document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201601>Question 1201601</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 5. The distribution of survival time of guinea pigs with cancer after treatment is typically strongly right-skewed. Assume that the mean of the guinea pig survival time for a population of guinea pigs with cancer is 100 days with a standard deviation of 10 days. Suppose we take a sample of size n=49 guinea pigs from the population of guinea pigs with cancer.\r<BR>\n" );
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document.write( "(a) Explain why even though the population in this case is known to be skewed right we can model the sample mean using a Normal distribution.\r<BR>\n" );
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document.write( "(b) What is the approximate probability that the sample mean survival time will be less than 95 days?\r<BR>\n" );
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document.write( "(c) Determine the required sample size such that the probability that the sample mean will be within 2 days of the population mean survival time is 95%. </SPAN>\n" );
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document.write( "<font color=red>Answers:</font><BR>\n" );
document.write( "(a) See explanation below.<BR>\n" );
document.write( "(b) 0.00023 (approximate)<BR>\n" );
document.write( "(c) n = 96 (approximate)\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "Part(a)\r<BR>\n" );
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document.write( "Right-skewed means the main cluster of individuals are to the left while there are a few large outliers to the right (thereby forming a long stretched tail to the right).<BR>\n" );
document.write( "In this context, it means that the vast majority of guinea pigs with cancer unfortunately do not survive too long after treatment. A few lucky individuals that are large outliers do survive for quite a while longer.<BR>\n" );
document.write( "Another term for \"right-skewed\" is \"positively skewed\".\r<BR>\n" );
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document.write( "n = 49 = sample size<BR>\n" );
document.write( "Because n > 30 is the case, the xbar distribution of sample means will be approximately normally distributed even if the population distribution isn't symmetric.<BR>\n" );
document.write( "For more information, check out the Central Limit Theorem.\r<BR>\n" );
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document.write( "See this article for more detail<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.statology.org/central-limit-theorem/\">https://www.statology.org/central-limit-theorem/</a><BR>\n" );
document.write( "Quote from the page: <BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\"If the population distribution is skewed, generally a sample size of at least 30 is needed.\r\n" );
document.write( "If the population distribution is extremely skewed, then a sample size of 40 or higher may be necessary\"</PRE>\r<BR>\n" );
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document.write( "This article<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.whatissixsigma.net/confidence-intervals-why-n30-is-acceptable-as-population-representative/\">https://www.whatissixsigma.net/confidence-intervals-why-n30-is-acceptable-as-population-representative/</a><BR>\n" );
document.write( "talks about why the number 30 is so special when it comes to the n > 30 criteria.\r<BR>\n" );
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document.write( "Part(b)\r<BR>\n" );
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document.write( "mu = 100 = population mean survival time<BR>\n" );
document.write( "sigma = 10 = population standard deviation of the survival times (tells us how spread out the population is)<BR>\n" );
document.write( "n = 49 = sample size<BR>\n" );
document.write( "xbar = sample mean survival time<BR>\n" );
document.write( "Each survival time is measured in days.\r<BR>\n" );
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document.write( "We wish to compute <BR>\n" );
document.write( "P(xbar < 95)\r<BR>\n" );
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document.write( "Let's find the standard error<BR>\n" );
document.write( "SE = Standard error<BR>\n" );
document.write( "SE = sigma/sqrt(n)<BR>\n" );
document.write( "SE = 10/sqrt(49)<BR>\n" );
document.write( "SE = 1.428571 which is approximate\r<BR>\n" );
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document.write( "Now determine the z score when xbar = 95<BR>\n" );
document.write( "z = (xbar - mu)/SE<BR>\n" );
document.write( "z = (95 - 100)/1.428571<BR>\n" );
document.write( "z = -3.50000105000031<BR>\n" );
document.write( "z = -3.50<BR>\n" );
document.write( "The task of finding P(xbar < 95) is equivalent to P(Z < -3.50)\r<BR>\n" );
