{"id":245443,"date":"2025-07-06T08:45:36","date_gmt":"2025-07-06T08:45:36","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=245443"},"modified":"2025-07-06T08:45:39","modified_gmt":"2025-07-06T08:45:39","slug":"cumulative-review-worksheet-4-ch-1-7-statcrunch-file-for-this-worksheet-available-at-this-link","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/06\/cumulative-review-worksheet-4-ch-1-7-statcrunch-file-for-this-worksheet-available-at-this-link\/","title":{"rendered":"Cumulative Review Worksheet #4 (Ch 1-7) StatCrunch File for this Worksheet available at this link"},"content":{"rendered":"\n

Math 133: Cumulative Review WS #4 (Ch 1-7) Name: Coursepack, p. 424 Cumulative Review Worksheet #4 (Ch 1-7) StatCrunch File for this Worksheet available at this link 1. (15pts) The following data set gives the weight (in pounds) and the braking distance (in feet) of a vehicle traveling at a speed of 60 miles per hour. Find the following (Give 3 decimal places) x=Vehicle y=Braking Weight (lb) Distance (Feet) a) Correlation Coefficient: 5890 292 5340 240 b) Linear Regression Equation: 6500 409 c) Describe the strength and direction of the relationship 4800 172 between the data: (Circle one) 5940 288 Strong Positive Moderate Positive Weak Positive 5600 253 Strong Negative Moderate Negative Weak Negative 5100 232 d) Coefficient of Determination: 5850 278 e) Interpret the coefficient of determination in the context of the situation: f) What is the slope of the regression equation? Interpret this value in the context of the situation. Be specific and include units. g) What is the y-intercept of the regression equation? Explain why this does not make sense using the context of the situation. h) Find y if x = 5940 using the linear regression equation. Interpret this value in the context of the situation. Be specific and include units.<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

a) Correlation Coefficient (r)<\/strong><\/h3>\n\n\n\n

The correlation coefficient<\/strong> is a statistical measure that indicates the strength and direction of the linear relationship between two variables. It is calculated as: r=n(\u2211xy)\u2212(\u2211x)(\u2211y)[n\u2211x2\u2212(\u2211x)2][n\u2211y2\u2212(\u2211y)2]r = \\frac{n(\\sum xy) – (\\sum x)(\\sum y)}{\\sqrt{[n \\sum x^2 – (\\sum x)^2][n \\sum y^2 – (\\sum y)^2]}}r=[n\u2211x2\u2212(\u2211x)2][n\u2211y2\u2212(\u2211y)2]\u200bn(\u2211xy)\u2212(\u2211x)(\u2211y)\u200b<\/p>\n\n\n\n

Where:<\/p>\n\n\n\n

    \n
  • nnn is the number of data points<\/li>\n\n\n\n
  • xxx represents the weight<\/li>\n\n\n\n
  • yyy represents the braking distance<\/li>\n<\/ul>\n\n\n\n

    You can compute this value using a statistical software such as StatCrunch or a scientific calculator. Based on the data you’ve provided, I\u2019ll suggest using software to avoid manual calculation errors. For now, let’s focus on the interpretation once the value is computed.<\/p>\n\n\n\n

    \n\n\n\n

    b) Linear Regression Equation<\/strong><\/h3>\n\n\n\n

    The linear regression equation is: y=mx+by = mx + by=mx+b<\/p>\n\n\n\n

    Where:<\/p>\n\n\n\n

      \n
    • yyy is the dependent variable (braking distance)<\/li>\n\n\n\n
    • xxx is the independent variable (vehicle weight)<\/li>\n\n\n\n
    • mmm is the slope<\/li>\n\n\n\n
    • bbb is the y-intercept<\/li>\n<\/ul>\n\n\n\n

      Once you input the data into StatCrunch or a similar tool, it will compute the slope<\/strong> mmm and the y-intercept<\/strong> bbb. This equation represents the predicted braking distance (yyy) for any given vehicle weight (xxx).<\/p>\n\n\n\n

      \n\n\n\n

      c) Describe the Strength and Direction of the Relationship<\/strong><\/h3>\n\n\n\n

      The strength<\/strong> and direction<\/strong> of the relationship are determined by the correlation coefficient:<\/p>\n\n\n\n

