Chapter 5 LSRL

Bivariate Data x: explanatory (independent) variable y: response (dependent) variable Use x to predict y

Be sure to put the hat on the y! – (y-hat) means predicted y-value b – slope the amount y increases when x increases by 1 unit a – y-intercept height of the line when x = 0 in some situations, y-intercept has no meaning Be sure to put the hat on the y!

Least Squares Regression Line (LSRL) Line that gives the best fit to the data Minimizes the sum of the squared deviations from the line

(0,0) (3,10) (6,2) 4.5 -5 -4 (0,0) y =.5(6) + 4 = 7 2 – 7 = -5 y = 4 0 – 4 = -4 -5 y =.5(3) + 4 = 5.5 10 – 5.5 = 4.5 -4 (0,0) Sum of the squares = 61.25

What is the sum of the deviations from the line? Use a calculator to find the line of best fit (0,0) (3,10) (6,2) Will it always be zero? 6 -3 The line that minimizes the sum of the squared deviations from the line is the LSRL -3 Sum of the squares = 54

Interpretations Slope: For each unit increase in x, there is an approximate increase/decrease of b in y. Correlation coefficient: There is a strength, direction, linear relationship between x and y.

The ages (in months) and heights (in inches) of seven children are given. Find the LSRL. Interpret the slope and correlation coefficient in the context of the problem.

Slope: For each increase of one month in age, there is an approximate increase of .34 inches in heights of children. Correlation coefficient: There is a strong, positive, linear relationship between the age and height of children.

Predict the height of a child who is 4.5 years old. The ages (in months) and heights (in inches) of seven children are given. x 16 24 42 60 75 102 120 y 24 30 35 40 48 56 60 Predict the height of a child who is 4.5 years old. Predict the height of someone who is 20 years old. Graph, find lsrl, also examine mean of x & y

Extrapolation LSRL should not be used to predict y for x-values outside the data set We don’t know if the pattern in the scatterplot continues

This point is always on the LSRL Age 16 24 42 60 75 102 120 Ht. 24 30 35 40 48 56 60 Calculate x & y. Find the point (x, y) on the scatterplot. This point is always on the LSRL Graph, find lsrl, also examine mean of x & y

 Both r and the LSRL are non-resistant measures

Formulas on green sheet

The following statistics are found for the variables posted speed limit and the average number of accidents. Find the LSRL & predict the number of accidents for a posted speed limit of 50 mph.