document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208012>Question 1208012</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose that a word-association experiment is conducted using eight people as blocks and making a comparison of reaction times within each person; that is, each person is subjected to both stimuli in a random order. The reaction times (in seconds) for the experiment are as follows.<BR>\n" );
document.write( "Person__________Stimulus 1__________Stimulus 2  (IGNORE THE LINES - space)<BR>\n" );
document.write( "1__________________2__________________5  (IGNORE THE LINES - space)<BR>\n" );
document.write( "2__________________1__________________3 (IGNORE THE LINES - space)<BR>\n" );
document.write( "3__________________1__________________4  (IGNORE THE LINES - space)<BR>\n" );
document.write( "4__________________2__________________2  (IGNORE THE LINES - space)<BR>\n" );
document.write( "5__________________1__________________3  (IGNORE THE LINES - space)<BR>\n" );
document.write( "6__________________2__________________3  (IGNORE THE LINES - space)<BR>\n" );
document.write( "7__________________3__________________4  (IGNORE THE LINES - space)<BR>\n" );
document.write( "8__________________2__________________3  (IGNORE THE LINES - space)\r<BR>\n" );
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document.write( "Do the data present sufficient evidence to indicate a difference in mean reaction times for the two stimuli? Test using 𝛼 = 0.05. (Use 𝜇1 − 𝜇2 = 𝜇d.)\r<BR>\n" );
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document.write( "a) State the test statistic. (Round your answer to three decimal places.)<BR>\n" );
document.write( "t = __________\r<BR>\n" );
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document.write( "b) State the rejection region. (If the test is one-tailed, enter NONE for the unused region. Round your answers to three decimal places.)<BR>\n" );
document.write( "t > __________<BR>\n" );
document.write( "t < ___________<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 #846155 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=846155\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answers:<BR>\n" );
document.write( "(a)  <font color=red>-4.333</font><BR>\n" );
document.write( "(b)  <font color=red>t > 2.365  or  t < -2.365</font><BR>\n" );
document.write( "All decimal values mentioned are approximate.\r<BR>\n" );
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document.write( "-------------------------------------------------------------------------------------------------------\r<BR>\n" );
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document.write( "Part (a)\r<BR>\n" );
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document.write( "Given data<BR>\n" );
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document.write( "<tr><TD >Person</TD><TD >Stimulus 1</TD><TD >Stimulus 2 </TD></tr><tr><TD >1</TD><TD >2</TD><TD >5</TD></tr><tr><TD >2</TD><TD >1</TD><TD >3</TD></tr><tr><TD >3</TD><TD >1</TD><TD >4</TD></tr><tr><TD >4</TD><TD >2</TD><TD >2</TD></tr><tr><TD >5</TD><TD >1</TD><TD >3</TD></tr><tr><TD >6</TD><TD >2</TD><TD >3</TD></tr><tr><TD >7</TD><TD >3</TD><TD >4</TD></tr><tr><TD >8</TD><TD >2</TD><TD >3</TD></tr>\n" );
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document.write( "There are n = 8 people which is the sample size.\r<BR>\n" );
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document.write( "Subtract the first two columns to form a new column of values.<BR>\n" );
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document.write( "<tr><TD >Person</TD><TD >Stimulus 1</TD><TD >Stimulus 2 </TD><TD >Stim1-Stim2</TD></tr><tr><TD >1</TD><TD >2</TD><TD >5</TD><TD >-3</TD></tr><tr><TD >2</TD><TD >1</TD><TD >3</TD><TD >-2</TD></tr><tr><TD >3</TD><TD >1</TD><TD >4</TD><TD >-3</TD></tr><tr><TD >4</TD><TD >2</TD><TD >2</TD><TD >0</TD></tr><tr><TD >5</TD><TD >1</TD><TD >3</TD><TD >-2</TD></tr><tr><TD >6</TD><TD >2</TD><TD >3</TD><TD >-1</TD></tr><tr><TD >7</TD><TD >3</TD><TD >4</TD><TD >-1</TD></tr><tr><TD >8</TD><TD >2</TD><TD >3</TD><TD >-1</TD></tr>\n" );
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document.write( "Example: 2-5 = -3 in the first row of this new column.\r<BR>\n" );
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document.write( "Add up the values in that new column to get -13.<BR>\n" );
document.write( "Divide this sum over the sample size n = 8 to get -13/8 = -1.625 which is the value of xbar<sub>d</sub><BR>\n" );
document.write( "xbar = x with a horizontal bar over top = sample mean<BR>\n" );
document.write( "xbar with a subscript \"d\" indicates we're looking at the sample mean of the differences.\r<BR>\n" );
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document.write( "If you want, you can compute the sample standard deviation of the differences by hand, but it's much better to use technology. <BR>\n" );
document.write( "Grab your favorite calculator. I used the spreadsheet's calculator since the data is already in spreadsheet form.<BR>\n" );
document.write( "You should find that<BR>\n" );
document.write( "s<sub>d</sub> = 1.06066017 approximately<BR>\n" );
document.write( "Caution: Do <u>not</u> mix up sample standard deviation with population standard deviation. You'll be using the sample version.\r<BR>\n" );
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document.write( "Then,<BR>\n" );
document.write( "Test Statistic = (xbar<sub>d</sub>)/(  (s<sub>d</sub>)/sqrt(n)  )<BR>\n" );
document.write( "Test Statistic = (-1.625)/(  (1.06066017)/sqrt(8)  )<BR>\n" );
document.write( "Test Statistic = -4.333333340605<BR>\n" );
document.write( "Test Statistic = <font color=red>-4.333</font>\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "We're doing a paired T-test.<BR>\n" );
document.write( "𝜇 = greek letter mu = population mean<BR>\n" );
document.write( "𝜇<sub>1</sub> = population mean of stimulus 1.<BR>\n" );
document.write( "𝜇<sub>2</sub> = population mean of stimulus 2.<BR>\n" );
document.write( "𝜇<sub>d</sub> = 𝜇<sub>1</sub>-𝜇<sub>2</sub> = population mean of the differences.\r<BR>\n" );
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document.write( "Hypotheses<BR>\n" );
document.write( "Null: 𝜇<sub>d</sub> = 0<BR>\n" );
document.write( "Alternative: 𝜇<sub>d</sub> =/= 0\r<BR>\n" );
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document.write( "The alternative hypothesis has the \"not equal\" sign because of the phrasing \"Do the data present sufficient evidence to <font color=red>indicate a <u>difference</u></font> in mean reaction times for the two stimuli?\"<BR>\n" );
document.write( "If the answer to this question was \"no\", then we'd stick with the null.<BR>\n" );
document.write( "The \"not equal\" sign in the alternative means we have a two-tailed test.\r<BR>\n" );
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document.write( "As mentioned, we're doing a paired T-test.<BR>\n" );
document.write( "We'll be using the T distribution.<BR>\n" );
document.write( "n = 8 = sample size<BR>\n" );
document.write( "df = degrees of freedom<BR>\n" );
document.write( "df = n-1<BR>\n" );
document.write( "df = 8-1<BR>\n" );
document.write( "df = 7\r<BR>\n" );
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document.write( "I'll now use this T table<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( "A similar table should be found in your textbook somewhere. Likely in the back (appendix section). Alternatively you can use a stats calculator such as a TI83.\r<BR>\n" );
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document.write( "In that table, highlight the df = 7 row and \"two tail = 0.05\" column.<BR>\n" );
document.write( "The 0.05 refers to the alpha value. <BR>\n" );
document.write( "This row and column intersect to give us the value 2.365 which is approximate.\r<BR>\n" );
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document.write( "It tells us that P(-2.365 < t < 2.365) = 0.95 when df = 7.<BR>\n" );
document.write( "i.e. the area under the curve between t = -2.365 and t = 2.365 is roughly 0.95<BR>\n" );
document.write( "This applies only when df = 7.<BR>\n" );
document.write( "This corresponds to a 95% confidence interval.<BR>\n" );
document.write( "The remaining 1-0.95 = 0.05 is the combined area of both tails.\r<BR>\n" );
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document.write( "If the test statistic is in the interval -2.365 < t < 2.365, then we fail to reject the null.<BR>\n" );
document.write( "Otherwise the test statistic is in the rejection region and we would reject the null.\r<BR>\n" );
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document.write( "Therefore the rejection region is:<BR>\n" );
document.write( "<font color=red>t > 2.365 or t < -2.365</font>\r<BR>\n" );
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document.write( "Extra info:<BR>\n" );
document.write( "Test Statistic = -4.333 was found earlier in part (a).<BR>\n" );
document.write( "It is outside the interval -2.365 < t < 2.365 and fits with t < -2.365 instead.<BR>\n" );
document.write( "We're in the rejection region.<BR>\n" );
document.write( "This means we'll reject the null. We'll conclude that there appears to be a difference in the reaction times for the two stimuli.<BR>\n" );
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