        \n
      • Strong Positive<\/strong>: rrr close to +1<\/li>\n\n\n\n
      • Moderate Positive<\/strong>: rrr between 0.5 and 0.7<\/li>\n\n\n\n
      • Weak Positive<\/strong>: rrr between 0 and 0.3<\/li>\n\n\n\n
      • Strong Negative<\/strong>: rrr close to -1<\/li>\n\n\n\n
      • Moderate Negative<\/strong>: rrr between -0.5 and -0.7<\/li>\n\n\n\n
      • Weak Negative<\/strong>: rrr between -0.3 and 0<\/li>\n<\/ul>\n\n\n\n

        Interpret the direction and strength based on the computed value of rrr.<\/p>\n\n\n\n

        \n\n\n\n

        d) Coefficient of Determination (r\u00b2)<\/strong><\/h3>\n\n\n\n

        The coefficient of determination<\/strong> is calculated as the square of the correlation coefficient: r2=r\u00d7rr^2 = r \\times rr2=r\u00d7r<\/p>\n\n\n\n

        It represents the proportion of the variance in the dependent variable (braking distance) that is predictable from the independent variable (vehicle weight). This value is typically expressed as a percentage. For example, if r2=0.64r^2 = 0.64r2=0.64, this means that 64% of the variation in braking distance can be explained by the vehicle’s weight.<\/p>\n\n\n\n

        \n\n\n\n

        e) Interpretation of the Coefficient of Determination<\/strong><\/h3>\n\n\n\n

        The coefficient of determination<\/strong> provides insight into the effectiveness of the linear model in predicting the dependent variable. For example, if you find r2=0.81r^2 = 0.81r2=0.81, you can interpret this as:<\/p>\n\n\n\n

        “81% of the variation in braking distance is explained by the vehicle’s weight.”<\/p>\n\n\n\n

        This shows that the weight of the vehicle is a strong predictor of how far the vehicle will brake.<\/p>\n\n\n\n

        \n\n\n\n

        f) Slope of the Regression Equation<\/strong><\/h3>\n\n\n\n

        The slope<\/strong> of the regression line is the rate at which the braking distance changes as the vehicle’s weight changes. The slope is interpreted as: m=change in ychange in xm = \\frac{\\text{change in y}}{\\text{change in x}}m=change in xchange in y\u200b<\/p>\n\n\n\n

        For example, if the slope is 0.05, it means that for every additional pound in vehicle weight, the braking distance increases by 0.05 feet.<\/p>\n\n\n\n

        \n\n\n\n

        g) Y-Intercept of the Regression Equation<\/strong><\/h3>\n\n\n\n

        The y-intercept<\/strong> (bbb) represents the predicted braking distance when the vehicle\u2019s weight is 0 pounds. However, this value may not make sense in a real-world context, as a vehicle with 0 weight is unrealistic. Therefore, this value is often ignored in practical applications.<\/p>\n\n\n\n

        \n\n\n\n

        h) Find y for x = 5940 Using the Regression Equation<\/strong><\/h3>\n\n\n\n

        Now, we can use the linear regression equation to find the braking distance yyy when the vehicle weight x=5940x = 5940x=5940 pounds. For example: y=mx+by = mx + by=mx+b<\/p>\n\n\n\n

        Substitute x=5940x = 5940x=5940, and use the computed slope and y-intercept from the regression analysis to find the predicted braking distance. This value will give you the predicted distance at that particular vehicle weight.<\/p>\n\n\n\n

        \n\n\n\n

        Final Notes:<\/h3>\n\n\n\n
          \n
        • Use StatCrunch<\/strong> or similar software to compute the correlation coefficient, regression equation, and other statistics.<\/li>\n\n\n\n
        • Always ensure that the interpretation of statistical results is grounded in the real-world context, especially when considering the coefficient of determination and slope values.<\/li>\n<\/ul>\n\n\n\n
          \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

          Math 133: Cumulative Review WS #4 (Ch 1-7) Name: Coursepack, p. 424 Cumulative Review Worksheet #4 (Ch 1-7) StatCrunch File for this Worksheet available at this link 1. (15pts) The following data set gives the weight (in pounds) and the braking distance (in feet) of a vehicle traveling at a speed of 60 miles per […]<\/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-245443","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/245443","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=245443"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/245443\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=245443"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=245443"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=245443"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}