{"id":553,"date":"2015-07-18T18:34:09","date_gmt":"2015-07-18T22:34:09","guid":{"rendered":"http:\/\/sites.berry.edu\/vbissonnette\/?page_id=553"},"modified":"2016-06-23T10:08:14","modified_gmt":"2016-06-23T14:08:14","slug":"correlation-solution","status":"publish","type":"page","link":"https:\/\/sites.berry.edu\/vbissonnette\/index\/stats-homework\/documentation\/correlation\/correlation-solution\/","title":{"rendered":"Correlation Solution"},"content":{"rendered":"

Example homework problem:<\/h3>\n

You work for an automotive magazine, and you are investigating the relationship between a car’s gas mileage (in miles-per-gallon) and the amount of horsepower produced by a car’s engine. You collect the following data:<\/p>\n\n\n\n\n\n
Automobile:<\/td>\n1<\/td>\n2<\/td>\n3<\/td>\n4<\/td>\n5<\/td>\n6<\/td>\n7<\/td>\n8<\/td>\n9<\/td>\n10<\/td>\n<\/tr>\n
Horsepower:<\/td>\n95<\/td>\n135<\/td>\n120<\/td>\n167<\/td>\n210<\/td>\n146<\/td>\n245<\/td>\n110<\/td>\n160<\/td>\n130<\/td>\n<\/tr>\n
MPG:<\/td>\n37<\/td>\n19<\/td>\n26<\/td>\n20<\/td>\n24<\/td>\n30<\/td>\n15<\/td>\n32<\/td>\n23<\/td>\n33<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Is there a significant correlation between horsepower and MPG (alpha = .05)?<\/p>\n

\n

The goal of your analysis is to assess the naturally-occurring relationship between these two variables. To do this, we will compute the Pearson Product Moment Correlation Coefficient<\/u>:<\/p>\n

\"corr1\"<\/a>The correlation is equal to the ratio of the covariance between your variables to the variance within your variables. n<\/i> is equal to the number of pairs of scores.<\/p>\n

The Covariance<\/u> between the two variables is equal to:<\/p>\n

\"corr2\"<\/a>where \u03a3XY is equal to the sum of the crossproducts: you multiply each pair of scores and then add up the products.<\/p>\n

Compute test statistic<\/b>. Begin by computing the crossproduct scores, and then compute \u03a3XY, \u03a3X, and \u03a3Y:<\/p>\n\n\n\n\n\n\n
Automobile:<\/td>\n1<\/td>\n2<\/td>\n3<\/td>\n4<\/td>\n5<\/td>\n6<\/td>\n7<\/td>\n8<\/td>\n9<\/td>\n10<\/td>\nSum<\/b><\/td>\n<\/tr>\n
Horsepower:<\/td>\n95<\/td>\n135<\/td>\n120<\/td>\n167<\/td>\n210<\/td>\n146<\/td>\n245<\/td>\n110<\/td>\n160<\/td>\n130<\/td>\n1518<\/b><\/td>\n<\/tr>\n
MPG:<\/td>\n37<\/td>\n19<\/td>\n26<\/td>\n20<\/td>\n24<\/td>\n30<\/td>\n15<\/td>\n32<\/td>\n23<\/td>\n33<\/td>\n259<\/b><\/td>\n<\/tr>\n
Crossproduct:<\/td>\n3515<\/td>\n2565<\/td>\n3120<\/td>\n3340<\/td>\n5040<\/td>\n4380<\/td>\n3675<\/td>\n3520<\/td>\n3680<\/td>\n4290<\/td>\n37125<\/b><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Here are the preliminary sums: \u03a3X = 1518, \u03a3Y = 259, and \u03a3XY = 37125. n<\/i> = 10.<\/p>\n

Compute the variance for each variable. You will find that the variance of X (horsepower) is equal to 2127.51, and the variance of Y (gas mileage) is equal to 48.99. If you would like help with these computations, see the documentation for descriptive statistics<\/a>. Now, compute the covariance between the two variables:<\/p>\n

\"corr3\"<\/a>Let’s pause here for a moment, and place these three statistics into a Variance \/ Covariance Matrix<\/u>:<\/p>\n\n\n\n\n\n
Variable:<\/td>\nHorsepower
\n(X)<\/td>\n		MPG
\n(Y)<\/td>\n<\/tr>\n
Horsepower (X):<\/td>\n2127.5111<\/td>\n-243.4667<\/td>\n<\/tr>\n
MPG (Y):<\/td>\n<\/td>\n48.9889<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

where the values along the diagonal of the matrix are the variances of X and Y, and the value off the diagonal is the covariance between X and Y. It is a good habit to construct a variance\/covariance matrix when you are working on correlation and regression problems.<\/p>\n

