document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1189881>Question 1189881</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The IRS claims that it takes 90 minutes to prepare a tax form.  You believe it takes longer than 90 minutes.  You ask 22 tax filers and find the following preparation times:<BR>\n" );
document.write( "66, 69, 78, 79, 87, 87, 89, 96, 96, 97, 98, 98, 99, 100, 100, 105, 108, 115, 117, 118, 122, 125\r<BR>\n" );
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document.write( "Determine if the preparation time is significantly longer than 90 minutes.  Use  α  = .025.<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 #821408 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=821408\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Hypotheses:<BR>\n" );
document.write( "H0: mu = 90<BR>\n" );
document.write( "H1: mu > 90<BR>\n" );
document.write( "mu = population mean of the tax prep times (in minutes)<BR>\n" );
document.write( "This is a right-tailed test due directly to the inequality sign in H1\r<BR>\n" );
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document.write( "Given Data Set<BR>\n" );
document.write( "{66, 69, 78, 79, 87, 87, 89, 96, 96, 97, 98, 98, 99, 100, 100, 105, 108, 115, 117, 118, 122, 125}<BR>\n" );
document.write( "sample size = n = 22\r<BR>\n" );
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document.write( "Since we don't know sigma, and n is not larger than 30, this means we'll use the T distribution (instead of the Z distribution).<BR>\n" );
document.write( "The T distribution has degrees of freedom = df = n-1 = 22-1 = 21\r<BR>\n" );
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document.write( "Add up the values in the data set and divide by n = 22 to compute the sample mean xbar.<BR>\n" );
document.write( "You should find that the values add to 2149\r<BR>\n" );
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document.write( "So, xbar = 2149/22 = 97.68182 approximately<BR>\n" );
document.write( "Based on this xbar value alone, it appears the mean tax prep time is over 90 minutes. <BR>\n" );
document.write( "However, this might be due to random chance and bad luck. <BR>\n" );
document.write( "Perhaps we could do the experiment again to randomly select 22 other people and get some xbar 90 or smaller?<BR>\n" );
document.write( "We'll have to take into account the standard deviation (and subsequently the standard error).\r<BR>\n" );
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document.write( "Use your calculator to compute the sample standard deviation of that original set. You should get roughly <BR>\n" );
document.write( "s = 16.13746<BR>\n" );
document.write( "Make sure you use the sample standard deviation and <u>not</u> the population standard deviation. There's a subtle difference.\r<BR>\n" );
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document.write( "This leads to a standard error (SE) of...<BR>\n" );
document.write( "SE = s/sqrt(n)<BR>\n" );
document.write( "SE = 16.13746/sqrt(22)<BR>\n" );
document.write( "SE = 3.44052<BR>\n" );
document.write( "which is also approximate\r<BR>\n" );
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document.write( "We found that<BR>\n" );
document.write( "mu = 90<BR>\n" );
document.write( "xbar = 97.68182<BR>\n" );
document.write( "SE = 3.44052\r<BR>\n" );
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document.write( "The test statistic is<BR>\n" );
document.write( "t = (xbar - mu)/SE<BR>\n" );
document.write( "t = (97.68182 - 90)/3.44052<BR>\n" );
document.write( "t = 2.23274970062664<BR>\n" );
document.write( "t = 2.23\r<BR>\n" );
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document.write( "Now to calculate the p-value. <BR>\n" );
document.write( "You can use a T Table such as this one (or anything similar found in the back of your stats textbook)<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"http://www.ttable.org/\">http://www.ttable.org/</a><BR>\n" );
document.write( "Look in the row df = 21<BR>\n" );
document.write( "Try to see if 2.23 is in that row. Unfortunately it isn't. But it's between 2.080 and 2.518<BR>\n" );
document.write( "Trace back up those respective columns and locate the \"one tail\" values in bold.<BR>\n" );
document.write( "For the 2.080, that one tail value is 0.025<BR>\n" );
document.write( "For the 2.518, that one tail value is 0.01\r<BR>\n" );
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document.write( "While we can't nail down the p-value exactly, we know that it's somewhere between 0.01 and 0.025<BR>\n" );
document.write( "In other words,<BR>\n" );
document.write( "0.01 < pvalue < 0.025\r<BR>\n" );
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document.write( "The level of significance was alpha = 0.025<BR>\n" );
document.write( "Whatever the p-value is, it's definitely smaller than alpha. <BR>\n" );
document.write( "We reject the null based on this. \r<BR>\n" );
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document.write( "The rule is if pvalue < alpha, then you reject the null. <BR>\n" );
document.write( "A good way to remember the rule is to think of the phrase \"if the p-value is low, then the null must go\".\r<BR>\n" );
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document.write( "If you were to use a p-value calculator such as this<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.omnicalculator.com/statistics/p-value\">https://www.omnicalculator.com/statistics/p-value</a><BR>\n" );
document.write( "then you'd find the p-value is roughly 0.018395 which helps confirm our decision.\r<BR>\n" );
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document.write( "Here's a screenshot of a piece of that page to show the inputs and output.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" width = \"35%\" src = \"https://i.imgur.com/PQlPVEo.png\">\r<BR>\n" );
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document.write( "If you were to use a t-value calculator such as this<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://calculators.io/t-value-calculator/\">https://calculators.io/t-value-calculator/</a><BR>\n" );
document.write( "Then you should find the T-value (right-tailed) is roughly 2.0796. Anything larger than this critical cut-off point will be in the rejection region. <BR>\n" );
document.write( "Since t = 2.23 is larger than 2.0796, we'll reject the null.\r<BR>\n" );
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document.write( "So there are a few approaches we can take to see that we reject the null hypothesis.<BR>\n" );
document.write( "We side with the alternative that mu > 90.<BR>\n" );
document.write( "It appears that the mean tax prep time is longer than 90 minutes.<BR>\n" );
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