document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1169101>Question 1169101</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You start a new job answering phones, and when you’re hired, your boss tells you<BR>\n" );
document.write( "that you should expect about 4 calls each hour. You notice that you are regularly<BR>\n" );
document.write( "getting more than 8 calls each hour. If the average number of calls per hour<BR>\n" );
document.write( "is 4, what is the probability that you get more than 8 calls in an hour? Do you<BR>\n" );
document.write( "think your boss is being honest with you? </SPAN>\n" );
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document.write( "mu = average number of calls per hour = 4\r<BR>\n" );
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document.write( "We'll use the Poisson Distribution<BR>\n" );
document.write( "The formula is <BR>\n" );
document.write( "P(x) = (mu^x*e^(-mu))/(x!)<BR>\n" );
document.write( "where the exclamation mark indicates factorial. The variable x is any positive whole number.\r<BR>\n" );
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document.write( "So if we wanted to find the probability of getting exactly x = 0 calls per hour, then we would say<BR>\n" );
document.write( "P(x) = (mu^x*e^(-mu))/(x!)<BR>\n" );
document.write( "P(0) = (4^0*e^(-4))/(0!)<BR>\n" );
document.write( "P(0) = 0.0183156388887\r<BR>\n" );
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document.write( "If we wanted the probability of getting 1 call per hour, then,<BR>\n" );
document.write( "P(x) = (mu^x*e^(-mu))/(x!)<BR>\n" );
document.write( "P(1) = (4^1*e^(-4))/(1!)<BR>\n" );
document.write( "P(1) = 0.07326255555483\r<BR>\n" );
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document.write( "We continue this for x = 2, x = 3, all the way up to x = 8<BR>\n" );
document.write( "Once we have all of those P(x) values, we add up the results to get P(X <= 8) which represents the probability of getting at most 8 calls per hour. \r<BR>\n" );
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document.write( "Doing this is a bit tediuous, but you can use the poissonCDF function in your TI83 or TI84 calculator to speed up the process. You can also use an online calculator like this<BR>\n" );
document.write( "https://stattrek.com/online-calculator/poisson.aspx<BR>\n" );
document.write( "to get the job done\r<BR>\n" );
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document.write( "Whichever method you pick you should find that P(X <= 8) is roughly 0.97864, so 1-0.97864 = 0.02136 is the approximate probability of getting more than 8 calls in an hour.\r<BR>\n" );
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document.write( "There's roughly a 2.136% chance of getting more than 8 calls per hour, assuming the claim \"the mean number of calls per hour is 4\" is correct. \r<BR>\n" );
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document.write( "Because this event of getting more than 8 calls per hour happens regularly, I would say that your boss is not being honest; or perhaps they are just using the wrong information. Either way, the average number of calls appears to be greater than 4. \r<BR>\n" );
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document.write( "Side note: if you're doing a hypothesis test with significance level alpha = 0.05, then we would reject the null hypothesis and conclude that the mean is larger than 4. Recall that you always reject the null if the p value is smaller than alpha. The p value is equal to the probability of getting some event, plus any extreme beyond that event. The p value in this case is roughly  0.02136<BR>\n" );
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document.write( " </SPAN>\n" );
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