Now that you have the variances and the covariance, you will find that the correlation is equal to:<\/p>\n

\"corr4\"<\/a>Conduct hypothesis test<\/b>. Our correlation will have df<\/i> equal to the number of pairs of scores minus 2. In our case, we have 10 automobiles, so we would have df<\/i> = 10 – 2 = 8.<\/p>\n

Alpha was set at .05 and we will conduct a two-tailed test. When you consult your table of critical values<\/a> for r<\/i>, you will find that if our obtained value of r<\/i> is greater than .632, then we would conclude that the relationship between horsepower and gas mileage is significant — i.e., that r<\/i> is significantly different from zero.<\/p>\n

Since the obtained r<\/i> (-.75) is greater in absolute value than the critical r<\/i> (.632), we would conclude that there is a significant negative relationship between the horsepower level of the automobile and its gas mileage — i.e., the greater the horsepower, the lower the gas
\nmileage.<\/p>\n

If you are not testing r<\/i> directly in your class, then you are probably testing the significance of the correlation coefficient with the t<\/i> distribution:<\/p>\n

\"corr5\"<\/a>This t<\/i> test will have df<\/i> equal to n – 2, the same as r<\/i>. In our case, df<\/i> = 8. With alpha = .02 and assuming a two-tailed test, we would compare t<\/i> to a critical value<\/a> of\u00a02.306.<\/p>\n

Since our obtained t<\/i> (-3.25) is greater in absolute value than our critical value of t<\/i> (2.306), we would conclude that the correlation between horsepower and gas mileage was significant. Note that these two approaches are perfectly equivalent, and will always result in the same decision.<\/p>\n

Effect Size<\/b>. The correlation is an effective descriptive statistic in its own right. You know that it ranges from 0 to +1 and from 0 to -1, where 0 = no relationship and +1\/-1 = perfect positive\/negative relationship. In Statistical Power Analysis<\/i>, Cohen suggested that we interpret a correlation of .10 as a small effect, a correlation of .30 as a moderate effect, and a correlation of .50 or greater as a large effect.<\/p>\n

In addition, the square of the correlation (r<\/i>\u00b2) is a very useful effect size estimate called the Coefficient of Determination<\/u>. r<\/i>\u00b2 represents the proportion of variance in X and Y that is shared — i.e., the proportion of variance in one of the variables that can be accounted for by variation in the other variable. In our case, r<\/i>\u00b2 = -.7541\u00b2 = .57. Thus, if you were predicting gas milage from horsepower, you could account for 57% of the variance in gas milage by variations in horsepower.<\/p>\n

\n

Return to Correlation Procedure<\/a><\/p>\n

Return to Table of Contents<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

Example homework problem: You work for an automotive magazine, and you are investigating the relationship between a car’s gas mileage (in miles-per-gallon) and the amount of horsepower produced by a […]<\/p>\n","protected":false},"author":34,"featured_media":0,"parent":548,"menu_order":0,"comment_status":"open","ping_status":"open","template":"","meta":{"site-container-style":"default","site-container-layout":"default","site-sidebar-layout":"default","site-transparent-header":"default","disable-article-header":"default","disable-site-header":"default","disable-site-footer":"default","disable-content-area-spacing":"default","footnotes":""},"class_list":["post-553","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sites.berry.edu\/vbissonnette\/wp-json\/wp\/v2\/pages\/553","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.berry.edu\/vbissonnette\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sites.berry.edu\/vbissonnette\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sites.berry.edu\/vbissonnette\/wp-json\/wp\/v2\/users\/34"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.berry.edu\/vbissonnette\/wp-json\/wp\/v2\/comments?post=553"}],"version-history":[{"count":4,"href":"https:\/\/sites.berry.edu\/vbissonnette\/wp-json\/wp\/v2\/pages\/553\/revisions"}],"predecessor-version":[{"id":950,"href":"https:\/\/sites.berry.edu\/vbissonnette\/wp-json\/wp\/v2\/pages\/553\/revisions\/950"}],"up":[{"embeddable":true,"href":"https:\/\/sites.berry.edu\/vbissonnette\/wp-json\/wp\/v2\/pages\/548"}],"wp:attachment":[{"href":"https:\/\/sites.berry.edu\/vbissonnette\/wp-json\/wp\/v2\/media?parent=553"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}