document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193085>Question 1193085</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two-sample t test versus matched pairs t test. Consider the following data set. The data were actually collected in pairs, and each row represents a pair.\r<BR>\n" );
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document.write( "Group 1  Group 2\r<BR>\n" );
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document.write( "48.86       48.88\r<BR>\n" );
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document.write( "50.60       52.63\r<BR>\n" );
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document.write( "51.02       52.55\r<BR>\n" );
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document.write( "47.99       50.94\r<BR>\n" );
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document.write( "54.20       53.02\r<BR>\n" );
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document.write( "50.66       50.66\r<BR>\n" );
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document.write( "45.91       47.78\r<BR>\n" );
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document.write( "48.79       48.44\r<BR>\n" );
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document.write( "47.76       48.92\r<BR>\n" );
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document.write( "51.13       51.63\r<BR>\n" );
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document.write( "(A)Suppose that we ignore the fact that the data were collected in pairs and mistakenly treat this as a two-sample problem. Compute the sample mean and variance for each group. Then compute the two-sample t statistic, degrees of freedom, and P-value for the two-sided alternative.<BR>\n" );
document.write( "(B)Now analyze the data in the proper way. Compute the sample mean and variance of the differences. Then compute the t statistic, degrees of freedom, and P-value.<BR>\n" );
document.write( "(C)Describe the differences in the two test results </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #825078 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=825078\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Given Data<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >Group 1</TD><TD >Group 2</TD></tr><tr><TD >48.86</TD><TD >48.88</TD></tr><tr><TD >50.60</TD><TD >52.63</TD></tr><tr><TD >51.02</TD><TD >52.55</TD></tr><tr><TD >47.99</TD><TD >50.94</TD></tr><tr><TD >54.20</TD><TD >53.02</TD></tr><tr><TD >50.66</TD><TD >50.66</TD></tr><tr><TD >45.91</TD><TD >47.78</TD></tr><tr><TD >48.79</TD><TD >48.44</TD></tr><tr><TD >47.76</TD><TD >48.92</TD></tr><tr><TD >51.13</TD><TD >51.63</TD></tr>\n" );
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document.write( "Part (A)\r<BR>\n" );
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document.write( "Let's compute the sample mean of Group 1.<BR>\n" );
document.write( "To get the sample mean, we first add up the data values<BR>\n" );
document.write( "48.86+50.6+51.02+47.99+54.2+50.66+45.91+48.79+47.76+51.13 = 496.92\r<BR>\n" );
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document.write( "Then we divide that over the sample size n = 10<BR>\n" );
document.write( "496.92/n = 496.92/10 = 49.692<BR>\n" );
document.write( "This is the sample mean (xbar) for group 1.<BR>\n" );
document.write( "I'll refer to this as xbar1.\r<BR>\n" );
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document.write( "Follow similar steps for group 2 to find that<BR>\n" );
document.write( "xbar2 = 50.545<BR>\n" );
document.write( "Note: this does NOT mean xbar squared.\r<BR>\n" );
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document.write( "Now let's calculate the variance for group 1.<BR>\n" );
document.write( "Here's the data values for group 1 only, which I'll call X.<BR>\n" );
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document.write( "<tr><TD >X</TD></tr><tr><TD >48.86</TD></tr><tr><TD >50.6</TD></tr><tr><TD >51.02</TD></tr><tr><TD >47.99</TD></tr><tr><TD >54.2</TD></tr><tr><TD >50.66</TD></tr><tr><TD >45.91</TD></tr><tr><TD >48.79</TD></tr><tr><TD >47.76</TD></tr><tr><TD >51.13</TD></tr>\n" );
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document.write( "For each X value, subtract off the value of xbar = 49.692<BR>\n" );
document.write( "Then square the difference.<BR>\n" );
document.write( "For example, we have (X-xbar)^2 = (48.86-49.692)^2 = 0.692224 in the first row.<BR>\n" );
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document.write( "<tr><TD >X</TD><TD >(X-xbar)^2</TD></tr><tr><TD >48.86</TD><TD >0.692224</TD></tr><tr><TD >50.6</TD><TD >0.824464</TD></tr><tr><TD >51.02</TD><TD >1.763584</TD></tr><tr><TD >47.99</TD><TD >2.896804</TD></tr><tr><TD >54.2</TD><TD >20.322064</TD></tr><tr><TD >50.66</TD><TD >0.937024</TD></tr><tr><TD >45.91</TD><TD >14.303524</TD></tr><tr><TD >48.79</TD><TD >0.813604</TD></tr><tr><TD >47.76</TD><TD >3.732624</TD></tr><tr><TD >51.13</TD><TD >2.067844</TD></tr>\n" );
