{"id":2351,"date":"2021-10-11T19:37:06","date_gmt":"2021-10-11T19:37:06","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=2351"},"modified":"2023-12-05T09:48:55","modified_gmt":"2023-12-05T09:48:55","slug":"outliers-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/outliers-2\/","title":{"raw":"How does the outlier affect the best-fit line?","rendered":"How does the outlier affect the best-fit line?"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<section>\r\n<ul>\r\n \t<li>Identify outliers numerically by comparing residuals to two standard errors<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<div><section id=\"eip-212\" data-depth=\"1\">\r\n<p class=\" \"><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">How does the outlier affect the best-fit line?<\/span><\/p>\r\n\r\n<\/section><section id=\"fs-idm166882672\" data-depth=\"1\">\r\n<p id=\"element-8021\" class=\" \">Numerically and graphically, we have identified the point (65, 175) as an outlier. We should re-examine the data for this point to see if there are any problems with the data. If there is an error, we should fix the error if possible, or delete the data. If the data is correct, we would leave it in the data set. For this problem, we will suppose that we examined the data and found that this outlier data was an error. Therefore, we will continue on and delete the outlier so that we can explore how it affects the results as a learning experience.<\/p>\r\n<p id=\"element-6691\" class=\"finger \"><strong>Compute a new best-fit line and correlation coefficient using the ten remaining points:<\/strong><\/p>\r\n<p class=\"finger \">On the TI-83, TI-83+, and TI-84+ calculators, delete the outlier from L1 and L2. Using the LinRegTTest, the new line of best fit and the correlation coefficient is:<\/p>\r\n<p id=\"element-669\" class=\" \">[latex]\\hat{y}[\/latex] = \u2013355.19 + 7.39<em data-effect=\"italics\">x<\/em>\u00a0and\u00a0<em data-effect=\"italics\">r<\/em>\u00a0= 0.9121<\/p>\r\n<p id=\"element-101\" class=\" \">The new line with\u00a0<em data-effect=\"italics\">r<\/em>\u00a0= 0.9121 is a stronger correlation than the original (<em data-effect=\"italics\">r<\/em>\u00a0= 0.6631) because\u00a0<em data-effect=\"italics\">r<\/em>\u00a0= 0.9121 is closer to one. This means that the new line is a better fit for the ten remaining data values. The line can better predict the final exam score given the third exam score.<\/p>\r\n\r\n<\/section><section id=\"fs-idm24143888\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Numerical identification of outliers: Calculating\u00a0<em data-effect=\"italics\">s<\/em>\u00a0and finding outliers manually<\/h2>\r\n<p id=\"eip-819\" class=\" \">If you do not have the function LinRegTTest, then you can calculate the outlier in the first example by doing the following.<span data-type=\"newline\">\r\n<\/span><\/p>\r\n<p id=\"eip-819\" class=\" \">First,\u00a0<strong>square each |<em data-effect=\"italics\">y<\/em>\u00a0\u2013\u00a0<em data-effect=\"italics\">\u0177\u00a0<\/em>|<\/strong><\/p>\r\n<p id=\"eip-152\" class=\" \">The squares are\u00a0<span id=\"list1\" data-type=\"list\" data-list-type=\"labeled-item\" data-display=\"inline\"><span data-type=\"item\">35<sup>2<\/sup><span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">17<sup>2<\/sup><span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">16<sup>2<\/sup><span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">6<sup>2<\/sup><span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">19<sup>2<\/sup><span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">9<sup>2<\/sup><span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">3<sup>2<\/sup><span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">1<sup>2<\/sup><span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">10<sup>2<\/sup><span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">9<sup>2<\/sup><span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">1<sup>2<\/sup><\/span><\/span><\/p>\r\n<p id=\"eip-912\" class=\" \"><strong>Then, add (sum) all the |<em data-effect=\"italics\">y<\/em>\u00a0\u2013\u00a0<em data-effect=\"italics\">\u0177\u00a0<\/em>| squared terms<\/strong>\u00a0using the formula<\/p>\r\n[latex]{\\sum_{i=1}^{11}}{\\left ( |{y_i} - {\\hat y_i}| \\right )^2}[\/latex] = [latex]{\\sum_{i=1}^{11}}{\\epsilon_i}^2[\/latex] (Recall that [latex]{y_i} - {\\hat y_i}[\/latex] = [latex]{\\epsilon_i}[\/latex].\r\n<p id=\"eip-977\" class=\" \">= 35<sup>2<\/sup>\u00a0+ 17<sup>2<\/sup>\u00a0+ 16<sup>2<\/sup>\u00a0+ 6<sup>2<\/sup>\u00a0+ 19<sup>2<\/sup>\u00a0+ 9<sup>2<\/sup>\u00a0+ 3<sup>2<\/sup>\u00a0+ 1<sup>2<\/sup>\u00a0+ 10<sup>2<\/sup>\u00a0+ 9<sup>2<\/sup>\u00a0+ 1<sup>2<\/sup><\/p>\r\n<p id=\"eip-684\" class=\" \">= 2440 =\u00a0<strong>SSE<\/strong>. The result,\u00a0<strong>SSE<\/strong>\u00a0is the Sum of Squared Errors.<\/p>\r\n<p id=\"eip-961\" class=\" \"><strong>Next, calculate\u00a0<em data-effect=\"italics\">s<\/em>, the standard deviation of all the\u00a0<em data-effect=\"italics\">y<\/em>\u00a0\u2013\u00a0<em data-effect=\"italics\">\u0177<\/em>\u00a0=\u00a0<em data-effect=\"italics\">\u03b5<\/em>\u00a0values where\u00a0<em data-effect=\"italics\">n<\/em>\u00a0= the total number of data points.<\/strong><\/p>\r\nThe calculation is [latex]s = {\\sqrt{\\dfrac{SSE}{n - 2}}}[\/latex].\r\n\r\nFor the third exam\/final exam problem,\u00a0[latex]s = {\\sqrt{\\frac{2440}{11 - 2}}} = 16.47[\/latex].\r\n<p id=\"eip-799\" class=\" \">Next, multiply\u00a0<em data-effect=\"italics\">s<\/em>\u00a0by 2:<span data-type=\"newline\">\r\n<\/span>(2)(16.47) = 32.94<span data-type=\"newline\">\r\n<\/span>32.94 is 2 standard deviations away from the mean of the\u00a0<em data-effect=\"italics\">y<\/em>\u00a0\u2013\u00a0<em>[latex]\\hat {y}[\/latex]<\/em>\u00a0values.<\/p>\r\n<p id=\"eip-28\" class=\" \">If we were to measure the vertical distance from any data point to the corresponding point on the line of best fit and that distance is at least 2<em data-effect=\"italics\">s<\/em>, then we would consider the data point to be \"too far\" from the line of best fit. We call that point a\u00a0<strong><span id=\"term228\" data-type=\"term\">potential outlier<\/span><\/strong>.