{"id":183018,"date":"2025-01-15T10:43:42","date_gmt":"2025-01-15T10:43:42","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=183018"},"modified":"2025-01-15T10:43:44","modified_gmt":"2025-01-15T10:43:44","slug":"the-accompanying-minitab-regression-output-is-based-on-data-that-appeared-in-the-article-application-of-design-of-experiments-for-modeling-surface-roughness-in-ultrasonic-vibration-turning","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/15\/the-accompanying-minitab-regression-output-is-based-on-data-that-appeared-in-the-article-application-of-design-of-experiments-for-modeling-surface-roughness-in-ultrasonic-vibration-turning\/","title":{"rendered":"The accompanying Minitab regression output is based on data that appeared in the article \u201cApplication of Design of Experiments for Modeling Surface Roughness in Ultrasonic Vibration Turning\u201d (J. of Engr. Manuf., 2009: 641\u2013652)"},"content":{"rendered":"\n<p>The accompanying Minitab regression output is based on data that appeared in the article \u201cApplication of Design of Experiments for Modeling Surface Roughness in Ultrasonic Vibration Turning\u201d (J. of Engr. Manuf., 2009: 641\u2013652). The response variable is surface roughness (mm), and the independent variables are vibration amplitude (mm), depth of cut (mm), feed rate (mm\/rev), and cutting speed (m\/min), respectively<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/01\/image-229.png\" alt=\"\" class=\"wp-image-183019\"\/><\/figure>\n\n\n\n<p>a. Predict the value of surface roughness when amplitude is 10, depth of cut is .5, feed rate is .25, and cutting speed is 50.<\/p>\n\n\n\n<p>b. What proportion of observed variation in surface roughness can be explained by the approximate relationship between surface roughness and the four predictors?<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>Given the regression equation for predicting surface roughness (Ra) in ultrasonic vibration turning:<\/p>\n\n\n\n<p>Ra = -0.972 &#8211; 0.0312a + 0.557d + 18.3f + 0.00282v<\/p>\n\n\n\n<p>where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>a = vibration amplitude (mm)<\/li>\n\n\n\n<li>d = depth of cut (mm)<\/li>\n\n\n\n<li>f = feed rate (mm\/rev)<\/li>\n\n\n\n<li>v = cutting speed (m\/min)<\/li>\n<\/ul>\n\n\n\n<p><strong>a. Predicting Surface Roughness<\/strong><\/p>\n\n\n\n<p>To predict the surface roughness when the input parameters are:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Amplitude (a) = 10 mm<\/li>\n\n\n\n<li>Depth of cut (d) = 0.5 mm<\/li>\n\n\n\n<li>Feed rate (f) = 0.25 mm\/rev<\/li>\n\n\n\n<li>Cutting speed (v) = 50 m\/min<\/li>\n<\/ul>\n\n\n\n<p>Substitute these values into the regression equation:<\/p>\n\n\n\n<p>Ra = -0.972 &#8211; 0.0312(10) + 0.557(0.5) + 18.3(0.25) + 0.00282(50)<\/p>\n\n\n\n<p>Calculating each term:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Constant: -0.972<\/li>\n\n\n\n<li>Amplitude: -0.0312 \u00d7 10 = -0.312<\/li>\n\n\n\n<li>Depth of cut: 0.557 \u00d7 0.5 = 0.2785<\/li>\n\n\n\n<li>Feed rate: 18.3 \u00d7 0.25 = 4.575<\/li>\n\n\n\n<li>Cutting speed: 0.00282 \u00d7 50 = 0.141<\/li>\n<\/ul>\n\n\n\n<p>Summing these contributions:<\/p>\n\n\n\n<p>Ra = -0.972 &#8211; 0.312 + 0.2785 + 4.575 + 0.141<\/p>\n\n\n\n<p>Ra \u2248 3.7105 mm<\/p>\n\n\n\n<p>Therefore, the predicted surface roughness is approximately <strong>3.7105 mm<\/strong>.<\/p>\n\n\n\n<p><strong>b. Proportion of Variation Explained<\/strong><\/p>\n\n\n\n<p>The coefficient of determination, denoted as R\u00b2, indicates the proportion of the variance in the dependent variable (surface roughness) that is predictable from the independent variables.<\/p>\n\n\n\n<p>From the regression output:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>R\u00b2 = 88.6%<\/li>\n<\/ul>\n\n\n\n<p>This means that <strong>88.6%<\/strong> of the observed variation in surface roughness can be explained by the linear relationship with the four predictors: vibration amplitude, depth of cut, feed rate, and cutting speed.<\/p>\n\n\n\n<p><strong>Explanation<\/strong><\/p>\n\n\n\n<p>In multiple linear regression, R\u00b2 measures the goodness of fit of the model. An R\u00b2 value of 88.6% suggests that the model explains a substantial portion of the variability in surface roughness, indicating a strong relationship between the predictors and the response variable.<\/p>\n\n\n\n<p>However, it&#8217;s important to consider the adjusted R\u00b2, especially when multiple predictors are involved. The adjusted R\u00b2 accounts for the number of predictors relative to the number of data points and provides a more accurate measure of model performance.<\/p>\n\n\n\n<p>In this case, the adjusted R\u00b2 is 88.0%, which is slightly lower than the R\u00b2 but still indicates a strong model fit.<\/p>\n\n\n\n<p>While a high R\u00b2 value indicates that the model explains a large portion of the variance, it doesn&#8217;t guarantee that the model is appropriate or that the predictors are significant.<\/p>\n\n\n\n<p>Statistical tests, such as the F-test for overall model significance and t-tests for individual predictors, should be conducted to assess the validity and significance of the model and its predictors.<\/p>\n\n\n\n<p>Additionally, examining residual plots can help verify the assumptions of linearity, homoscedasticity, and normality, ensuring the reliability of the regression analysis.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The accompanying Minitab regression output is based on data that appeared in the article \u201cApplication of Design of Experiments for Modeling Surface Roughness in Ultrasonic Vibration Turning\u201d (J. of Engr. Manuf., 2009: 641\u2013652). The response variable is surface roughness (mm), and the independent variables are vibration amplitude (mm), depth of cut (mm), feed rate (mm\/rev), [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-183018","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/183018","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=183018"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/183018\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=183018"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=183018"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=183018"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}