Least Squares Regression Lines Text: Chapter 3.3 Unit 4: Notes page 58

Bivariate data x – variable: is the independent or explanatory variable y- variable: is the dependent or response variable Use x to predict 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 - (y-hat) means the predicted y Be sure to put the hat on the y

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

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

(0,0) (3,10) (6,2) Sum of the squares = 54 Use a calculator to find the line of best fit STAT EDIT L1, L2 STAT CALC 4 LinReg (ax+b) Find y - y -3 6 What is the sum of the deviations from the line? Will it always be zero? minimizes LSRL The line that minimizes the sum of the squares of the deviations from the line is the LSRL.

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

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

Ans: r =.994, Correlation coefficient: Slope: age of one month increase.34 inches in heights of children. For an increase in age of one month, there is an approximate increase of.34 inches in heights of children. strong, positive, linear age and height of children There is a strong, positive, linear association between the age and height of children.

The ages (in months) and heights (in inches) of seven children are given. x y Predict the height of a child who is 4.5 years old. (4.5 yrs = 54 months) Predict the height of someone who is 20 years old. (240 months)

Extrapolation should not outsideThe 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.

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? LinReg L2,L1 Calculate: LinReg L2,L1 Do you get the same LSRL? However, statisticians will always use this equation to predict x from y 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. x y Calculate x & y. Will this point always be on the LSRL? Plot the point (x, y) on the LSRL , YES!

non-resistant 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. (Hint: Find b 1, then b 0, then LSRL)

Predict the number of accidents for a posted speed limit of 50 mph.

Homework: Packet page 64, “Linear Regression Activity” Packet page 68