document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201653>Question 1201653</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The life expectancy of light bulbs whose lifetimes are normally distributed with a mean life of 750 hours and with a standard deviation of 80 hours. Show or explain how you determine the appropriate z-score and related percentage. \r<BR>\n" );
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document.write( "What percent of light bulbs will last longer than 870 hours? \r<BR>\n" );
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document.write( "What percent of light bulbs will last between 730 hours and 850 hours? \r<BR>\n" );
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document.write( "What percent of light bulbs will last less than 770 hours? \r<BR>\n" );
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document.write( "Inverse Normal Distribution: If the quality control program of the company can consistently eliminate the worst 10% of the bulbs manufactured, the manufacturer can safely offer customers a money-back guarantee on all lights that fail before [_____] hours of burning time.\r<BR>\n" );
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document.write( "Thank you! </SPAN>\n" );
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document.write( "<font color=red>Answers:</font><ol><li>percentage = 6.681%</li><li>percentage = 49.306%</li><li>percentage = 59.871%</li><li>lifespan = 647 hours</li></ol>Each value is approximate. <BR>\n" );
document.write( "Round each value according to the instructions your teacher provides.\r<BR>\n" );
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document.write( "Explanations:\r<BR>\n" );
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document.write( "Problem 1\r<BR>\n" );
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document.write( "x = life span measured in hours<BR>\n" );
document.write( "This is some positive real number.\r<BR>\n" );
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document.write( "The goal is to find P(x > 870) which computes the probability of getting a random light bulb that has a life span longer than 870 hours. <BR>\n" );
document.write( "This is equivalent to finding the percentage of light bulbs that last longer than 870 hours. \r<BR>\n" );
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document.write( "mu = 750 = population mean lifetime<BR>\n" );
document.write( "sigma = 80 = population standard deviation of the lifetimes\r<BR>\n" );
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document.write( "Find the z score when x = 870<BR>\n" );
document.write( "z = (x-mu)/sigma<BR>\n" );
document.write( "z = (870-750)/80<BR>\n" );
document.write( "z = 120/80<BR>\n" );
document.write( "z = 1.50\r<BR>\n" );
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document.write( "The task of finding P(x > 870) is equivalent to P(z > 1.50) for the mu and sigma values mentioned.\r<BR>\n" );
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document.write( "I'll be using this Z table<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.ztable.net/\">https://www.ztable.net/</a><BR>\n" );
document.write( "A similar table can be found in the back of your stats textbook.<BR>\n" );
document.write( "On that link, scroll down the page to see a few examples how to read that table.\r<BR>\n" );
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document.write( "Use such a table to find that<BR>\n" );
document.write( "P(z < 1.50) = 0.93319<BR>\n" );
document.write( "which means<BR>\n" );
document.write( "P(z > 1.50) = 1-P(z < 1.50)<BR>\n" );
document.write( "P(z > 1.50) = 1-0.93319<BR>\n" );
document.write( "P(z > 1.50) = 0.06681<BR>\n" );
document.write( "That leads back to<BR>\n" );
document.write( "P(x > 870) = 0.06681<BR>\n" );
document.write( "This value is approximate. <BR>\n" );
document.write( "The same will apply to nearly every other decimal value I mention, with the exception of something like a z score of z = 1.50 (that decimal value is exact).\r<BR>\n" );
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document.write( "Roughly <font color=red>6.681%</font> of the light bulbs will last longer than 870 hours.\r<BR>\n" );
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document.write( "You could use a specialized stats 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( "as an alternative to using a Z table.\r<BR>\n" );
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document.write( "This article goes over a few examples of how to calculate normal distribution probabilities on a TI84 calculator.<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>\r<BR>\n" );
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document.write( "Problem 2\r<BR>\n" );
