document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208428>Question 1208428</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Good afternoon, please help me solve this problem.<BR>\n" );
document.write( "During the baking of a certain type of bread roll, each bread goes through a series of heat process. The length of time spent under this heat treatment is related to the lifespan of the bread rolls. A sample of eight bread rolls that underwent different baking times was selected and the life span (hours) of each roll was recorded as follows:\r<BR>\n" );
document.write( "\n" );
document.write( "Length of time	18	13	18	15	10	12	8	4<BR>\n" );
document.write( "Life span	23	20	18	16	14	11	10	7\r<BR>\n" );
document.write( "\n" );
document.write( "a)	Calculate the Pearson’s correlation coefficient and explain the result 										[10 Marks<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #846839 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=846839\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <font color=red size=4>0.8703</font><BR>\n" );
document.write( "This value is approximate.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Explanation\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This value can be quickly calculated using technology.<BR>\n" );
document.write( "Here are some options:<ul><li>Use 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). Make sure that Diagnostic is set to \"on\".</li><li>Use 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.</li><li>Input the data into the spreadsheet mode of <a  rel=nofollow HREF=\"https://www.geogebra.org/\">GeoGebra</a>. Refer to <a  rel=nofollow HREF=\"https://www.youtube.com/watch?v=UEjrcVy7zwg\">this video</a> for an example.</li><li>Search out \"pearson correlation coefficient calculator\" to find many free tools online. Two search results I found were <a  rel=nofollow HREF=\"https://www.socscistatistics.com/tests/pearson/\">this</a> and <a  rel=nofollow HREF=\"https://www.gigacalculator.com/calculators/correlation-coefficient-calculator.php\">this</a> (among many other results).</li><li>Use a programming language such as R or Python.</li><li>There are many other technology options. I encourage the student to explore his/her favorite.</li></ul>If you aren't allowed to use a specialized correlation coefficient calculator, then check out any of the three methods shown below.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "----------------------------------------------------------------------------------------<BR>\n" );
document.write( "Method 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Define the following<BR>\n" );
document.write( "x = length of baking time (in minutes)<BR>\n" );
document.write( "y = life span (in hours)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The given table is <BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >x</TD><TD >18</TD><TD >13</TD><TD >18</TD><TD >15</TD><TD >10</TD><TD >12</TD><TD >8</TD><TD >4</TD></tr><tr><TD >y</TD><TD >23</TD><TD >20</TD><TD >18</TD><TD >16</TD><TD >14</TD><TD >11</TD><TD >10</TD><TD >7</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There are n = 8 columns of values.<BR>\n" );
document.write( "In other words, there are 8 pairs of (x,y) values.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Add up the x values and divide by n = 8 to find the sample mean of the x values.<BR>\n" );
document.write( "xbar = sample mean of x values = x with a horizontal bar over top<BR>\n" );
document.write( "xbar = (sum of x values)/(number of x values)<BR>\n" );
document.write( "xbar = (18+13+18+15+10+12+8+4)/8<BR>\n" );
document.write( "xbar = 98/8<BR>\n" );
document.write( "xbar = 12.25<BR>\n" );
document.write( "If you are using spreadsheet software, then the function you can use is called <a  rel=nofollow HREF=\"https://support.microsoft.com/en-us/office/average-function-047bac88-d466-426c-a32b-8f33eb960cf6\">Average</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Apply the same idea to the y values.<BR>\n" );
document.write( "ybar = (sum of y values)/(number of y values)<BR>\n" );
document.write( "ybar = (23+20+18+16+14+11+10+7)/8<BR>\n" );
document.write( "ybar = 119/8<BR>\n" );
document.write( "ybar = 14.875\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Introduce a new row labeled xy. This is the product of the x and y values.<BR>\n" );
document.write( "For example, we'll have x*y = 18*23 = 414 as the first entry of this row.<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >x</TD><TD >18</TD><TD >13</TD><TD >18</TD><TD >15</TD><TD >10</TD><TD >12</TD><TD >8</TD><TD >4</TD></tr><tr><TD >y</TD><TD >23</TD><TD >20</TD><TD >18</TD><TD >16</TD><TD >14</TD><TD >11</TD><TD >10</TD><TD >7</TD></tr><tr><TD >xy</TD><TD >414</TD><TD >260</TD><TD >324</TD><TD >240</TD><TD >140</TD><TD >132</TD><TD >80</TD><TD >28</TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "Add up the items in this new row to get Sum(xy) = 414+260+324+240+140+132+80+28 = 1618<BR>\n" );