<\/p>\r\n<p id=\"eip-130\" class=\" \">For the example, if any of the |<em data-effect=\"italics\">y<\/em>\u00a0\u2013\u00a0<em>[latex]\\hat {y}[\/latex]<\/em>| values are\u00a0<strong>at least<\/strong>\u00a032.94, the corresponding (<em data-effect=\"italics\">x<\/em>,\u00a0<em data-effect=\"italics\">y<\/em>) data point is a potential outlier.<\/p>\r\n<p id=\"eip-696\" class=\" \">For the third exam\/final exam problem, all the |<em data-effect=\"italics\">y<\/em>\u00a0\u2013\u00a0<em>[latex]\\hat {y}[\/latex]<\/em>|'s are less than 31.29 except for the first one which is 35.<\/p>\r\n<p id=\"eip-994\" class=\" \">35 &gt; 31.29 That is, |<em data-effect=\"italics\">y<\/em>\u00a0\u2013\u00a0<em>[latex]\\hat {y}[\/latex]<\/em>| \u2265 (2)(s)<\/p>\r\n<p id=\"eip-980\" class=\" \">The point which corresponds to |<em data-effect=\"italics\">y<\/em>\u00a0\u2013\u00a0<em>[latex]\\hat {y}[\/latex]<\/em>| = 35 is (65, 175).\u00a0<strong>Therefore, the data point (65,175) is a potential outlier.<\/strong>\u00a0For this example, we will delete it. (Remember, we do not always delete an outlier).<\/p>\r\n\r\n<h4>Note<\/h4>\r\n<div id=\"fs-idp12863088\" class=\"ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-idp101044864\" class=\" \">When outliers are deleted, the researcher should either record that data was deleted, and why, or the researcher should provide results both with and without the deleted data. If data is erroneous and the correct values are known (e.g., student one actually scored a 70 instead of a 65), then this correction can be made to the data.<\/p>\r\n<p id=\"fs-idm274617664\" class=\" \">The next step is to compute a new best-fit line using the ten remaining points. The new line of best fit and the correlation coefficient is:<\/p>\r\n<p id=\"eip-451\" class=\" \"><em>[latex]\\hat{y}[\/latex]<\/em>\u00a0= \u2013355.19 + 7.39<em data-effect=\"italics\">x<\/em>\u00a0and\u00a0<em data-effect=\"italics\">r<\/em>\u00a0= 0.9121<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div class=\"textbox exercises\">\r\n<h3>Example 2<\/h3>\r\n<div id=\"id1171155161094\" data-type=\"problem\">\r\n<div class=\"os-problem-container \">\r\n<p id=\"element-802\" class=\" \">Using this new line of best fit (based on the remaining ten data points in the\u00a0<a href=\"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/the-regression-equation\/\" target=\"_blank\" rel=\"noopener\" data-page-slug=\"12-3-the-regression-equation\" data-page-uuid=\"fd60680f-dbb7-4a97-84b8-f352c3a6c141\" data-page-fragment=\"element-22\">third exam\/final exam example<\/a>\u00a0(example 2)), what would a student who receives a 73 on the third exam expect to receive on the final exam? Is this the same as the prediction made using the original line?\r\n[reveal-answer q=\"285238\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"285238\"]<\/p>\r\nUsing the new line of best fit, <em>[latex]\\hat{y}[\/latex]\u00a0<\/em>= \u2013355.19 + 7.39(73) = 184.28. A student who scored 73 points on the third exam would expect to earn 184 points on the final exam.<span data-type=\"newline\">\r\n<\/span><span data-type=\"newline\">\r\n<\/span>The original line predicted\u00a0<em>[latex]\\hat{y}[\/latex]<\/em>\u00a0= \u2013173.51 + 4.83(73) = 179.08 so the prediction using the new line with the outlier eliminated differs from the original prediction.\r\n<p class=\" \">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it 2<\/h3>\r\n<div id=\"fs-idp77690080\" class=\"statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"eip-846\" class=\" unnumbered\" data-type=\"exercise\"><section>\r\n<div id=\"eip-372\" data-type=\"problem\">\r\n<div class=\"os-problem-container \">\r\n<p id=\"eip-353\" class=\" \">The data points for the graph from the\u00a0<a href=\"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/the-regression-equation\/\" target=\"_blank\" rel=\"noopener\" data-page-slug=\"12-3-the-regression-equation\" data-page-uuid=\"fd60680f-dbb7-4a97-84b8-f352c3a6c141\" data-page-fragment=\"element-22\">third exam\/final exam example<\/a>\u00a0(example 2 on the <a href=\"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/the-regression-equation\/\" target=\"_blank\" rel=\"noopener\">page linked<\/a>) are as follows: (1, 5), (2, 7), (2, 6), (3, 9), (4, 12), (4, 13), (5, 18), (6, 19), (7, 12), and (7, 21). Remove the outlier and recalculate the line of best fit. Find the value of\u00a0<em>[latex]\\hat{y}[\/latex]<\/em>\u00a0when\u00a0<em data-effect=\"italics\">x<\/em>\u00a0= 10.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example 3<\/h3>\r\nThe Consumer Price Index (CPI) measures the average change over time in the prices paid by urban consumers for consumer goods and services. The CPI affects nearly all Americans because of the many ways it is used. One of its biggest uses is as a measure of inflation. By providing information about price changes in the Nation's economy to government, business, and labor, the CPI helps them to make economic decisions. The President, Congress, and the Federal Reserve Board use the CPI's trends to formulate monetary and fiscal policies. In the following table,\u00a0<em data-effect=\"italics\">x<\/em>\u00a0is the year and\u00a0<em data-effect=\"italics\">y<\/em>\u00a0is the CPI.