3 Chapter Exploring the relationship between two variables, an explanatory variable, and a response variable. The response variable depends on, or can be explained by the explanatory variable. Section 3.1: Two Categorical Variables Sections 3.2-3.4: Two Quantitative Variables
Association between two Quantitative Variables What is the association between high school GPA and college GPA? Can high school GPA be used to predict college GPA? Identify the explanatory and response variables.
Is there an association between the size (carat) of a diamond and the cost of a diamond?
Cut Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good Good color J J J J J J J J J J J J J J J J J clarity SI2 SI2 SI2 SI2 SI2 SI2 SI2 SI2 SI2 SI2 SI2 SI2 SI2 SI2 SI2 SI2 SI2 report GIA GIA GIA GIA GIA GIA GIA GIA GIA GIA GIA GIA GIA GIA GIA GIA GIA carat 1.51 1.50 1.50 1.39 1.27 1.23 1.21 1 21 1.21 1.21 1.27 1.29 1.33 1.50 1.50 1.51 1.53 1.62 price 5763 6283 6396 5415 4580 4587 4556 4572 4475 4374 4654 5041 5386 5814 5980 6126 6415 Vender B B B B B B B B A A A A A A A A A
A data point is given by two values, (x, y). Explanatory variable, x Response variable, y Can the Explantory variable, x be used to predict the Response variable, y? Can the size of a diamond be used to predict its cost?
Monopoly
Monopoly Sample
Property Reading Railroad # spaces from GO x 5 14 24 32 Cost y $200 $160 $ $240 $300
Explanatory Variable: x = # Spaces from GO Response Variable: y = Cost of Property
Virginia Ave. Illinois Ave. North Carolina Ave.
1
Scatterplot
A graphical display for two quantitative variables. The data pairs, (x,y) are plotted with x values on the horizontal axis and the values on the vertical axis.
300
Examining a Scatterplot Does there appear to be an association? Do certain values of x seem to be associated with certain values of y? Is the association positive or negative? positive: as x i iti increases, y i increases negative: as x increases, y decreases Does there appear to be a linear association?
250
Cost
200 150 10 20 30
Spaces GO
Examine the scatterplot below. Does there appear to be an association? What is it? Is the association positive or negative? Does the association appear to be linear?
Blue Nile and Adiamore
price = - 1777 + 5115 carat 6500
S R-Sq R-Sq(adj) 251.048 89.7% 89.0%
Example of a scatterplot showing an association that is NOT linear.
6000
price
5500
5000
4500 1.2 1.3 1.4 carat 1.5 1.6
Horizontal Axis: respondents age at the time the song was popular. Vertical Axis: rating the respondent gave the song.
Would the following pairs of variables would be positively or negatively associated? The speed of a car and the distance required to come to a complete stop. The degree of job satisfaction and the number of absences from work. The size (carat) of a diamond and the cost.
A categorical variable can be added to a scatterplot by using a symbol.
Blue Nile and Adiamore
6500
Vender A B
6000
price
5500
5000
4500 1.2 1.3 1.4 carat 1.5 1.6
2
Scatterplot What is the Direction and Strength of the Linear Association? Direction: positive or negative Strength: determined by how close the dots are to the line. Correlation, r A measure of the direction and strength of the linear association between two quantitative variables.
Properties of the Correlation, r
The correlation is between negative one and one. 1 r +1. r > 0 indicates a positive association r < 0 indicates a negative association r = 0 indicates no linear association r = 1 indicates a perfect positive linear association. r = -1 indicates a perfect negative linear association.
A value of r close to 1 or -1 indicates a strong linear association while a value of r close to 0 represents a weak linear association. -1 -.5 0 .5 1
The correlation does not depend on a variables units. The correlation does not depend on which variable is the response variable.
Calculating r xi x yi y )( ) ( sx sy ( z x )( z y ) r= = n 1 n 1
300
Property
x = 18.75
250
y = 225
200
Reading Railroad Virginia 14 Avenue Illinois 24 Avenue North Carolina 32 Avenue
x y z for x z for y Product # Spcs Cost 5 200 160 240 300
Cost
150 10 20 30
Spaces GO
X: mean = 18.75, sd = 11.758 Y: mean = 225, sd = 59.7216
3
Predicting the value of the response variable Least Squares Regression
A method of determining a line of best fit by minimizing the squares of the vertical distances of a point from the line.
Regression Plot
Y = 147 016 + 4 15913X 147.016 4.15913X R-Sq = 67.1 %
300
Regression equation:
y = a + bx
The y-intercept is a, and the slope is b. For the monopoly example example,
250
y = 147.02 + 4.159 x
Where
10 20 30
Cost
200
x the is number of spaces from go
150
Spaces GO
and
y
is the estimated cost.