document.write( "Spreadsheet software is strongly recommended.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The last two things we need are:<BR>\n" );
document.write( "SD(x)<BR>\n" );
document.write( "SD(y)<BR>\n" );
document.write( "which represent the sample standard deviations of the x and y values respectively.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "You could find these by hand (see part d of <a  rel=nofollow HREF=\"https://www.algebra.com/algebra/homework/word/numbers/Numbers_Word_Problems.faq.question.1196832.html\">this page</a>) but I'll use technology to speed things up.<BR>\n" );
document.write( "SD(x) = 4.862392<BR>\n" );
document.write( "SD(y) = 5.409978<BR>\n" );
document.write( "These values are approximate. I rounded each to 6 decimal places.<BR>\n" );
document.write( "Make sure you use the sample standard deviation and NOT the population standard deviation.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's calculate the correlation coefficient.<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" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{r} = \frac{ (1618) - 8*12.25*14.875}{(8-1)*4.862392*5.409978}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{r} \approx 0.8703\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Many textbooks will round to 4 decimal places. If your teacher instructs otherwise, then be sure to follow said instructions.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "----------------------------------------------------------------------------------------<BR>\n" );
document.write( "Method 2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Return back to this table.<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >x</TD><TD >18</TD><TD >13</TD><TD >18</TD><TD >15</TD><TD >10</TD><TD >12</TD><TD >8</TD><TD >4</TD></tr><tr><TD >y</TD><TD >23</TD><TD >20</TD><TD >18</TD><TD >16</TD><TD >14</TD><TD >11</TD><TD >10</TD><TD >7</TD></tr><tr><TD >xy</TD><TD >414</TD><TD >260</TD><TD >324</TD><TD >240</TD><TD >140</TD><TD >132</TD><TD >80</TD><TD >28</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's introduce rows labeled x^2 and y^2.<BR>\n" );
document.write( "As their name suggests, we'll square each value.<BR>\n" );
document.write( "Eg: x^2 = 18^2 = 324<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >x</TD><TD >18</TD><TD >13</TD><TD >18</TD><TD >15</TD><TD >10</TD><TD >12</TD><TD >8</TD><TD >4</TD></tr><tr><TD >y</TD><TD >23</TD><TD >20</TD><TD >18</TD><TD >16</TD><TD >14</TD><TD >11</TD><TD >10</TD><TD >7</TD></tr><tr><TD >xy</TD><TD >414</TD><TD >260</TD><TD >324</TD><TD >240</TD><TD >140</TD><TD >132</TD><TD >80</TD><TD >28</TD></tr><tr><TD >x^2</TD><TD >324</TD><TD >169</TD><TD >324</TD><TD >225</TD><TD >100</TD><TD >144</TD><TD >64</TD><TD >16</TD></tr><tr><TD >y^2</TD><TD >529</TD><TD >400</TD><TD >324</TD><TD >256</TD><TD >196</TD><TD >121</TD><TD >100</TD><TD >49</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Define the following<BR>\n" );
document.write( "A = sum of the x values = 98<BR>\n" );
document.write( "B = sum of the y values = 119<BR>\n" );
document.write( "C = sum of the xy values = 1618<BR>\n" );
document.write( "D = sum of the x^2 values = 1366<BR>\n" );
document.write( "E = sum of the y^2 values = 1975<BR>\n" );
document.write( "Spreadsheet software can quickly compute each of these.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We can now compute the correlation coefficient.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{r} = \frac{ n*C - A*B }{ \sqrt{ (n*D - A^2)*(n*E - B^2) } }\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{r} = \frac{ 8*1618 - 98*119 }{ \sqrt{ (8*1366 - 98^2)*(8*1975 - 119^2) } }\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{r} \approx 0.8703\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "----------------------------------------------------------------------------------------<BR>\n" );
document.write( "Method 3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's return back to the original table.<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >x</TD><TD >18</TD><TD >13</TD><TD >18</TD><TD >15</TD><TD >10</TD><TD >12</TD><TD >8</TD><TD >4</TD></tr><tr><TD >y</TD><TD >23</TD><TD >20</TD><TD >18</TD><TD >16</TD><TD >14</TD><TD >11</TD><TD >10</TD><TD >7</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Recall these previous calculations<BR>\n" );
document.write( "xbar = 12.25<BR>\n" );
document.write( "SD(x) = 4.862392\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We'll use them to determine the z score of each x value.<BR>\n" );
