document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193902>Question 1193902</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A publisher wants to estimate the mean length of time (in minutes) all adults spend reading newspapers. To determine this estimate, the publisher takes a random sample of 15 people and obtains the results below. From past studies, the publisher assumes that the population of times is normally distributed. Construct the 95% confidence interval for the population mean. \r<BR>\n" );
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document.write( "12    8    11    10    11    6    6    9    6    11    11    11    7    7    9 <BR>\n" );
document.write( "The sample mean is   <BR>\n" );
document.write( "The sample standard deviation rounded to 2 decimal places is   <BR>\n" );
document.write( "The number of degrees of freedom is   <BR>\n" );
document.write( "According to the t-distribution table in the formula sheets t c rounded to 3 decimal places is   <BR>\n" );
document.write( "The margin of error E rounded to two decimal places is   <BR>\n" );
document.write( "The confidence interval is   < 𝜇 <  </SPAN>\n" );
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document.write( "There are n = 15 values in the list, which is the sample size.\r<BR>\n" );
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document.write( "To get the sample mean, first we add up the values<BR>\n" );
document.write( "12+8+11+10+11+6+6+9+6+11+11+11+7+7+9 = 135<BR>\n" );
document.write( "Then divide this over the sample size (n = 15) to get<BR>\n" );
document.write( "135/15 = 9<BR>\n" );
document.write( "The sample mean is 9<BR>\n" );
document.write( "We call this xbar because x has a horizontal bar over top.<BR>\n" );
document.write( "xbar = 9\r<BR>\n" );
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document.write( "You could calculate the sample standard deviation by hand, but it's preferable to use technology instead. There are many free calculators online to help with that if you don't have a TI83/84.<BR>\n" );
document.write( "You should get a sample standard deviation of roughly s = 2.17<BR>\n" );
document.write( "The sample standard deviation s estimates the population standard deviation sigma. <BR>\n" );
document.write( "s = sample standard deviation = 2.17 approximately<BR>\n" );
document.write( "sigma = population standard deviation = unknown\r<BR>\n" );
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document.write( "Because we don't know sigma and because n > 30 is not true, we must use a T distribution. <BR>\n" );
document.write( "The degrees of freedom (df) is equal to the sample size minus 1<BR>\n" );
document.write( "df = n-1<BR>\n" );
document.write( "df = 15-1<BR>\n" );
document.write( "df = 14\r<BR>\n" );
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document.write( "Use the T table in the back of your textbook, or use a free online resource like this<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf\">https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf</a><BR>\n" );
document.write( "Start by looking at the row labeled \"two tails\". The value 0.05 corresponds to a confidence interval of 95% since 1 - 0.05 = 0.95<BR>\n" );
document.write( "The 0.05 refers to the combined area of both tails.<BR>\n" );
document.write( "Highlight this entire column.<BR>\n" );
document.write( "Then look in the row that has df = 14 at the very left.<BR>\n" );
document.write( "The intersection of the row and column leads to the approximate t critical value of 2.145<BR>\n" );
document.write( "What is this value saying? It says that P(-2.145 < T < 2.145) = 0.95 approximately when df = 14.<BR>\n" );
document.write( "95% of the area under the T curve (df = 14) is between roughly t = -2.145 and t = 2.145\r<BR>\n" );
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document.write( "Let's compute the margin of error<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E+=+t%5Bc%5D%2Aexpr%28s%2Fsqrt%28n%29%29 ALIGN=MIDDLE  ALT=\"E+=+t%5Bc%5D%2Aexpr%28s%2Fsqrt%28n%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E+=++2.145%2Aexpr%282.17%2Fsqrt%2815%29%29 ALIGN=MIDDLE  ALT=\"E+=++2.145%2Aexpr%282.17%2Fsqrt%2815%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E+=++1.20182546216162 ALIGN=MIDDLE  ALT=\"E+=++1.20182546216162\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E+=++1.20 ALIGN=MIDDLE  ALT=\"E+=++1.20\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "And lastly, let's compute the confidence interval for the mean mu (symbol 𝜇)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xbar+-+E+%3C+mu+%3C+xbar+%2B+E ALIGN=MIDDLE  ALT=\"xbar+-+E+%3C+mu+%3C+xbar+%2B+E\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9+-+1.20+%3C+mu+%3C+9+%2B+1.20 ALIGN=MIDDLE  ALT=\"9+-+1.20+%3C+mu+%3C+9+%2B+1.20\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7.80+%3C+mu+%3C+10.20 ALIGN=MIDDLE  ALT=\"7.80+%3C+mu+%3C+10.20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We could shorten this to (7.80, 10.20) which is common notation for confidence intervals. <BR>\n" );
document.write( "Some books and research papers use the notation [7.80, 10.20] to mean the same thing. <BR>\n" );
document.write( "We are 95% confident the true population mean (mu) is somewhere between 7.80 and 10.20\r<BR>\n" );
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document.write( "Answers:<BR>\n" );
document.write( "Sample mean = 9<BR>\n" );
document.write( "Sample standard deviation = 2.17<BR>\n" );
document.write( "Degrees of Freedom = 14<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5Bc%5D ALIGN=MIDDLE  ALT=\"t%5Bc%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > (t critical value) is roughly  2.145<BR>\n" );
document.write( "Margin of error: E = 1.20<BR>\n" );
document.write( "The 95% confidence interval is roughly  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7.80+%3C+mu+%3C+10.20 ALIGN=MIDDLE  ALT=\"7.80+%3C+mu+%3C+10.20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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