "z = (166000 - 169000)/( 36000/sqrt(91) )<BR>\n" );
document.write( "z = -0.79494933451412<BR>\n" );
document.write( "z = -0.79<BR>\n" );
document.write( "The result is approximate.\r<BR>\n" );
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document.write( "Use a table to find that<BR>\n" );
document.write( "P(Z < -0.79) = 0.21476<BR>\n" );
document.write( "which leads to<BR>\n" );
document.write( "P(Z > -0.79) = 1-P(Z < -0.79)<BR>\n" );
document.write( "P(Z > -0.79) = 1-0.21476<BR>\n" );
document.write( "P(Z > -0.79) = 0.78524<BR>\n" );
document.write( "P(Z > -0.79) = 0.7852<BR>\n" );
document.write( "and ultimately leads back to<BR>\n" );
document.write( "P(xbar > 166000) = 0.7852\r<BR>\n" );
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document.write( "If using a TI83, then <BR>\n" );
document.write( "<font color=red>normalCDF(-0.7949, 99, 0, 1)</font> = 0.786 664 228 9<BR>\n" );
document.write( "and<BR>\n" );
document.write( "<font color=red>normalCDF(-0.7949493, 99, 0, 1)</font> = 0.786 678 568 6<BR>\n" );
document.write( "Play around with the precision and you'll see the more accurate the input, the results tend to approach 0.7867<BR>\n" );
document.write( "It's not too far from the table result 0.7852<BR>\n" );
document.write( "Similar calculator results should happen with other calculators.<BR>\n" );
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