document.write( "For instance, if x = 18, then,<BR>\n" );
document.write( "z = (x-xbar)/( SD(x) )<BR>\n" );
document.write( "z = (18-12.25)/( 4.862392 )<BR>\n" );
document.write( "z = 1.182545545485<BR>\n" );
document.write( "z = 1.182546<BR>\n" );
document.write( "Repeat similar calculations for the other x values. This new row will be labeled Zx to indicate the z scores of the x values.<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >x</TD><TD >18</TD><TD >13</TD><TD >18</TD><TD >15</TD><TD >10</TD><TD >12</TD><TD >8</TD><TD >4</TD></tr><tr><TD >y</TD><TD >23</TD><TD >20</TD><TD >18</TD><TD >16</TD><TD >14</TD><TD >11</TD><TD >10</TD><TD >7</TD></tr><tr><TD >Zx</TD><TD >1.182546</TD><TD >0.154245</TD><TD >1.182546</TD><TD >0.565565</TD><TD >-0.462735</TD><TD >-0.051415</TD><TD >-0.874055</TD><TD >-1.696696</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "For the y values we have<BR>\n" );
document.write( "ybar = 14.875<BR>\n" );
document.write( "SD(y) = 5.409978  (approximate)<BR>\n" );
document.write( "Here's what it looks like to introduce the z scores for the y values.<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >x</TD><TD >18</TD><TD >13</TD><TD >18</TD><TD >15</TD><TD >10</TD><TD >12</TD><TD >8</TD><TD >4</TD></tr><tr><TD >y</TD><TD >23</TD><TD >20</TD><TD >18</TD><TD >16</TD><TD >14</TD><TD >11</TD><TD >10</TD><TD >7</TD></tr><tr><TD >Zx</TD><TD >1.182546</TD><TD >0.154245</TD><TD >1.182546</TD><TD >0.565565</TD><TD >-0.462735</TD><TD >-0.051415</TD><TD >-0.874055</TD><TD >-1.696696</TD></tr><tr><TD >Zy</TD><TD >1.501855</TD><TD >0.947324</TD><TD >0.577636</TD><TD >0.207949</TD><TD >-0.161738</TD><TD >-0.716269</TD><TD >-0.901113</TD><TD >-1.455644</TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "As mentioned earlier, spreadsheet software is strongly recommended.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The next thing we'll do is multiply each Zx and Zy value to form a new row.<BR>\n" );
document.write( "Example: 1.182546*1.501855 = 1.776013 approximately<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >x</TD><TD >18</TD><TD >13</TD><TD >18</TD><TD >15</TD><TD >10</TD><TD >12</TD><TD >8</TD><TD >4</TD></tr><tr><TD >y</TD><TD >23</TD><TD >20</TD><TD >18</TD><TD >16</TD><TD >14</TD><TD >11</TD><TD >10</TD><TD >7</TD></tr><tr><TD >Zx</TD><TD >1.182546</TD><TD >0.154245</TD><TD >1.182546</TD><TD >0.565565</TD><TD >-0.462735</TD><TD >-0.051415</TD><TD >-0.874055</TD><TD >-1.696696</TD></tr><tr><TD >Zy</TD><TD >1.501855</TD><TD >0.947324</TD><TD >0.577636</TD><TD >0.207949</TD><TD >-0.161738</TD><TD >-0.716269</TD><TD >-0.901113</TD><TD >-1.455644</TD></tr><tr><TD >Zx*Zy</TD><TD >1.776013</TD><TD >0.14612</TD><TD >0.683081</TD><TD >0.117609</TD><TD >0.074842</TD><TD >0.036827</TD><TD >0.787622</TD><TD >2.469785</TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "The decimal values are approximate.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The values in that bottom row add to 6.091899 which is also approximate.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We can finally calculate the r value.<BR>\n" );
document.write( "r = Sum(Zx*Zy)/(n-1)<BR>\n" );
document.write( "r = 6.091899/(8-1)<BR>\n" );
document.write( "r = 0.870271285714<BR>\n" );
document.write( "<font color=red size=4>r = 0.8703</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Other methods are possible. <BR>\n" );
document.write( "I'll let the student explore them.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "----------------------------------------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Further Reading<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.statology.org/pearson-correlation-coefficient/\">https://www.statology.org/pearson-correlation-coefficient/</a><BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://statisticsbyjim.com/basics/correlation-coefficient-formula/\">https://statisticsbyjim.com/basics/correlation-coefficient-formula/</a><BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.statisticshowto.com/probability-and-statistics/correlation-coefficient-formula/\">https://www.statisticshowto.com/probability-and-statistics/correlation-coefficient-formula/</a><BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.howtoexcel.org/correlation-coefficient/\">https://www.howtoexcel.org/correlation-coefficient/</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "More Practice<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.algebra.com/algebra/homework/word/evaluation/Evaluation_Word_Problems.faq.question.1196643.html\">https://www.algebra.com/algebra/homework/word/evaluation/Evaluation_Word_Problems.faq.question.1196643.html</a><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );