document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196643>Question 1196643</A>: <I><SPAN STYLE=\"word-break: break-all;\"> TABLE: https://imagizer.imageshack.com/img924/3557/S7r4QJ.jpg\r<BR>\n" );
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document.write( "Disk drives have been getting larger. Their capacity is now often given in terabytes​ (TB) where 1TB=1000 ​gigabytes, or about a trillion bytes. A search of prices for external disk drives on a large shopping website in a recent year found the accompanying data. Find and interpret the value of R^2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #829582 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=829582\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "If you are in a hurry, then you can use technology to quickly compute the r and r^2 values. <BR>\n" );
document.write( "Two examples would be the <a  rel=nofollow HREF=\"https://www.mathbootcamps.com/find-the-correlation-coefficient-r-on-your-calculator-ti83-84/\">LinReg</a> command on a TI83 (or similar) and using the <a  rel=nofollow HREF=\"https://support.microsoft.com/en-us/office/correl-function-995dcef7-0c0a-4bed-a3fb-239d7b68ca92\">the CORREL command</a> in a spreadsheet.<BR>\n" );
document.write( "There are many other options to choose from. Feel free to search out your favorite.<BR>\n" );
document.write( "You should find these approximations<BR>\n" );
document.write( "r = 0.9878<BR>\n" );
document.write( "<font color=red>r^2 = 0.9757</font><BR>\n" );
document.write( "Since r^2 is very close to 1, this makes the linear regression a good fit. Approximately 97.57% of the variation in x explains the variation in y.\r<BR>\n" );
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document.write( "If you have more time to go over the math, then there are various ways to calculate the correlation coefficient. <BR>\n" );
document.write( "I'll go over two slightly different methods.\r<BR>\n" );
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document.write( "Method 1\r<BR>\n" );
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document.write( "x = capacity of hard drive in terabytes (TB)<BR>\n" );
document.write( "y = price in dollars\r<BR>\n" );
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document.write( "Given info:<BR>\n" );
document.write( "n = 9 = sample size = number of x,y pairs<BR>\n" );
document.write( "xbar = 7.611<BR>\n" );
document.write( "ybar = 786.49<BR>\n" );
document.write( "SD(x) = 9.854<BR>\n" );
document.write( "SD(y) = 1417.82\r<BR>\n" );
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document.write( "The term \"xbar\" refers to the horizontal bar over the x, i.e. <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \overline{\text{x}}\"><BR>\n" );
document.write( "A similar story is with ybar as well.\r<BR>\n" );
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document.write( "Given Data<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >x</TD><TD >y</TD></tr><tr><TD >0.5</TD><TD >60.99</TD></tr><tr><TD >1</TD><TD >77.99</TD></tr><tr><TD >2</TD><TD >112.97</TD></tr><tr><TD >3</TD><TD >110.99</TD></tr><tr><TD >4</TD><TD >151.99</TD></tr><tr><TD >6</TD><TD >425.34</TD></tr><tr><TD >8</TD><TD >597.11</TD></tr><tr><TD >12</TD><TD >1081.99</TD></tr><tr><TD >32</TD><TD >4459</TD></tr>\n" );
document.write( "</TABLE>Form a third column which is the product of the x and y columns<BR>\n" );
document.write( "Eg: 0.5*60.99 = 30.495 in the first row<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >x</TD><TD >y</TD><TD >xy</TD></tr><tr><TD >0.5</TD><TD >60.99</TD><TD >30.495</TD></tr><tr><TD >1</TD><TD >77.99</TD><TD >77.99</TD></tr><tr><TD >2</TD><TD >112.97</TD><TD >225.94</TD></tr><tr><TD >3</TD><TD >110.99</TD><TD >332.97</TD></tr><tr><TD >4</TD><TD >151.99</TD><TD >607.96</TD></tr><tr><TD >6</TD><TD >425.34</TD><TD >2552.04</TD></tr><tr><TD >8</TD><TD >597.11</TD><TD >4776.88</TD></tr><tr><TD >12</TD><TD >1081.99</TD><TD >12983.88</TD></tr><tr><TD >32</TD><TD >4459</TD><TD >142688</TD></tr>\n" );
document.write( "</TABLE>I strongly recommend using spreadsheet software. <BR>\n" );
document.write( "It's not only fast and efficient, but also something that is expected in real world applications.<BR>\n" );
document.write( "I'm using LibreOffice but you could use Excel or Google Sheets or whichever app you prefer most.\r<BR>\n" );
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document.write( "Add up the values in the xy column to get 164,276.155<BR>\n" );
document.write( "Then we subtract off the value of n*xbar*ybar = 9*7.611*786.49 = 53,873.77851\r<BR>\n" );
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document.write( "So we have:<BR>\n" );
document.write( "Sum(xy) - n*xbar*ybar = 164,276.155 - 53,873.77851 = 110,402.37649\r<BR>\n" );
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document.write( "We'll divide that result over the product of the given standard deviation values, multiplied with (n-1)<BR>\n" );
document.write( "So,<BR>\n" );
document.write( "(n-1)*SD(x)*SD(y) = (9-1)*(9.854)*(1417.82) = 111,769.58624\r<BR>\n" );
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document.write( "Therefore,<BR>\n" );
document.write( "r = (110,402.37649)/(111,769.58624)<BR>\n" );
document.write( "r = 0.98776760480204<BR>\n" );
document.write( "r^2 = (0.98776760480204)^2<BR>\n" );
document.write( "r^2 = 0.97568484109636<BR>\n" );
document.write( "<font color=red>r^2 = 0.9757</font><BR>\n" );
document.write( "which is approximate.\r<BR>\n" );
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document.write( "Since r^2 is very close to 1, this makes the linear regression a good fit. Approximately 97.57% of the variation in x explains the variation in y.\r<BR>\n" );
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document.write( "Note: The formula I used just now is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{r} = \frac{ (\sum \text{x}\text{y}) - \text{n}*\overline{\text{x}}*\overline{\text{y}}}{(\text{n}-1)*SD(\text{x})*SD(\text{y})}\">\r<BR>\n" );
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document.write( "-------------------------------------------------------------------------------------------------------------\r<BR>\n" );
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document.write( "Method 2\r<BR>\n" );
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document.write( "x = capacity of hard drive in terabytes (TB)<BR>\n" );
document.write( "y = price in dollars\r<BR>\n" );
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document.write( "Given info:<BR>\n" );
document.write( "n = 9 = sample size = number of x,y pairs<BR>\n" );
document.write( "xbar = 7.611<BR>\n" );
document.write( "ybar = 786.49<BR>\n" );
document.write( "SD(x) = 9.854<BR>\n" );
document.write( "SD(y) = 1417.82\r<BR>\n" );
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document.write( "Given Data<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >x</TD><TD >y</TD></tr><tr><TD >0.5</TD><TD >60.99</TD></tr><tr><TD >1</TD><TD >77.99</TD></tr><tr><TD >2</TD><TD >112.97</TD></tr><tr><TD >3</TD><TD >110.99</TD></tr><tr><TD >4</TD><TD >151.99</TD></tr><tr><TD >6</TD><TD >425.34</TD></tr><tr><TD >8</TD><TD >597.11</TD></tr><tr><TD >12</TD><TD >1081.99</TD></tr><tr><TD >32</TD><TD >4459</TD></tr>\n" );
document.write( "</TABLE>Instead of an xy column, we'll form the Zx column<BR>\n" );
document.write( "Zx = (x - xbar)/(SD(x))<BR>\n" );
document.write( "We're computing the z score for each x term\r<BR>\n" );
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document.write( "For instance, in the first row we have<BR>\n" );
document.write( "Zx = (x-xbar)/(SD(x))<BR>\n" );
document.write( "Zx = (0.5-7.611)/(9.854)<BR>\n" );
document.write( "Zx = -0.72163588390501<BR>\n" );
document.write( "Zx = -0.721636<BR>\n" );
document.write( "Do the same thing for each item in the x column. The values of xbar and SD(x) will remain constant.\r<BR>\n" );