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document.write( "I'm rounding to two decimal places so I can then use a table such as this one<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.\r<BR>\n" );
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document.write( "Use a table to find that:<BR>\n" );
document.write( "P(Z < -3.50) = 0.00023<BR>\n" );
document.write( "this leads back to<BR>\n" );
document.write( "P(xbar < 95) = 0.00023\r<BR>\n" );
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document.write( "There is roughly a 0.023% chance of getting a sample mean survival time of less than 95 days.\r<BR>\n" );
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document.write( "Part(c)\r<BR>\n" );
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document.write( "mu = 100 = population mean survival time<BR>\n" );
document.write( "Add and subtract off 2 from the population mean<BR>\n" );
document.write( "mu-2 = 100-2 = 98<BR>\n" );
document.write( "mu+2 = 100+2 = 102\r<BR>\n" );
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document.write( "The goal is to find a sample size n that will lead to this statement<BR>\n" );
document.write( "P(98 < xbar < 102) = 0.95\r<BR>\n" );
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document.write( "Let's find the standard error<BR>\n" );
document.write( "SE = Standard error<BR>\n" );
document.write( "SE = sigma/sqrt(n)<BR>\n" );
document.write( "SE = 10/sqrt(n)<BR>\n" );
document.write( "Right now we don't know what n is, so we leave the SE as shown above.\r<BR>\n" );
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document.write( "Now determine the z score when xbar = 98<BR>\n" );
document.write( "z = (xbar - mu)/SE<BR>\n" );
document.write( "z = (98 - 100)/(10/sqrt(n))<BR>\n" );
document.write( "z = -2/(10/sqrt(n))<BR>\n" );
document.write( "z = -2*sqrt(n)/10<BR>\n" );
document.write( "z = -sqrt(n)/5<BR>\n" );
document.write( "Repeat for xbar = 102 and you should get: z = sqrt(n)/5\r<BR>\n" );
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document.write( "P(98 < xbar < 102) <BR>\n" );
document.write( "is the same as <BR>\n" );
document.write( "P(-sqrt(n)/5 < Z < sqrt(n)/5)<BR>\n" );
document.write( "which is in the format<BR>\n" );
document.write( "P(-k < Z < k)\r<BR>\n" );
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document.write( "We need to find a value of k such that<BR>\n" );
document.write( "P(-k < Z < k) = 0.95\r<BR>\n" );
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document.write( "You could use the table in reverse but it would take a lot of trial and error.<BR>\n" );
document.write( "A better approach is to 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 you could use a TI84 (or similar).\r<BR>\n" );
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document.write( "You should find that k = 1.96 approximately<BR>\n" );
document.write( "P(-1.96 < Z < 1.96) = 0.95 approximately\r<BR>\n" );
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document.write( "We can now determine n.<BR>\n" );
document.write( "k = 1.96<BR>\n" );
document.write( "sqrt(n)/5 = 1.96<BR>\n" );
document.write( "sqrt(n) = 5*1.96<BR>\n" );
document.write( "sqrt(n) = 9.8<BR>\n" );
document.write( "n = (9.8)^2<BR>\n" );
document.write( "n = 96.04<BR>\n" );
document.write( "n = 96\r<BR>\n" );
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document.write( "If n = 96, then the SE is<BR>\n" );
document.write( "SE = Standard error<BR>\n" );
document.write( "SE = sigma/sqrt(n)<BR>\n" );
document.write( "SE = 10/sqrt(96)<BR>\n" );
document.write( "SE = 1.020621 which is approximate\r<BR>\n" );
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document.write( "Now determine the z score when xbar = 98<BR>\n" );
document.write( "z = (xbar - mu)/SE<BR>\n" );
document.write( "z = (98 - 100)/1.020621<BR>\n" );
document.write( "z = -1.95959126845322<BR>\n" );
document.write( "z = -1.96\r<BR>\n" );
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document.write( "Repeat for xbar = 102<BR>\n" );
document.write( "z = (xbar - mu)/SE<BR>\n" );
document.write( "z = (102 - 100)/1.020621<BR>\n" );
document.write( "z = 1.95959126845322<BR>\n" );
document.write( "z = 1.96\r<BR>\n" );
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document.write( "Then you should find that P(-1.96 < z < 1.96) = 0.95 approximately. <BR>\n" );
document.write( "Use the table or a calculator. <BR>\n" );
document.write( "This helps confirm we have the correct value of n.\r<BR>\n" );
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document.write( "Conclusion: <BR>\n" );
document.write( "Having a sample size of n = 96 will lead to P(98 < xbar < 102) = 0.95 approximately.<BR>\n" );
document.write( "Sampling 96 guinea pigs will have their sample mean survival time between 98 and 102 days with 95% probability.<BR>\n" );
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