\r\n<table style=\"border-collapse: collapse; width: 99.8836%;\" border=\"1\">\r\n<thead>\r\n<tr>\r\n<th style=\"width: 24.9127%;\" scope=\"col\" data-align=\"center\"><em data-effect=\"italics\">x<\/em><\/th>\r\n<th style=\"width: 22.8172%;\" scope=\"col\" data-align=\"center\"><em data-effect=\"italics\">y<\/em><\/th>\r\n<th style=\"width: 24.9127%;\" scope=\"col\" data-align=\"center\"><em data-effect=\"italics\">x<\/em><\/th>\r\n<th style=\"width: 27.241%;\" scope=\"col\" data-align=\"center\"><em data-effect=\"italics\">y<\/em><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 24.9127%;\">1915<\/td>\r\n<td style=\"width: 22.8172%;\">10.1<\/td>\r\n<td style=\"width: 24.9127%;\">1969<\/td>\r\n<td style=\"width: 27.241%;\">36.7<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 24.9127%;\">1926<\/td>\r\n<td style=\"width: 22.8172%;\">17.7<\/td>\r\n<td style=\"width: 24.9127%;\">1975<\/td>\r\n<td style=\"width: 27.241%;\">49.3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 24.9127%;\">1935<\/td>\r\n<td style=\"width: 22.8172%;\">13.7<\/td>\r\n<td style=\"width: 24.9127%;\">1979<\/td>\r\n<td style=\"width: 27.241%;\">72.6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 24.9127%;\">1940<\/td>\r\n<td style=\"width: 22.8172%;\">14.7<\/td>\r\n<td style=\"width: 24.9127%;\">1980<\/td>\r\n<td style=\"width: 27.241%;\">82.4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 24.9127%;\">1947<\/td>\r\n<td style=\"width: 22.8172%;\">24.1<\/td>\r\n<td style=\"width: 24.9127%;\">1986<\/td>\r\n<td style=\"width: 27.241%;\">109.6<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 24.9127%;\">1952<\/td>\r\n<td style=\"width: 22.8172%;\">26.5<\/td>\r\n<td style=\"width: 24.9127%;\">1991<\/td>\r\n<td style=\"width: 27.241%;\">130.7<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 24.9127%;\">1964<\/td>\r\n<td style=\"width: 22.8172%;\">31.0<\/td>\r\n<td style=\"width: 24.9127%;\">1999<\/td>\r\n<td style=\"width: 27.241%;\">166.6<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol>\r\n \t<li>Draw a scatterplot of the data.<\/li>\r\n \t<li>Calculate the least-squares line. Write the equation in the form\u00a0<em data-effect=\"italics\">\u0177<\/em>\u00a0=\u00a0<em data-effect=\"italics\">a<\/em>\u00a0+\u00a0<em data-effect=\"italics\">bx<\/em>.<\/li>\r\n \t<li>Draw the line on the scatterplot.<\/li>\r\n \t<li>Find the correlation coefficient. Is it significant?<\/li>\r\n \t<li>What is the average CPI for the year 1990?<\/li>\r\n<\/ol>\r\n<h4>Note<\/h4>\r\n<div id=\"eip-413\" class=\"ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-idp122850352\" class=\" \">In the example, notice the pattern of the points compared to the line. Although the correlation coefficient is significant, the pattern in the scatterplot indicates that a curve would be a more appropriate model to use than a line. In this example, a statistician should prefer to use other methods to fit a curve to this data, rather than model the data with the line we found. In addition to doing the calculations, it is always important to look at the scatterplot when deciding whether a linear model is appropriate.<\/p>\r\n\r\n<div id=\"element-986\" class=\"ui-has-child-title\" data-type=\"example\"><section>\r\n<div class=\"body\">\r\n<p id=\"eip-983\" class=\" \">If you are interested in seeing more years of data, visit the Bureau of Labor Statistics CPI website ftp:\/\/ftp.bls.gov\/pub\/special.requests\/cpi\/cpiai.txt; our data is taken from the column entitled \"Annual Avg.\" (third column from the right). For example you could add more current years of data. Try adding the more recent years: 2004: CPI = 188.9; 2008: CPI = 215.3; 2011: CPI = 224.9. See how it affects the model. (Check:\u00a0<em>[latex]\\hat{y}[\/latex]<\/em>\u00a0= \u20134436 + 2.295<em data-effect=\"italics\">x<\/em>;\u00a0<em data-effect=\"italics\">r<\/em>\u00a0= 0.9018. Is\u00a0<em data-effect=\"italics\">r<\/em>\u00a0significant? Is the fit better with the addition of the new points?)\r\n[reveal-answer q=\"732815\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"732815\"]<\/p>\r\n\r\n<ol>\r\n \t<li><img class=\"alignnone wp-image-1603\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/07\/25225800\/Example-21-300x137.jpeg\" alt=\"Scatter plot and line of best fit of the consumer price index data, on the y-axis, and year data, on the x-axis.\" width=\"500\" height=\"228\" \/><\/li>\r\n \t<li><em>[latex]\\hat{y}[\/latex]<\/em> = \u20133204 + 1.662<em data-effect=\"italics\">x<\/em>\u00a0is the equation of the line of best fit.<\/li>\r\n \t<li><em data-effect=\"italics\">r<\/em>\u00a0= 0.8694<\/li>\r\n \t<li>The number of data points is\u00a0<em data-effect=\"italics\">n<\/em>\u00a0= 14. Use the 95% Critical Values of the Sample Correlation Coefficient table at the end of Chapter 12.\u00a0<em data-effect=\"italics\">n<\/em>\u00a0\u2013 2 = 12. The corresponding critical value is 0.532. Since 0.8694 &gt; 0.532,\u00a0<em data-effect=\"italics\">r<\/em>\u00a0is significant.<span data-type=\"newline\">\r\n<\/span><em>[latex]\\hat{y}[\/latex]<\/em> = \u20133204 + 1.662(1990) = 103.4 CPI<\/li>\r\n \t<li>Using the calculator LinRegTTest, we find that\u00a0<em data-effect=\"italics\">s<\/em>\u00a0= 25.4 ; graphing the lines Y2 = \u20133204 + 1.662X \u2013 2(25.4) and Y3 = \u20133204 + 1.662X + 2(25.4) shows that no data values are outside those lines, identifying no outliers. (Note that the year 1999 was very close to the upper line, but still inside it).<\/li>\r\n<\/ol>\r\n<p class=\" \">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it 3<\/h3>\r\n<p id=\"eip-987\" class=\" \">The following table shows economic development measured in per capita income PCINC.<\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 99.8836%;\" border=\"1\">\r\n<thead>\r\n<tr>\r\n<th style=\"width: 21.8859%;\" scope=\"col\">Year<\/th>\r\n<th style=\"width: 28.0559%;\" scope=\"col\">PCINC<\/th>\r\n<th style=\"width: 21.8859%;\" scope=\"col\">Year<\/th>\r\n<th style=\"width: 28.0559%;\" scope=\"col\">PCINC<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 21.8859%;\">1870<\/td>\r\n<td style=\"width: 28.0559%;\">340<\/td>\r\n<td style=\"width: 21.8859%;\">1920<\/td>\r\n<td style=\"width: 28.0559%;\">1050<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 21.8859%;\">1880<\/td>\r\n<td style=\"width: 28.0559%;\">499<\/td>\r\n<td style=\"width: 21.8859%;\">1930<\/td>\r\n<td style=\"width: 28.0559%;\">1170<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 21.8859%;\">1890<\/td>\r\n<td style=\"width: 28.0559%;\">592<\/td>\r\n<td style=\"width: 21.8859%;\">1940<\/td>\r\n<td style=\"width: 28.0559%;\">1364<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 21.8859%;\">1900<\/td>\r\n<td style=\"width: 28.0559%;\">757<\/td>\r\n<td style=\"width: 21.8859%;\">1950<\/td>\r\n<td style=\"width: 28.0559%;\">1836<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 21.8859%;\">1910<\/td>\r\n<td style=\"width: 28.0559%;\">927<\/td>\r\n<td style=\"width: 21.8859%;\">1960<\/td>\r\n<td style=\"width: 28.0559%;\">2132<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol>\r\n \t<li>What are the independent and dependent variables?