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document.write( "This is what the updated table looks like<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >x</TD><TD >y</TD><TD >Zx</TD></tr><tr><TD >0.5</TD><TD >60.99</TD><TD >-0.721636</TD></tr><tr><TD >1</TD><TD >77.99</TD><TD >-0.670895</TD></tr><tr><TD >2</TD><TD >112.97</TD><TD >-0.569413</TD></tr><tr><TD >3</TD><TD >110.99</TD><TD >-0.467932</TD></tr><tr><TD >4</TD><TD >151.99</TD><TD >-0.36645</TD></tr><tr><TD >6</TD><TD >425.34</TD><TD >-0.163487</TD></tr><tr><TD >8</TD><TD >597.11</TD><TD >0.039476</TD></tr><tr><TD >12</TD><TD >1081.99</TD><TD >0.445403</TD></tr><tr><TD >32</TD><TD >4459</TD><TD >2.475036</TD></tr>\n" );
document.write( "</TABLE>Follow similar steps for the Zy column<BR>\n" );
document.write( "For example, we'll have the following calculation for the 1st row.<BR>\n" );
document.write( "Zy = (y - ybar)/(SD(y))<BR>\n" );
document.write( "Zy = (60.99 - 786.49)/(1417.82)<BR>\n" );
document.write( "Zy = -0.511701\r<BR>\n" );
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document.write( "We have this so far<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >x</TD><TD >y</TD><TD >Zx</TD><TD >Zy</TD></tr><tr><TD >0.5</TD><TD >60.99</TD><TD >-0.721636</TD><TD >-0.511701</TD></tr><tr><TD >1</TD><TD >77.99</TD><TD >-0.670895</TD><TD >-0.499711</TD></tr><tr><TD >2</TD><TD >112.97</TD><TD >-0.569413</TD><TD >-0.475039</TD></tr><tr><TD >3</TD><TD >110.99</TD><TD >-0.467932</TD><TD >-0.476436</TD></tr><tr><TD >4</TD><TD >151.99</TD><TD >-0.36645</TD><TD >-0.447518</TD></tr><tr><TD >6</TD><TD >425.34</TD><TD >-0.163487</TD><TD >-0.254722</TD></tr><tr><TD >8</TD><TD >597.11</TD><TD >0.039476</TD><TD >-0.133571</TD></tr><tr><TD >12</TD><TD >1081.99</TD><TD >0.445403</TD><TD >0.208419</TD></tr><tr><TD >32</TD><TD >4459</TD><TD >2.475036</TD><TD >2.590251</TD></tr>\n" );
document.write( "</TABLE>Then we'll multiply the Zx and Zy items for each row.<BR>\n" );
document.write( "Eg: Zx*Zy = (-0.721636)*(-0.511701) = 0.369262 in row one\r<BR>\n" );
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document.write( "This is what the fully completed table looks like<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >x</TD><TD >y</TD><TD >Zx</TD><TD >Zy</TD><TD >ZxZy</TD></tr><tr><TD >0.5</TD><TD >60.99</TD><TD >-0.721636</TD><TD >-0.511701</TD><TD >0.369262</TD></tr><tr><TD >1</TD><TD >77.99</TD><TD >-0.670895</TD><TD >-0.499711</TD><TD >0.335254</TD></tr><tr><TD >2</TD><TD >112.97</TD><TD >-0.569413</TD><TD >-0.475039</TD><TD >0.270493</TD></tr><tr><TD >3</TD><TD >110.99</TD><TD >-0.467932</TD><TD >-0.476436</TD><TD >0.22294</TD></tr><tr><TD >4</TD><TD >151.99</TD><TD >-0.36645</TD><TD >-0.447518</TD><TD >0.163993</TD></tr><tr><TD >6</TD><TD >425.34</TD><TD >-0.163487</TD><TD >-0.254722</TD><TD >0.041644</TD></tr><tr><TD >8</TD><TD >597.11</TD><TD >0.039476</TD><TD >-0.133571</TD><TD >-0.005273</TD></tr><tr><TD >12</TD><TD >1081.99</TD><TD >0.445403</TD><TD >0.208419</TD><TD >0.09283</TD></tr><tr><TD >32</TD><TD >4459</TD><TD >2.475036</TD><TD >2.590251</TD><TD >6.410964</TD></tr>\n" );
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document.write( "Add up the values in the final column and you should get roughly 7.902107\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "r = Sum(ZxZy)/(n-1)<BR>\n" );
document.write( "r = 7.902107/(9-1)<BR>\n" );
document.write( "r = 7.902107/8<BR>\n" );
document.write( "r = 0.987763375<BR>\n" );
document.write( "r^2 = (0.987763375)^2<BR>\n" );
document.write( "r^2 = 0.9756764849914<BR>\n" );
document.write( "r^2 = 0.9757\r<BR>\n" );
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document.write( "Since r^2 is very close to 1, this makes the linear regression a good fit. Approximately 97.57% of the variation in x explains the variation in y.\r<BR>\n" );
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document.write( "<font color=red size=4>Answer: r^2 = 0.9757 approximately</font><BR>\n" );
document.write( "</font> </SPAN>\n" );
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