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# Further7 regression analysis

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### Further7 regression analysis

1. 1. Further MathsRegression Analysis K McMullen 2012
2. 2. Regression AnalysisThe process of fitting a straight line to bivariatedata is known as linear regression This can be done either by: Calculating and plotting the least squares regression line Drawing the three median lineA regression analysis involves a range ofstatistics to summarise the relationship betweentwo numerical variables K McMullen 2012
3. 3. Regression AnalysisIncluded in a regression analysis: Scatterplot: form, direction, outliers Correlation coefficient: measures strength Regression line: models the relationship Interpreting slope and y-intercept Coefficient of determination: measures the predictive power of the relationship Residual plot: tests linearity Final report K McMullen 2012
4. 4. Regression AnalysisWe have already covered: Scatterplot Correlation coefficient (r) Coefficient of determination (r2)What we will cover here: Regression line Interpreting slope and y-intercept Residual plots K McMullen 2012
5. 5. Regression AnalysisLeast Squares Regression Line (also referred to asRegression Line) The least squares line is like the mean- it is a line that best fits the data The line balances out the data values on either side of the line The distance between the data value and the regression line is known as the residual The least squares line is the line where the sum of the squares of the residuals is the least (we take the squares so that the negative residuals don’t cancel out the positive residuals- remember that squaring a negative makes a positive) K McMullen 2012
6. 6. Regression AnalysisThe equation of the least squares regression lineis given by: K McMullen 2012
7. 7. Regression AnalysisInterpreting slope: remember that the slopeindicates what happens to the DV as the IVincreases (as x increases y either increases(positive gradient) or decreases (negativegradient) Comment: “On average, the DV will increase/decrease by slope for every 1 increase in IV” Eg. On average, life expectancies (DV) in countries will decrease by 1.44years for an increase in birth rate (x) of one birth per 1000 people. K McMullen 2012
8. 8. Regression AnalysisInterpreting intercept: the regression line gives usthe value of the y-intercept and what the DV iswhen the IV is 0. Comment: “On average, when IV is 0 the DV is intercept” Eg. On average, the life expectancy for countries with a zero birth rate is 105.4 years K McMullen 2012
9. 9. Regression AnalysisThe residual plot: shows how far the data values are from theregression line The regression line basically becomes the line x=0 and the data values remain the same distance away from this line (basically think of moving the regression line and all the data values so that the line is now horizontal- the data values remain the same distance away from the line) Remember: Residual value= data value- predicted valueIf a residual plot is randomly scattered and the residual valuesare close to 0 then we can confirm our assumption of a linearrelationshipComment: “The assumption that there is a linear relationshipbetween IV and DV is confirmed by the residual plot” K McMullen 2012
10. 10. Regression AnalysisExtrapolation and interpolation: we use theregression line to predict values where: Extrapolation: is where we predict outside the range of data Interpolation is where we predict inside the range of data K McMullen 2012