<\/li>\r\n \t<li>Draw a scatter plot.<\/li>\r\n \t<li>Use regression to find the line of best fit and the correlation coefficient.<\/li>\r\n \t<li>Interpret the significance of the correlation coefficient.<\/li>\r\n \t<li>Is there a linear relationship between the variables?<\/li>\r\n \t<li>Find the coefficient of determination and interpret it.<\/li>\r\n \t<li>What is the slope of the regression equation? What does it mean?<\/li>\r\n \t<li>Use the line of best fit to estimate PCINC for 1900, for 2000.<\/li>\r\n \t<li>Determine if there are any outliers.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<h2>95% critical values of the sample correlation coefficient table<\/h2>\r\n<table style=\"border-collapse: collapse; width: 99.8908%; height: 182px;\" border=\"1\">\r\n<thead>\r\n<tr style=\"height: 14px;\">\r\n<th style=\"width: 49.8908%; height: 14px;\" scope=\"col\">Degrees of Freedom:\u00a0<em data-effect=\"italics\">n<\/em>\u00a0\u2013 2<\/th>\r\n<th style=\"width: 50%; height: 14px;\" scope=\"col\">Critical Values: (+ and \u2013)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 49.8908%; height: 14px;\" data-align=\"center\">1<\/td>\r\n<td style=\"width: 50%; height: 14px;\" data-align=\"center\">0.997<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 49.8908%; height: 14px;\" data-align=\"center\">2<\/td>\r\n<td style=\"width: 50%; height: 14px;\" data-align=\"center\">0.950<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 49.8908%; height: 14px;\" data-align=\"center\">3<\/td>\r\n<td style=\"width: 50%; height: 14px;\" data-align=\"center\">0.878<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 49.8908%; height: 14px;\" data-align=\"center\">4<\/td>\r\n<td style=\"width: 50%; height: 14px;\" data-align=\"center\">0.811<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 49.8908%; height: 14px;\" data-align=\"center\">5<\/td>\r\n<td style=\"width: 50%; height: 14px;\" data-align=\"center\">0.754<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 49.8908%; height: 14px;\" data-align=\"center\">6<\/td>\r\n<td style=\"width: 50%; height: 14px;\" data-align=\"center\">0.707<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 49.8908%; height: 14px;\" data-align=\"center\">7<\/td>\r\n<td style=\"width: 50%; height: 14px;\" data-align=\"center\">0.666<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 49.8908%; height: 14px;\" data-align=\"center\">8<\/td>\r\n<td style=\"width: 50%; height: 14px;\" data-align=\"center\">0.632<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">9<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.602<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">10<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.576<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">11<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.555<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">12<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.532<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">13<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.514<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">14<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.497<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">15<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.482<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">16<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.468<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">17<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.456<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">18<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.444<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">19<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.433<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">20<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.423<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">21<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.413<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">22<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.404<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">23<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.396<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">24<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.388<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">25<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.381<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">26<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.374<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">27<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.367<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">28<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.361<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">29<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.355<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">30<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.349<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">40<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.304<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">50<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.273<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">60<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.250<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">70<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.232<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">80<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.217<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">90<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.205<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.8908%;\" data-align=\"center\">100<\/td>\r\n<td style=\"width: 50%;\" data-align=\"center\">0.195<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<section>\n<ul>\n<li>Identify outliers numerically by comparing residuals to two standard errors<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<div>\n<section id=\"eip-212\" data-depth=\"1\">\n<p class=\"\"><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">How does the outlier affect the best-fit line?