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document.write( "Sum everything in the second column and you should get 48.35376<BR>\n" );
document.write( "This is the Sum of the Squared Errors (SSE)<BR>\n" );
document.write( "Divide the SSE value over n-1 = 10-1 = 9 to compute the sample variance\r<BR>\n" );
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document.write( "sample variance = (SSE)/(n-1)<BR>\n" );
document.write( "sample variance = (48.35376)/9<BR>\n" );
document.write( "sample variance = 5.37264<BR>\n" );
document.write( "I'll refer to this as V1 to represent the variance of group 1.\r<BR>\n" );
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document.write( "Follow similar steps to find that V2 = 3.70316 is the approximate sample variance of group 2.<BR>\n" );
document.write( "Use of a calculator with a built-in standard deviation function will make quick work of finding the variance.\r<BR>\n" );
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document.write( "Now onto the standard error (SE)<BR>\n" );
document.write( "SE = sqrt( (V1)/(n1) + (V2)/(n2) )<BR>\n" );
document.write( "SE = sqrt( (5.37264)/(10) + (3.70316)/(10) )<BR>\n" );
document.write( "SE = 0.95266993234803\r<BR>\n" );
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document.write( "Which helps us find the t statistic<BR>\n" );
document.write( "t = ((xbar1 - xbar2) - (mu1 - mu2))/(SE)<BR>\n" );
document.write( "t = ((49.692 - 50.545) - (0))/(0.95266993234803)<BR>\n" );
document.write( "t = -0.89537831628381<BR>\n" );
document.write( "t = -0.8954\r<BR>\n" );
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document.write( "The degrees of freedom is the smaller of n1-1 or n2-1<BR>\n" );
document.write( "Because n1 = n2 = 10, we just simply can think of it as n-1<BR>\n" );
document.write( "The degrees of freedom is df = n-1 = 10-1 = 9\r<BR>\n" );
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document.write( "Use a calculator like this one<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://stattrek.com/online-calculator/t-distribution.aspx\">https://stattrek.com/online-calculator/t-distribution.aspx</a><BR>\n" );
document.write( "to find that P(T < -0.8954) = 0.1969 approximately when we have df = 9<BR>\n" );
document.write( "This doubles to 2*0.1969 = 0.3938 due to the fact that we have a two-sided test (because the phrasing \"P-value for the two-sided alternative.\")<BR>\n" );
document.write( "The result 0.3938 is the approximate P-value.\r<BR>\n" );
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document.write( "------------------------------\r<BR>\n" );
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document.write( "Summary:\r<BR>\n" );
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document.write( "xbar1 = 49.692 and xbar2 = 50.545 are the sample means<BR>\n" );
document.write( "V1 = 5.37264 and V2 = 3.70316 are the sample variances<BR>\n" );
document.write( "t = -0.8954 is the test statistic<BR>\n" );
document.write( "df = 9 is the degrees of freedom<BR>\n" );
document.write( "p-value = 0.3938\r<BR>\n" );
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document.write( "==============================================================================================================<BR>\n" );
document.write( "Part (B)\r<BR>\n" );
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document.write( "Here are the two original groups of data.<BR>\n" );
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document.write( "<tr><TD >Group 1</TD><TD >Group 2</TD></tr><tr><TD >48.86</TD><TD >48.88</TD></tr><tr><TD >50.60</TD><TD >52.63</TD></tr><tr><TD >51.02</TD><TD >52.55</TD></tr><tr><TD >47.99</TD><TD >50.94</TD></tr><tr><TD >54.20</TD><TD >53.02</TD></tr><tr><TD >50.66</TD><TD >50.66</TD></tr><tr><TD >45.91</TD><TD >47.78</TD></tr><tr><TD >48.79</TD><TD >48.44</TD></tr><tr><TD >47.76</TD><TD >48.92</TD></tr><tr><TD >51.13</TD><TD >51.63</TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "For each row, subtract the values in the form X1 - X2<BR>\n" );
document.write( "X1 is from group 1<BR>\n" );
document.write( "X2 is from group 2<BR>\n" );
document.write( "For instance, the first row has 48.86 - 48.88 = -0.02 as the difference<BR>\n" );
document.write( "We'll list the differences in the column labeled \"d\"<BR>\n" );
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document.write( "<tr><TD >Group 1</TD><TD >Group 2</TD><TD >d</TD></tr><tr><TD >48.86</TD><TD >48.88</TD><TD >-0.02</TD></tr><tr><TD >50.6</TD><TD >52.63</TD><TD >-2.03</TD></tr><tr><TD >51.02</TD><TD >52.55</TD><TD >-1.53</TD></tr><tr><TD >47.99</TD><TD >50.94</TD><TD >-2.95</TD></tr><tr><TD >54.2</TD><TD >53.02</TD><TD >1.18</TD></tr><tr><TD >50.66</TD><TD >50.66</TD><TD >0</TD></tr><tr><TD >45.91</TD><TD >47.78</TD><TD >-1.87</TD></tr><tr><TD >48.79</TD><TD >48.44</TD><TD >0.35</TD></tr><tr><TD >47.76</TD><TD >48.92</TD><TD >-1.16</TD></tr><tr><TD >51.13</TD><TD >51.63</TD><TD >-0.5</TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "If you were to compute the sample mean of the d column, you should find that the mean is -0.853<BR>\n" );
