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Scatter Plots Best-Fitting Lines Residuals

Scatter Plots Best-Fitting Lines Residuals. Notes 21. Essential Learnings. Students will understand and be able to create scatter plots and determine lines of best fit .

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Scatter Plots Best-Fitting Lines Residuals

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  1. Scatter PlotsBest-Fitting LinesResiduals Notes 21

  2. Essential Learnings • Students will understand and be able to create scatter plots and determine lines of best fit. • Students will understand and be able to calculate residuals to determine how well the line of best fit represents the data.

  3. Vocabulary • Scatter plot - a graph of a set of ordered pairs.

  4. Positive Correlation • The relationship between x & y – as x increases y increases.

  5. Negative Correlation • The relationship between x & y – as x increases y decreases.

  6. Correlation Coefficient A number, denoted by r, that measures how well a line fits a scatter plot. r = -1 r = 0 r = 1 Points don’t lie near any line Points lie near a line with a negative slope Points lie near a line with a positive slope

  7. Correlation Coefficient r = -1 → Strong Negative Correlation r = -.5 → Weak Negative Correlation r = 0 → No Correlation r = .5 → Weak Positive Correlation r = 1 → Strong Positive Correlation

  8. Examples

  9. Examples

  10. Best-Fitting Lines A line that is close to most of the points in the scatter plot and goes through at least 2 points.

  11. Steps for finding Best-Fit Equation • Draw the scatter plot (label axes and scale) • Draw a best-fit line that goes through 2 points • Find the slope between the 2 points • Using the slope, choose a point & use the point-slope form equation • Simplify the point-slope form equation to slope-intercept form (solve for y)

  12. Example • The table shows the number y (in thousands) of alternative-fuel vehicles in use in the U.S. after 1997. Approximate the best-fitting line from the data.

  13. Example • Estimate the correlation coefficient (-1 to 1) for the data: r = _________ • Describe the correlation:

  14. Residuals • Residual – the difference between an observed y-value and its predicted y-value found using the line of best fit. • Residual = measured value – predicted value • Using the line of best fit you found in the example, determine the predicted y-value by plugging in each x-value then calculate the residuals.

  15. Table

  16. Residuals • Residual Sum of Squares - square each residual and add them up. • A smaller residual sum of squares figure represents a line of best fit which explains a greater amount of the data.

  17. Linear Regression • Linear regression is another way to analyze data and get a line of best fit. • For the example, using linear regression on the Nspire we get the equation: y = 41x + 262.6 • How close to the linear regression was your line of best fit?

  18. Assignment WS 7-3: Scatter Plots

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