REGRESSION Stats 1 with Liz

AIMS By the end of the lesson, you should be able to… o Understand the method of least squares to find a regression line o Use the regression line to estimate values & understand the limitations of that prediction o Use linear transformation to convert the variable in a regression line o Calculate residuals to identify outliers

Linear regression involves finding the equation of the line of best fit on a scatter graph. REGRESSION The best fitting line is the one that minimizes the sum of the squared deviations,, where d i is the vertical distance between the i th point and the line. d1d1 d2d2 d3d3 d4d4 d5d5 d6d6 The distances d i are sometimes referred to as residuals. Note: The best fitting line should pass through the mean point,. The equation can then be used to make an estimate of one variable given the value of the other variable.

The line that minimises the sum of squared deviations is formally known as the least squares regression line of y on x. The equation of the least squares regression line of y on x is: REGRESSION and : Recall : and y = a + bx where: y-intercept gradient

A BIT MORE ABOUT THE PMCC IN RELATION TO THE LINE OF BEST FIT… The fit of the line is related to the value of r (the PMCC). If r = 1or r = -1, it is considered a “perfect fit”. Being a perfect fit is also known as collinear. Example: If r = , the fit is “excellent”. If r = 0.854, the fit is “good”.

EXAMPLE 1 You can also find these values using your STAT feature on your calculator!

EXAMPLE 1

Look up the formulae on pg. 13. Enter data in calculator & list summarised data needed: Start by finding b: So we need:

EXAMPLE 1 Look up the formulae on pg. 13. Enter data in calculator & list summarised data needed: Start by finding b: and…

EXAMPLE 1 Look up the formulae on pg. 13. Enter data in calculator & list summarised data needed: Start by finding b: Next, find a:

EXAMPLE 1 Look up the formulae on pg. 13. Enter data in calculator & list summarised data needed: Bring it all together in the form: y = a + bx

EXAMPLE 1 When foot size, x = 7, When foot size, x = 14,

NOTE: A regression equation can only confidently be used to predict values of y that correspond to x values that lie within the range of the data values available. It is unwise to extrapolate beyond the given data because it is not guaranteed that the trend will continue. EXAMPLE 1

CALCULATING THE LEAST SQUARES REGRESSION LINE IN YOUR GDC… In your GDC, press MENU STAT enter data in lists 1 and 2 CALC (F2) REG (F3) X (F1) a + bx (F2) copy down your a and b values in the form y = a + bx

EXAM QUESTION JANUARY 2013, Q1 a represents the y-intercept (or the value when time, x = 0), so we can expect a to be 15cm.

EXAM QUESTION JANUARY 2013, Q1

b = which indicates that the candle length decreases by 0.64cm per hour. As time increases, the length of the candle decreases.

EXAM QUESTION JANUARY 2013, Q1 When time, x = 50, It is impossible to have length of -1 cm, so the claim is not justified.

SCALING & RESIDUALS EXAMPLE 2

Recalculate the least squares regression line without this piece of data.

INFLUENTIAL DATA POINTS A data point may be both influential and an outlier. An influential data point has an x-value much greater (or less) than the other x-values. However, an influential data point may not be an outlier.

INDEPENDENT STUDY (CHOOSE 3 OR MORE!) o mymaths – Least Squares Regression Line o mymaths – Scaling and Residuals o Stats Textbook – Pg. 171, Ex. 6D o Stats Textbook – Pg. 175, Ex. 6E o Stats Textbook – Pg. 178, Ex. 6F o Stats Textbook – Pg. 181, Revision Exercise 6 o SET HW – CORRELATION & REGRESSION