document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204309>Question 1204309</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the indicated probability given that Z is a random variable with a standard normal distribution. (Round your answer to four decimal places.)<BR>\n" );
document.write( "P(0 ≤ Z ≤ 1.23)<BR>\n" );
document.write( "P(0 ≤ Z ≤ 1.23) =  </SPAN>\n" );
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document.write( "Answer: <font color=red size=4>0.3907</font>\r<BR>\n" );
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document.write( "Explanation\r<BR>\n" );
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document.write( "There are a few methods we could use.\r<BR>\n" );
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document.write( "Method 1 is to use a Z table such as this<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.ztable.net/\">https://www.ztable.net/</a><BR>\n" );
document.write( "to find that<BR>\n" );
document.write( "P(Z < 0.00) = 0.5<BR>\n" );
document.write( "P(Z < 1.23) = 0.89065\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "P(a < Z < b) = P(Z < b) - P(Z < a)<BR>\n" );
document.write( "P(0 < Z < 1.23) = P(Z < 1.23) - P(Z < 0)<BR>\n" );
document.write( "P(0 < Z < 1.23) = 0.89065 - 0.5<BR>\n" );
document.write( "P(0 < Z < 1.23) = 0.39065<BR>\n" );
document.write( "P(0 < Z < 1.23) = <font color=red>0.3907</font>\r<BR>\n" );
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document.write( "Method 2 involves a calculator such as this one<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://davidmlane.com/normal.html\">https://davidmlane.com/normal.html</a><BR>\n" );
document.write( "The mean and standard deviation are 0 and 1 respectively.<BR>\n" );
document.write( "Click the \"between\" and fill in the values 0 and 1.23 to have <font color=red>0.3907</font> show up.\r<BR>\n" );
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document.write( "Another calculator you can use is a TI83 or TI84.<BR>\n" );
document.write( "Refer to this article.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.statology.org/normal-probabilities-ti-84-calculator/\">https://www.statology.org/normal-probabilities-ti-84-calculator/</a><BR>\n" );
document.write( "Use the normalCDF function to get the area under the curve.<BR>\n" );
document.write( "The input that you'll type in is:  <font color=red>normalCDF(0,1.23)</font><BR>\n" );
document.write( "Or optionally you could type in <font color=red>normalCDF(0,1.23,0,1)</font><BR>\n" );
document.write( "The TI84 should display the approximate result 0.3906513828 which rounds to <font color=red>0.3907</font>\r<BR>\n" );
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document.write( "Or you could use a spreadsheet.<BR>\n" );
document.write( "The spreadsheet function you'll use is called <font color=red>normDist</font><BR>\n" );
document.write( "Refer to this page for more info<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.statology.org/normal-distribution-probability-excel/\">https://www.statology.org/normal-distribution-probability-excel/</a>\r<BR>\n" );
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document.write( "Yet another method is WolframAlpha<BR>\n" );
document.write( "Type in <font color=red>P(0 < z < 1.23)</font> and you should get roughly 0.390651 which rounds to the answer shown above.<BR>\n" );
document.write( "Caution: WolframAlpha is a great resource, but it has a glaring flaw. The graph shown on the page is not correct. The normal distribution curve should be entirely above the x axis. I don't know why some parts of it are below the x axis. I would hope the people running that website would fix the error soon. Rely on the diagram produced by the davidmlane website.\r<BR>\n" );
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document.write( "As you can see, there are a lot of ways to calculate the normal distribution probabilities. <BR>\n" );
document.write( "Feel free to explore other options.<BR>\n" );
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