Chapter 4.2 Notes LSRL

Regression Lines A regression line is straight line that describes how a response variable y changes as the explanatory variable x changes. A regression line summarizes the relationship between two variables, only when one variable helps explain or predict the other. Least-Squares Regression Line: makes the sum of the squares of the vertical distances of the data points from the line as small as possible.

The equation y = a + bx b: slope a: y-intercept x: explanatory variable y: response variable

Calculator Info – Least Square Regression Line Put data into Lists 1 and 2 2nd Y= Turn on scatterplot Go to Catalog and make sure Diagnostic is turned ON STAT – CALC - #8 Store it in Y1

Example The accompanying table lists the number of registered automatic weapons (in thousands) along with the murder rate (in murders per 100,000) for 8 randomly selected states. The data are provided by the FBI and the Bureau of Alcohol, Tobacco, and Firearms. Create the Scatterplot Find the least squares regression line for this data using your calculator. Use as the variable and as the automatic weapons murder rate B variable. 𝑦 = 4.047 + .853 X

Formulas – on chart

Example Find the equation of the line

Making Predictions When making a prediction, we must make sure that the explanatory and response variables are in the correct place. We can use x to predict 𝑦 , but we cannot use 𝑦 to predict x. These are two different equations.

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.

Example A study measured the heights of husbands and wives to see if there was any relationship. The mean of the wives’ heights was 64.5 inches with a standard deviation of 2.5 inches. The mean of the husbands’ heights was 68.5 inches with a standard deviation of 2.7 inches. The resulting correlation was 0.5. Find the equation of the LSRL to predict the husband’s height. 𝑦 = What is the height of a husband if his wife is 63 inches tall?

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

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

Residuals Residual: the difference between an observed value of the response variable and the predicted value by the regression line Residual = observed y - predicted y Because the residuals show how far the data fall from our regression line, examining a residual plot helps determine how well the line describes the data.

Continued If the regression line captures the overall pattern of the data, then two things should be true The residual plot should show no obvious pattern The residuals should be relatively small in size A residual plot is a graphical tool for evaluating how well a linear model fits the data The square of the correlation r2 is the numerical quantity that tells us how well the least-squares line predicts values of the response variable.

Interpreting r2 r2 =.632 means that about 63% of the observed variation in the response variable that is explained by the straight-line pattern Perfect correlation (r=1 or r=-1) means that points lie exactly on a line. Then r =1 and all of the variation in one variable is accounted for by the straight-line relationship

Calculator Info – Residual Plot 2nd y= Scroll to Y List change L2 to residual by going to 2nd STAT list arrow down to RESID (#7) press ENTER To graph ZOOM 9

Causation: by changing x we can bring about change in y Lurking Variable is a variable that has an important effect on the relationship among the variables in a study but is not one of the explanatory variables studied.

Caution: Beware of Lurking Variables

Confounding Two variables are confounded when their effects on a response variable cannot be distinguished from each other. The confounded variables may be either explanatory variables or lurking variables. Beware of extrapolation. predicting outside of the range of x

Caution: Beware of Extrapolation

Causation

Criteria for establishing Causation The association is strong The association is consistent Higher doses are associated with stronger responses The alleged cause precedes the effect in time The alleged cause is plausible