LSRL Least Squares Regression Line Chapter 5 LSRL Least Squares Regression Line

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: Predicted heights = 20.4 + .342(age in months) 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. 38.5 inches Graph, find lsrl, also examine mean of x & y 102.48 inches or 8.5 feet?

Interpolation (good): Using a regression line for estimating predicted values between known values.

Extrapolation (bad) 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

Will this point always be on the LSRL? 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 and sx and sy : Plot the point (x, y) on the scatterplot. Graph, find lsrl, also examine mean of x & y Will this point always be on the LSRL?

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

Slope = correlation coefficient (st. dev. of y / st.dev. of x) Formulas – on chart Predicted y = y-intercept + slope(x) Slope formula Y-intercept = mean of y – slope (mean of x) Slope = correlation coefficient (st. dev. of y / st.dev. of x)

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.

b0 = 18 - .7228(40) = -10.92 For LSRL need slope and y-intercept: Predicted # of accidents = .723(posted speed limit) – 10.92

Describe the relationship between the two variables: Sleep Score 8 25 9 28 7 21 10 26 8.5 18 6.5 16 5.5 11 29 19 6 7.5 23 24 27 20 Describe the relationship between the two variables: (Score = a + b (Hours of Sleep)) What is the equation for the LSRL? What is the slope of this line? Interpret the slope in the context of the problem. What are the units? If a student got 10 hours of sleep the night before the exam, use the linear equation to approximate her score on the ACT.

Example: The average annual cost per person due to traffic delays for 70 US cities in 2000 was $298.96 with a standard deviation of $180.83. The peak period average freeway speed is 54.34 mph with a standard deviation of 4.494 mph. The correlation between cost per person and freeway speed is -0.90. Write a regression model to estimate costs per person associated with traffic delays. = = = = r =

c) The regression model is Price = 9.564 + 122.74 size Example: A scatterplot of house prices (in thousands of dollars) vs. house size (in thousands of square feet) shows a relationship that is straight, with only moderate scatter and no outliers. The correlation between house price and house size is 0.85. If a house is 1 SD above the mean in size (making it about 2170 sq ft), how many SDs above the mean would you predict its sale price to be? b) What would you predict about the sale price of a house that’s 2 SDs below average in size? c) The regression model is Price = 9.564 + 122.74 size What does the slope of 122.74 mean? d) What are the units? e) How much can a homeowner expect the value of his house to increase if he builds on an additional 2000 sq ft? f) How much would you expect to pay for a house of 3000 sq ft?