<\/span><\/p>\n<\/section>\n<section id=\"fs-idm166882672\" data-depth=\"1\">\n<p id=\"element-8021\" class=\"\">Numerically and graphically, we have identified the point (65, 175) as an outlier. We should re-examine the data for this point to see if there are any problems with the data. If there is an error, we should fix the error if possible, or delete the data. If the data is correct, we would leave it in the data set. For this problem, we will suppose that we examined the data and found that this outlier data was an error. Therefore, we will continue on and delete the outlier so that we can explore how it affects the results as a learning experience.<\/p>\n<p id=\"element-6691\" class=\"finger\"><strong>Compute a new best-fit line and correlation coefficient using the ten remaining points:<\/strong><\/p>\n<p class=\"finger\">On the TI-83, TI-83+, and TI-84+ calculators, delete the outlier from L1 and L2. Using the LinRegTTest, the new line of best fit and the correlation coefficient is:<\/p>\n<p id=\"element-669\" class=\"\">[latex]\\hat{y}[\/latex] = \u2013355.19 + 7.39<em data-effect=\"italics\">x<\/em>\u00a0and\u00a0<em data-effect=\"italics\">r<\/em>\u00a0= 0.9121<\/p>\n<p id=\"element-101\" class=\"\">The new line with\u00a0<em data-effect=\"italics\">r<\/em>\u00a0= 0.9121 is a stronger correlation than the original (<em data-effect=\"italics\">r<\/em>\u00a0= 0.6631) because\u00a0<em data-effect=\"italics\">r<\/em>\u00a0= 0.9121 is closer to one. This means that the new line is a better fit for the ten remaining data values. The line can better predict the final exam score given the third exam score.<\/p>\n<\/section>\n<section id=\"fs-idm24143888\" data-depth=\"1\">\n<h2 data-type=\"title\">Numerical identification of outliers: Calculating\u00a0<em data-effect=\"italics\">s<\/em>\u00a0and finding outliers manually<\/h2>\n<p id=\"eip-819\" class=\"\">If you do not have the function LinRegTTest, then you can calculate the outlier in the first example by doing the following.<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<p id=\"eip-819\" class=\"\">First,\u00a0<strong>square each |<em data-effect=\"italics\">y<\/em>\u00a0\u2013\u00a0<em data-effect=\"italics\">\u0177\u00a0<\/em>|<\/strong><\/p>\n<p id=\"eip-152\" class=\"\">The squares are\u00a0<span id=\"list1\" data-type=\"list\" data-list-type=\"labeled-item\" data-display=\"inline\"><span data-type=\"item\">35<sup>2<\/sup><span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">17<sup>2<\/sup><span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">16<sup>2<\/sup><span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">6<sup>2<\/sup><span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">19<sup>2<\/sup><span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">9<sup>2<\/sup><span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">3<sup>2<\/sup><span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">1<sup>2<\/sup><span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">10<sup>2<\/sup><span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">9<sup>2<\/sup><span class=\"-os-inline-list-separator\">;\u00a0<\/span><\/span><span data-type=\"item\">1<sup>2<\/sup><\/span><\/span><\/p>\n<p id=\"eip-912\" class=\"\"><strong>Then, add (sum) all the |<em data-effect=\"italics\">y<\/em>\u00a0\u2013\u00a0<em data-effect=\"italics\">\u0177\u00a0<\/em>| squared terms<\/strong>\u00a0using the formula<\/p>\n<p>[latex]{\\sum_{i=1}^{11}}{\\left ( |{y_i} - {\\hat y_i}| \\right )^2}[\/latex] = [latex]{\\sum_{i=1}^{11}}{\\epsilon_i}^2[\/latex] (Recall that [latex]{y_i} - {\\hat y_i}[\/latex] = [latex]{\\epsilon_i}[\/latex].<\/p>\n<p id=\"eip-977\" class=\"\">= 35<sup>2<\/sup>\u00a0+ 17<sup>2<\/sup>\u00a0+ 16<sup>2<\/sup>\u00a0+ 6<sup>2<\/sup>\u00a0+ 19<sup>2<\/sup>\u00a0+ 9<sup>2<\/sup>\u00a0+ 3<sup>2<\/sup>\u00a0+ 1<sup>2<\/sup>\u00a0+ 10<sup>2<\/sup>\u00a0+ 9<sup>2<\/sup>\u00a0+ 1<sup>2<\/sup><\/p>\n<p id=\"eip-684\" class=\"\">= 2440 =\u00a0<strong>SSE<\/strong>. The result,\u00a0<strong>SSE<\/strong>\u00a0is the Sum of Squared Errors.<\/p>\n<p id=\"eip-961\" class=\"\"><strong>Next, calculate\u00a0<em data-effect=\"italics\">s<\/em>, the standard deviation of all the\u00a0<em data-effect=\"italics\">y<\/em>\u00a0\u2013\u00a0<em data-effect=\"italics\">\u0177<\/em>\u00a0=\u00a0<em data-effect=\"italics\">\u03b5<\/em>\u00a0values where\u00a0<em data-effect=\"italics\">n<\/em>\u00a0= the total number of data points.<\/strong><\/p>\n<p>The calculation is [latex]s = {\\sqrt{\\dfrac{SSE}{n - 2}}}[\/latex].<\/p>\n<p>For the third exam\/final exam problem,\u00a0[latex]s = {\\sqrt{\\frac{2440}{11 - 2}}} = 16.47[\/latex].<\/p>\n<p id=\"eip-799\" class=\"\">Next, multiply\u00a0<em data-effect=\"italics\">s<\/em>\u00a0by 2:<span data-type=\"newline\"><br \/>\n<\/span>(2)(16.47) = 32.94<span data-type=\"newline\"><br \/>\n<\/span>32.94 is 2 standard deviations away from the mean of the\u00a0<em data-effect=\"italics\">y<\/em>\u00a0\u2013\u00a0<em>[latex]\\hat {y}[\/latex]<\/em>\u00a0values.<\/p>\n<p id=\"eip-28\" class=\"\">If we were to measure the vertical distance from any data point to the corresponding point on the line of best fit and that distance is at least 2<em data-effect=\"italics\">s<\/em>, then we would consider the data point to be &#8220;too far&#8221; from the line of best fit. We call that point a\u00a0<strong><span id=\"term228\" data-type=\"term\">potential outlier<\/span><\/strong>.<\/p>\n<p id=\"eip-130\" class=\"\">For the example, if any of the |<em data-effect=\"italics\">y<\/em>\u00a0\u2013\u00a0<em>[latex]\\hat {y}[\/latex]<\/em>| values are\u00a0<strong>at least<\/strong>\u00a032.94, the corresponding (<em data-effect=\"italics\">x<\/em>,\u00a0<em data-effect=\"italics\">y<\/em>) data point is a potential outlier.<\/p>\n<p id=\"eip-696\" class=\"\">For the third exam\/final exam problem, all the |<em data-effect=\"italics\">y<\/em>\u00a0\u2013\u00a0<em>[latex]\\hat {y}[\/latex]<\/em>|&#8217;s are less than 31.29 except for the first one which is 35.