Chapter 5 LSRL

Bivariate data x – variable: is the independent or explanatory variable y- variable: is the dependent or response variable Use x to predict y

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

Least Squares Regression Line LSRL The line that gives the best fit to the data set The line that minimizes the sum of the squares of the deviations from the line

(0,0) (3,10) (6,2) y =.5(6) + 4 = 7 2 – 7 = -5 4.5 y =.5(0) + 4 = 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? Will it always be zero? Use a calculator to find the line of best fit (0,0) (3,10) (6,2) 6 Find y - y -3 The line that minimizes the sum of the squares of the 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 direction, strength, linear of association between x and y.

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 Find the LSRL. Interpret the slope and correlation coefficient in the context of the problem.

Correlation coefficient: There is a strong, positive, linear association between the age and height of children. Slope: For an increase in age of one month, there is an approximate increase of .34 inches in heights 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 The LSRL should not be used to predict y for values of x outside the data set. It is unknown whether the pattern observed in the scatterplot continues outside this range.

For these data, this is the best equation to predict y from x. The ages (in months) and heights (in inches) of seven children are given. The LSRL is Can this equation be used to estimate the age of a child who is 50 inches tall? Calculate: LinReg L2,L1 For these data, this is the best equation to predict y from x. Do you get the same LSRL? However, statisticians will always use this equation to predict x from y

Plot the point (x, y) on the scatterplot. 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 Calculate x & y. Plot the point (x, y) on the scatterplot. Graph, find lsrl, also examine mean of x & y

The correlation coefficient and the LSRL are both non-resistant measures.

Formulas – on chart

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.