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document.write( "mu = 750<BR>\n" );
document.write( "sigma = 80\r<BR>\n" );
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document.write( "Compute the z score for x = 730<BR>\n" );
document.write( "z = (x-mu)/sigma<BR>\n" );
document.write( "z = (730-750)/80<BR>\n" );
document.write( "z = -20/80<BR>\n" );
document.write( "z = -0.25\r<BR>\n" );
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document.write( "Repeat for x = 850<BR>\n" );
document.write( "z = (x-mu)/sigma<BR>\n" );
document.write( "z = (850-750)/80<BR>\n" );
document.write( "z = 100/80<BR>\n" );
document.write( "z = 1.25\r<BR>\n" );
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document.write( "The task of finding P(730 < x < 850) is equivalent to P(-0.25 < z < 1.25)\r<BR>\n" );
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document.write( "Use the Z table to find<BR>\n" );
document.write( "P(Z < -0.25) = 0.40129<BR>\n" );
document.write( "and<BR>\n" );
document.write( "P(Z < 1.25) = 0.89435\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(-0.25 < Z < 1.25) = P(Z < 1.25) - P(Z < -0.25)<BR>\n" );
document.write( "P(-0.25 < Z < 1.25) = 0.89435 - 0.40129<BR>\n" );
document.write( "P(-0.25 < Z < 1.25) = 0.49306\r<BR>\n" );
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document.write( "This leads back to<BR>\n" );
document.write( "P(730 < x < 850) = 0.49306\r<BR>\n" );
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document.write( "Roughly <font color=red>49.306%</font> of the light bulbs will last between 730 hours and 850 hours.\r<BR>\n" );
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document.write( "Problem 3\r<BR>\n" );
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document.write( "mu = 750<BR>\n" );
document.write( "sigma = 80\r<BR>\n" );
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document.write( "Compute the z score for x = 770<BR>\n" );
document.write( "z = (x-mu)/sigma<BR>\n" );
document.write( "z = (770-750)/80<BR>\n" );
document.write( "z = 20/80<BR>\n" );
document.write( "z = 0.25\r<BR>\n" );
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document.write( "The task of finding P(x < 770) is equivalent to P(z < 0.25)\r<BR>\n" );
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document.write( "The Z table says that<BR>\n" );
document.write( "P(Z < 0.25) = 0.59871<BR>\n" );
document.write( "so,<BR>\n" );
document.write( "P(x < 770) = 0.59871\r<BR>\n" );
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document.write( "Therefore, roughly <font color=red>59.871%</font> of the light bulbs will last less than 770 hours.\r<BR>\n" );
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document.write( "Problem 4\r<BR>\n" );
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document.write( "The worst 10% is the lowest 10% of life spans.\r<BR>\n" );
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document.write( "We need to find a value of k such that<BR>\n" );
document.write( "P(Z < k) = 0.10<BR>\n" );
document.write( "This value of k is the marker that separates the bottom 10% from the top 90%.\r<BR>\n" );
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document.write( "Use 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( "or a TI84<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.statology.org/inverse-normal-distribution/\">https://www.statology.org/inverse-normal-distribution/</a>\r<BR>\n" );
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document.write( "If you use the first link, then click on the \"value from an area\" option.<BR>\n" );
document.write( "From there the instructions should be pretty straight forward. Let me know if you have questions.\r<BR>\n" );
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document.write( "You should find that P(Z < -1.282) = 0.10 approximately.\r<BR>\n" );
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document.write( "Convert the z score z = -1.282 into its equivalent raw x score.<BR>\n" );
document.write( "mu = 750<BR>\n" );
document.write( "sigma = 80<BR>\n" );
document.write( "z = (x-mu)/sigma<BR>\n" );
document.write( "-1.282 = (x-750)/80<BR>\n" );
document.write( "-1.282*80 = x-750<BR>\n" );
document.write( "-102.56 = x-750<BR>\n" );
document.write( "x = -102.56+750<BR>\n" );
document.write( "x = 647.44\r<BR>\n" );
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document.write( "Let's say we round that to the nearest whole number to get <font color=red>647</font><BR>\n" );
document.write( "We found that P(x < 647) = 0.10 approximately.<BR>\n" );
document.write( "About 10% of the light bulbs last less than 647 hours.<BR>\n" );
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