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Linear Regression

Linear Regression. The Science of Predicting Outcome. Least-Squares Regression. LSR is a method for finding a line that summarizes the relationship between two variables. Regression line is a straight line that describes how a response variable y changes as an explanatory variable x changes.

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Linear Regression

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  1. Linear Regression The Science of Predicting Outcome

  2. Least-Squares Regression LSR is a method for finding a line that summarizes the relationship between two variables Regression line is a straight line that describes how a response variable y changes as an explanatory variable x changes We often use a regression line to predict the value of y for a given value of x

  3. LSRL: Least Square Regression Line Y-intercept Slope

  4. Example #1 - Finding the LSRL • Consider the following data: • With this data, find the LSRL • Start by entering this data into list 1 and list 2

  5. Example #1 - Finding the LSRL We need our graphing calculator to solve the first Case for today

  6. This is the correlation coefficient for the scatterplot!!! Example #1 - Finding the LSRL • You should then see the results of the regression. • a=53.24 • b=1.65 • r-squared=.8422 • r=.9177

  7. Example #2 – Interpreting LSRL • Interpreting the intercept • When your shoe size is 0, you should be about 53.24 inches tall • (Of course this does not make much sense in the context of the problem) • Interpreting the slope • For each increase of 1 in the shoe size, we would expect the height to increase by 1.65 inches

  8. Example #3 – Using LSRL • Making predictions • How tall might you expect someone to be who has a shoe size of 12.5? • Just plug in 12.5 for the shoe size above, so… • Height = 53.24+1.65 (12.5)=73.865 inches • (this is a prediction and is therefore not exact.)

  9. Practice A. Find the strength of correlation between the 2 variables B. Write the linear model for this data set C. What will be your BAC level if you drink 6 bottle of beers.

  10. Coefficients a and b S-sub y and s-sub x are the sample standard deviations of y and x (kinda like rise over run) The slope is: The intercept is: y-bar and x-bar are the mean y and x respectively The equation of the least squares regression line is written as:

  11. This table describes a study that recorded data on number of beers consumed and blood alcohol content (BAC) for 16 students. Here is some partial computer output from Minitab relating to these data: Y-intercept Slope (a) Use the computer output to write the equation of the least-squares line. (b) Interpret the slope and y intercept of the equation in this setting. (c) What blood alcohol level would your equation predict for a student who consumed 6 beers?

  12. Answers (a) If y = blood alcohol content (BAC) and x = number of beers, BAC = −0.01270 + 0.017964(number of beers). (b)Slope: for every extra beer consumed, the BAC will increase by an average of 0.017964. Intercept: if no beers are consumed, the BAC will be, on average, −0.01270 (obviously meaningless). (c) Predicted BAC = 0.0951

  13. y-hat = -0.124 + 0.0179(x) Here’s a computer generated output of 2 bivariate data. Write a linear model that corresponds to these set of data.

  14. Class Activity: Arm-span vs Height “On predicting height given arm span “ Students will measure their height and arm span. Then they will write the LSRL from the data they collected and predict a person’s arm span with their height.

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