<\/p>\n<p id=\"eip-994\" class=\"\">35 &gt; 31.29 That is, |<em data-effect=\"italics\">y<\/em>\u00a0\u2013\u00a0<em>[latex]\\hat {y}[\/latex]<\/em>| \u2265 (2)(s)<\/p>\n<p id=\"eip-980\" class=\"\">The point which corresponds to |<em data-effect=\"italics\">y<\/em>\u00a0\u2013\u00a0<em>[latex]\\hat {y}[\/latex]<\/em>| = 35 is (65, 175).\u00a0<strong>Therefore, the data point (65,175) is a potential outlier.<\/strong>\u00a0For this example, we will delete it. (Remember, we do not always delete an outlier).<\/p>\n<h4>Note<\/h4>\n<div id=\"fs-idp12863088\" class=\"ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-idp101044864\" class=\"\">When outliers are deleted, the researcher should either record that data was deleted, and why, or the researcher should provide results both with and without the deleted data. If data is erroneous and the correct values are known (e.g., student one actually scored a 70 instead of a 65), then this correction can be made to the data.<\/p>\n<p id=\"fs-idm274617664\" class=\"\">The next step is to compute a new best-fit line using the ten remaining points. The new line of best fit and the correlation coefficient is:<\/p>\n<p id=\"eip-451\" class=\"\"><em>[latex]\\hat{y}[\/latex]<\/em>\u00a0= \u2013355.19 + 7.39<em data-effect=\"italics\">x<\/em>\u00a0and\u00a0<em data-effect=\"italics\">r<\/em>\u00a0= 0.9121<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div class=\"textbox exercises\">\n<h3>Example 2<\/h3>\n<div id=\"id1171155161094\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"element-802\" class=\"\">Using this new line of best fit (based on the remaining ten data points in the\u00a0<a href=\"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/the-regression-equation\/\" target=\"_blank\" rel=\"noopener\" data-page-slug=\"12-3-the-regression-equation\" data-page-uuid=\"fd60680f-dbb7-4a97-84b8-f352c3a6c141\" data-page-fragment=\"element-22\">third exam\/final exam example<\/a>\u00a0(example 2)), what would a student who receives a 73 on the third exam expect to receive on the final exam? Is this the same as the prediction made using the original line?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q285238\">Show Answer<\/span><\/p>\n<div id=\"q285238\" class=\"hidden-answer\" style=\"display: none\">\n<p>Using the new line of best fit, <em>[latex]\\hat{y}[\/latex]\u00a0<\/em>= \u2013355.19 + 7.39(73) = 184.28. A student who scored 73 points on the third exam would expect to earn 184 points on the final exam.<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span>The original line predicted\u00a0<em>[latex]\\hat{y}[\/latex]<\/em>\u00a0= \u2013173.51 + 4.83(73) = 179.08 so the prediction using the new line with the outlier eliminated differs from the original prediction.<\/p>\n<p class=\"\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it 2<\/h3>\n<div id=\"fs-idp77690080\" class=\"statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<section>\n<div class=\"os-note-body\">\n<div id=\"eip-846\" class=\"unnumbered\" data-type=\"exercise\">\n<section>\n<div id=\"eip-372\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"eip-353\" class=\"\">The data points for the graph from the\u00a0<a href=\"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/the-regression-equation\/\" target=\"_blank\" rel=\"noopener\" data-page-slug=\"12-3-the-regression-equation\" data-page-uuid=\"fd60680f-dbb7-4a97-84b8-f352c3a6c141\" data-page-fragment=\"element-22\">third exam\/final exam example<\/a>\u00a0(example 2 on the <a href=\"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/the-regression-equation\/\" target=\"_blank\" rel=\"noopener\">page linked<\/a>) are as follows: (1, 5), (2, 7), (2, 6), (3, 9), (4, 12), (4, 13), (5, 18), (6, 19), (7, 12), and (7, 21). Remove the outlier and recalculate the line of best fit. Find the value of\u00a0<em>[latex]\\hat{y}[\/latex]<\/em>\u00a0when\u00a0<em data-effect=\"italics\">x<\/em>\u00a0= 10.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example 3<\/h3>\n<p>The Consumer Price Index (CPI) measures the average change over time in the prices paid by urban consumers for consumer goods and services. The CPI affects nearly all Americans because of the many ways it is used. One of its biggest uses is as a measure of inflation. By providing information about price changes in the Nation&#8217;s economy to government, business, and labor, the CPI helps them to make economic decisions. The President, Congress, and the Federal Reserve Board use the CPI&#8217;s trends to formulate monetary and fiscal policies. In the following table,\u00a0<em data-effect=\"italics\">x<\/em>\u00a0is the year and\u00a0<em data-effect=\"italics\">y<\/em>\u00a0is the CPI.<\/p>\n<table style=\"border-collapse: collapse; width: 99.8836%;\">\n<thead>\n<tr>\n<th style=\"width: 24.9127%;\" scope=\"col\" data-align=\"center\"><em data-effect=\"italics\">x<\/em><\/th>\n<th style=\"width: 22.8172%;\" scope=\"col\" data-align=\"center\"><em data-effect=\"italics\">y<\/em><\/th>\n<th style=\"width: 24.9127%;\" scope=\"col\" data-align=\"center\"><em data-effect=\"italics\">x<\/em><\/th>\n<th style=\"width: 27.241%;\" scope=\"col\" data-align=\"center\"><em data-effect=\"italics\">y<\/em><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"width: 24.9127%;\">1915<\/td>\n<td style=\"width: 22.8172%;\">10.1<\/td>\n<td style=\"width: 24.9127%;\">1969<\/td>\n<td style=\"width: 27.241%;\">36.7<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 24.9127%;\">1926<\/td>\n<td style=\"width: 22.8172%;\">17.7<\/td>\n<td style=\"width: 24.9127%;\">1975<\/td>\n<td style=\"width: 27.241%;\">49.3<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 24.9127%;\">1935<\/td>\n<td style=\"width: 22.8172%;\">13.7<\/td>\n<td style=\"width: 24.9127%;\">1979<\/td>\n<td style=\"width: 27.241%;\">72.6<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 24.9127%;\">1940<\/td>\n<td style=\"width: 22.8172%;\">14.7<\/td>\n<td style=\"width: 24.9127%;\">1980<\/td>\n<td style=\"width: 27.241%;\">82.4<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 24.9127%;\">1947<\/td>\n<td style=\"width: 22.8172%;\">24.1<\/td>\n<td style=\"width: 24.9127%;\">1986<\/td>\n<td style=\"width: 27.241%;\">109.6<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 24.9127%;\">1952<\/td>\n<td style=\"width: 22.8172%;\">26.5<\/td>\n<td style=\"width: 24.9127%;\">1991<\/td>\n<td style=\"width: 27.241%;\">130.7<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 24.9127%;\">1964<\/td>\n<td style=\"width: 22.8172%;\">31.0<\/td>\n<td style=\"width: 24.9127%;\">1999<\/td>\n<td style=\"width: 27.241%;\">166.6<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol>\n<li>Draw a scatterplot of the data.