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Defining the Relationship Between Length and Weight: A Quantitative Data Analysis

This guide explores how to define the relationship between two sets of quantitative data, specifically focusing on the correlation between length and weight of channel iron. It explains the process of creating a scatter plot and finding the line of best fit using two chosen points. We derive the linear equation and examine the slope and y-intercept's significance. Additionally, we utilize the least squares regression method to model the relationship statistically. Practical examples demonstrate how to predict weight from length and vice versa, highlighting the importance of correlation coefficients.

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Defining the Relationship Between Length and Weight: A Quantitative Data Analysis

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Presentation Transcript

  1. Section 4.2 How Can We Define the Relationship between two sets of Quantitative Data?

  2. Consider the Relationship Between Length and Weight in some Lengths of Channel Iron:

  3. Scatter Plot

  4. Find an Equation by Hand: • Pick two points that define a line of best fit (the points do not have to be part of the data) … how about (19,160) & (56,518) • y = mx + b ( slope = m = ) … so m = • Find b by plugging in the slope and one of the two points … 160 = 9.68(19) + b … so b = -23.92 • So y = 9.68x – 23.92 models the relation ship between length (x) and weight (y) • What do m & b represent in the data beyond slope and y-intercept?

  5. Use the Equation: • How much will a 72 length of channel weigh? • y = 9.68(72) – 23.92 ≈ 673 lbs. • How long would a length of channel be weighing 250 lbs.? • 250 = 9.68x – 23.92, x ≈ 28.3 ft.

  6. Least Squares Regression: • A least squares regression line minimizes the sum of the squared vertical distance between the observed and predicted value. We name it , pronounced (y – hat) • = mx + b … where m = r & b = • = 35.8 & Sx= 15.1888 = 319.6 & Sy= 151.4754 r = .998 • = mx + b … where m = 9.953 & b = -36.72

  7. Three Cheers for Technology: • Stat … Calc … LinReg ( ax + b ) • Shazaam! … = 9.956x – 36.833 • Why is there a difference in the values from doing it by hand? • What is the correlation coefficient (r – value)? • r ≈ .998 • Web link to a good graphing calculator regression tutorial youtube: http://www.youtube.com/watch?v=nw6GOUtC2jY

  8. Temp/Elevation Correlation: • Find and its correlation coefficient • Graph the data and the line of best fit () on your calculator • Estimate the temperature on the top of Mt. Shasta … elevation 14,179 ft. • Estimate the elevation if the temperature is 40 degrees.

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