document.write( "We call this value dbar in much the same way xbar is denoted. The \"bar\" refers to the horizontal line up top.<BR>\n" );
document.write( "dbar = -0.853\r<BR>\n" );
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document.write( "The sample variance of the d column will follow the same type of steps as described in part (A) when I detailed how to compute the variance of group 1.<BR>\n" );
document.write( "You should get a sample variance of 1.610668 which leads to the sample standard deviation of sqrt(1.610668) = 1.269121<BR>\n" );
document.write( "I'll refer to this standard deviation as <font color=red>S</font><font color=blue>d</font> to indicate \"<font color=red>S</font>tandard deviation of the <font color=blue>d</font>ifferences\".\r<BR>\n" );
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document.write( "Now onto the standard error (SE)<BR>\n" );
document.write( "SE = Sd/sqrt(n)<BR>\n" );
document.write( "SE = 1.269121/sqrt(10)<BR>\n" );
document.write( "SE = 0.40133129863506<BR>\n" );
document.write( "SE = 0.401331\r<BR>\n" );
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document.write( "It allows us to compute the test statistic<BR>\n" );
document.write( "t = (dbar - mu_d)/SE<BR>\n" );
document.write( "t = (-0.853 - 0)/0.40133129863506<BR>\n" );
document.write( "t = -2.12542605797524<BR>\n" );
document.write( "t = -2.1254\r<BR>\n" );
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document.write( "The degrees of freedom is n-1 = 10-1 = 9\r<BR>\n" );
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document.write( "I'll then use this calculator again<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://stattrek.com/online-calculator/t-distribution.aspx\">https://stattrek.com/online-calculator/t-distribution.aspx</a><BR>\n" );
document.write( "to find that P(T < -2.1254) = 0.0312 when df = 9 which doubles to 2*0.0312 = 0.0624 since we're doing a two-tailed test.<BR>\n" );
document.write( "The result 0.0624 is the approximate P-value.\r<BR>\n" );
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document.write( "------------------------------\r<BR>\n" );
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document.write( "Summary:\r<BR>\n" );
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document.write( "dbar = -0.853 is the sample mean of the differences (d column)<BR>\n" );
document.write( "1.610668 is the approximate sample variance of the differences (d column)<BR>\n" );
document.write( "t = -2.1254 is the approximate test statistic<BR>\n" );
document.write( "df = 9 = degrees of freedom<BR>\n" );
document.write( "P-value = 0.0624\r<BR>\n" );
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document.write( "==============================================================================================================<BR>\n" );
document.write( "Part (C)\r<BR>\n" );
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document.write( "Admittedly there are a lot of numbers and variables to keep track of. <BR>\n" );
document.write( "It might be overwhelming if you aren't familiar with statistics too much.\r<BR>\n" );
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document.write( "Though if I had to pick one variable to focus on, I would say it's the P-value. <BR>\n" );
document.write( "In many scientific journals, the researchers report the P-value to the reader to indicate how (in)significant the results were. \r<BR>\n" );
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document.write( "In part (A), we got a P-value of roughly 0.3938<BR>\n" );
document.write( "In part (B), we got a P-value of roughly 0.0624<BR>\n" );
document.write( "That's quite a gap.\r<BR>\n" );
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document.write( "Recall that the P-value determines if you reject or fail to reject the null.<BR>\n" );
document.write( "Let's say the significance level is alpha = 0.05 which is the default level.\r<BR>\n" );
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document.write( "At this alpha value, we'd fail to reject the null for both part (A) and part (B). Why? Because the p-value for each is not less than alpha = 0.05<BR>\n" );
document.write( "We reject the null only if the p-value is smaller than alpha.\r<BR>\n" );
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document.write( "If we set alpha = 0.10, then we'd fail to reject in part (A) but reject the null in part (B)<BR>\n" );
document.write( "Sometimes you may see a significance level of alpha = 0.10 (of course it depends on the context).<BR>\n" );
document.write( "As you can see, part (B) has leads to a situation where we are more likely to reject the null.<BR>\n" );
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