<\/li>\n<li>Calculate the least-squares line. Write the equation in the form\u00a0<em data-effect=\"italics\">\u0177<\/em>\u00a0=\u00a0<em data-effect=\"italics\">a<\/em>\u00a0+\u00a0<em data-effect=\"italics\">bx<\/em>.<\/li>\n<li>Draw the line on the scatterplot.<\/li>\n<li>Find the correlation coefficient. Is it significant?<\/li>\n<li>What is the average CPI for the year 1990?<\/li>\n<\/ol>\n<h4>Note<\/h4>\n<div id=\"eip-413\" class=\"ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-idp122850352\" class=\"\">In the example, notice the pattern of the points compared to the line. Although the correlation coefficient is significant, the pattern in the scatterplot indicates that a curve would be a more appropriate model to use than a line. In this example, a statistician should prefer to use other methods to fit a curve to this data, rather than model the data with the line we found. In addition to doing the calculations, it is always important to look at the scatterplot when deciding whether a linear model is appropriate.<\/p>\n<div id=\"element-986\" class=\"ui-has-child-title\" data-type=\"example\">\n<section>\n<div class=\"body\">\n<p id=\"eip-983\" class=\"\">If you are interested in seeing more years of data, visit the Bureau of Labor Statistics CPI website ftp:\/\/ftp.bls.gov\/pub\/special.requests\/cpi\/cpiai.txt; our data is taken from the column entitled &#8220;Annual Avg.&#8221; (third column from the right). For example you could add more current years of data. Try adding the more recent years: 2004: CPI = 188.9; 2008: CPI = 215.3; 2011: CPI = 224.9. See how it affects the model. (Check:\u00a0<em>[latex]\\hat{y}[\/latex]<\/em>\u00a0= \u20134436 + 2.295<em data-effect=\"italics\">x<\/em>;\u00a0<em data-effect=\"italics\">r<\/em>\u00a0= 0.9018. Is\u00a0<em data-effect=\"italics\">r<\/em>\u00a0significant? Is the fit better with the addition of the new points?)<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q732815\">Show Answer<\/span><\/p>\n<div id=\"q732815\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1603\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/07\/25225800\/Example-21-300x137.jpeg\" alt=\"Scatter plot and line of best fit of the consumer price index data, on the y-axis, and year data, on the x-axis.\" width=\"500\" height=\"228\" \/><\/li>\n<li><em>[latex]\\hat{y}[\/latex]<\/em> = \u20133204 + 1.662<em data-effect=\"italics\">x<\/em>\u00a0is the equation of the line of best fit.<\/li>\n<li><em data-effect=\"italics\">r<\/em>\u00a0= 0.8694<\/li>\n<li>The number of data points is\u00a0<em data-effect=\"italics\">n<\/em>\u00a0= 14. Use the 95% Critical Values of the Sample Correlation Coefficient table at the end of Chapter 12.\u00a0<em data-effect=\"italics\">n<\/em>\u00a0\u2013 2 = 12. The corresponding critical value is 0.532. Since 0.8694 &gt; 0.532,\u00a0<em data-effect=\"italics\">r<\/em>\u00a0is significant.<span data-type=\"newline\"><br \/>\n<\/span><em>[latex]\\hat{y}[\/latex]<\/em> = \u20133204 + 1.662(1990) = 103.4 CPI<\/li>\n<li>Using the calculator LinRegTTest, we find that\u00a0<em data-effect=\"italics\">s<\/em>\u00a0= 25.4 ; graphing the lines Y2 = \u20133204 + 1.662X \u2013 2(25.4) and Y3 = \u20133204 + 1.662X + 2(25.4) shows that no data values are outside those lines, identifying no outliers. (Note that the year 1999 was very close to the upper line, but still inside it).<\/li>\n<\/ol>\n<p class=\"\"><\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it 3<\/h3>\n<p id=\"eip-987\" class=\"\">The following table shows economic development measured in per capita income PCINC.<\/p>\n<table style=\"border-collapse: collapse; width: 99.8836%;\">\n<thead>\n<tr>\n<th style=\"width: 21.8859%;\" scope=\"col\">Year<\/th>\n<th style=\"width: 28.0559%;\" scope=\"col\">PCINC<\/th>\n<th style=\"width: 21.8859%;\" scope=\"col\">Year<\/th>\n<th style=\"width: 28.0559%;\" scope=\"col\">PCINC<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"width: 21.8859%;\">1870<\/td>\n<td style=\"width: 28.0559%;\">340<\/td>\n<td style=\"width: 21.8859%;\">1920<\/td>\n<td style=\"width: 28.0559%;\">1050<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 21.8859%;\">1880<\/td>\n<td style=\"width: 28.0559%;\">499<\/td>\n<td style=\"width: 21.8859%;\">1930<\/td>\n<td style=\"width: 28.0559%;\">1170<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 21.8859%;\">1890<\/td>\n<td style=\"width: 28.0559%;\">592<\/td>\n<td style=\"width: 21.8859%;\">1940<\/td>\n<td style=\"width: 28.0559%;\">1364<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 21.8859%;\">1900<\/td>\n<td style=\"width: 28.0559%;\">757<\/td>\n<td style=\"width: 21.8859%;\">1950<\/td>\n<td style=\"width: 28.0559%;\">1836<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 21.8859%;\">1910<\/td>\n<td style=\"width: 28.0559%;\">927<\/td>\n<td style=\"width: 21.8859%;\">1960<\/td>\n<td style=\"width: 28.0559%;\">2132<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol>\n<li>What are the independent and dependent variables?<\/li>\n<li>Draw a scatter plot.<\/li>\n<li>Use regression to find the line of best fit and the correlation coefficient.<\/li>\n<li>Interpret the significance of the correlation coefficient.<\/li>\n<li>Is there a linear relationship between the variables?<\/li>\n<li>Find the coefficient of determination and interpret it.<\/li>\n<li>What is the slope of the regression equation? What does it mean?<\/li>\n<li>Use the line of best fit to estimate PCINC for 1900, for 2000.<\/li>\n<li>Determine if there are any outliers.<\/li>\n<\/ol>\n<\/div>\n<h2>95% critical values of the sample correlation coefficient table<\/h2>\n<table style=\"border-collapse: collapse; width: 99.8908%; height: 182px;\">\n<thead>\n<tr style=\"height: 14px;\">\n<th style=\"width: 49.8908%; height: 14px;\" scope=\"col\">Degrees of Freedom:\u00a0<em data-effect=\"italics\">n<\/em>\u00a0\u2013 2<\/th>\n<th style=\"width: 50%; height: 14px;\" scope=\"col\">Critical Values: (+ and \u2013)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 14px;\">\n<td style=\"width: 49.8908%; height: 14px;\" data-align=\"center\">1<\/td>\n<td style=\"width: 50%; height: 14px;\" data-align=\"center\">0.997<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 49.8908%; height: 14px;\" data-align=\"center\">2<\/td>\n<td style=\"width: 50%; height: 14px;\" data-align=\"center\">0.950<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 49.8908%; height: 14px;\" data-align=\"center\">3<\/td>\n<td style=\"width: 50%; height: 14px;\" data-align=\"center\">0.878<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 49.8908%; height: 14px;\" data-align=\"center\">4<\/td>\n<td style=\"width: 50%; height: 14px;\" data-align=\"center\">0.811<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 49.8908%; height: 14px;\" data-align=\"center\">5<\/td>\n<td style=\"width: 50%; height: 14px;\" data-align=\"center\">0.754<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 49.8908%; height: 14px;\" data-align=\"center\">6<\/td>\n<td style=\"width: 50%; height: 14px;\" data-align=\"center\">0.707<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 49.8908%; height: 14px;\" data-align=\"center\">7<\/td>\n<td style=\"width: 50%; height: 14px;\" data-align=\"center\">0.666<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 49.8908%; height: 14px;\" data-align=\"center\">8<\/td>\n<td style=\"width: 50%; height: 14px;\" data-align=\"center\">0.632<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">9<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.602<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">10<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.576<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">11<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.555<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">12<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.532<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">13<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.514<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">14<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.497<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">15<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.482<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">16<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.468<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">17<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.456<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">18<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.444<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">19<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.433<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">20<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.423<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">21<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.413<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">22<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.404<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">23<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.396<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">24<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.388<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">25<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.381<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">26<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.374<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">27<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.367<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">28<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.361<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">29<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.355<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">30<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.349<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">40<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.304<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">50<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.273<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">60<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.250<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">70<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.232<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">80<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.217<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">90<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.205<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.8908%;\" data-align=\"center\">100<\/td>\n<td style=\"width: 50%;\" data-align=\"center\">0.195<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-2351\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Measures of the Location of Data. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/2-3-measures-of-the-location-of-the-data\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/2-3-measures-of-the-location-of-the-data<\/a>. <strong>Project<\/strong>: Introductory Statistics. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/li><li>Introductory Statistics . <strong>Authored by<\/strong>: Barbara Illowsky, Susan Dean. <strong>Provided by<\/strong>: Open Stax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169134,"menu_order":26,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Measures of the Location of Data\",\"author\":\"\",\"organization\":\"\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/2-3-measures-of-the-location-of-the-data\",\"project\":\"Introductory Statistics\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"},{\"type\":\"cc\",\"description\":\"Introductory Statistics \",\"author\":\"Barbara Illowsky, Susan Dean\",\"organization\":\"Open Stax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at 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