Interpreting Slope and y-intercept
Slope is the amount the y-variable will change, on average, for every one unit change in the x-variable. For monopoly, this means that for each additional Space from Go, the property Cost increases by $4.159, or approximately $4.16. The y-intercept is the estimated value of the yvariable when the x-variable is 0. This value does not always have a meaningful interpretation.
Calculating Slope and y-intercept
The least squares method results in the following formulas for the slope, b and the y-intercept, a.
b = r
S S
y x
a = y bx
Note that the sign of the slope will be the same as the sign of r. If r > 0, the slope will be positive. If r <0, the slope will be negative.
Calculating Slope and y-intercept
Use the summary data below to calculate the slope and y-intercept for the monopoly data.
Predict the Cost of Tennessee Ave., 18 Spaces from Go:
b = r
S S
y x
x = 18.75 y = 225
y = 147.02 + 4.159 x
Estimated Cost = 147.02 + 4.159(#spcs) (p)
Sx = 11.758 Sy = 59 7216 59.7216
a = y bx
r = .8188
y = 147.02 + 4.159(18) = 221.88
Tennessee Ave. actually cost $180. Our equation overestimated by $41.88
4
Residual (prediction error) Observed Predicted
Residuals for Monopoly
y = 147.02 + 4.159 x
y y
The least squares method for determining the regression equation minimizes:
For Tennessee: Observed: (x = 18, y= $180) Predicted: (x = 18, = $221.88) Residual = 180 221.88 = -41.88
x 5 14 24 32 y 200 160 240 300
y
yy
( y y)
2
Example using Slope and y-intercept
A county real estate appraiser uses a regression equation to predict the appraised value of a house based on the number of rooms. The appraiser uses the equation
Give a practical interpretation of the slope of the regression equation.
y = 74.80 + 19.72 x
a. For each additional room in the house, we estimate the appraised value to increase $74,800. b. For each additional room in the house, we estimate the appraised value to increase $19,720. c. For each additional dollar of appraised value, we estimate the number of rooms in the house to increase by 19.72 rooms.
74.80 19.72 y = 74 80 + 19 72 x
where y is the appraised value (in thousands) and x = number of rooms. Slope = ________ y-intercept = _______
Give a practical interpretation of the yintercept of the regression equation.
y = 74.80 + 19.72 x
a. For each additional room in the house, we estimate the appraised value to increase $74,800. b. No practical interpretation, since a house with 0 rooms is nonsensical. c. We estimate the base appraised value for any house to be $74,800.
The number of people living on American farms has declined steadily during this century. Here are data on the farm population (in millions) from 1935 to 1980:
Year Population 1935 32.1 1940 30.5 1945 24.4 1950 23.0 1955 19.1 19 1 1960 15.6 1965 12.4 1970 9.7 1975 8.9 1980 7.2
5
Farm Population
Declining Farm Population (in millions) 1935-1980
Y = 1166.93 - 0.586788X R-Sq = 97.7 %
y
= 1166.93 0.5868(x)
30
Popula ation
Estimated Population = 1166.93 0.5868(year) According to the regression equation, how much did the farm population decline each year on the average during this period?
20
10
1940
1950
1960
1970
1980
Year
Farm Population
y = 1166.93 0.5868(x)
Estimated Population = 1166.93 0.5868(year) Could the regression equation be used to predict the number of people living on farms in 1990?
Y = 3.00009 + 0.500091X R = 66.7 % -Sq 11 10 9 8
Anscombe Data
Regression Plot
Regression Plot
Y = 2.888 +0.509818X R = 66.7 % -Sq 10 9 8 7
y1
7 6 5 4 4 5 6 7 8 9 10 11 12 13 14
y2
6 5 4 3 4 5 6 7 8 9 10 11 12 13 14
Extrapolation
Attempting to predict beyond the range of the data used to construct the equation.
x1
x2
Moral: GRAPH YOUR DATA
Influential Point
A point is said to be influential if its inclusion dramatically changes the regression line. Influential points are usually outliers in the x direction.
Regression Plot
Y = 3.00245 + 0.499727X R-Sq = 66.6 % 13 12 11 10
Lurking Variable
A lurking variable affects the response variable, but is not included in the study. Ex: The percent of students who take the SAT in each state affects the states average SAT score.
SAT Scores 2002
1200
10
Regression Plot
Y = 3.00173 + 0.499909X R-Sq = 66.7 % 13 12 11
y3
y4
9 8
9
Total Score
8 7 7 6 6 5 5 4 5 6 7 8 9 10 11 12 13 14
Total S AT score
1100
AL DC FL GA Other SC
SAT 2002
1400 1200 1000 800 600 400 200 0 0 20 40 60 80 100 Total Score
1000
10 15 20
x3
x4
0 10 20 30 40 50 60 70 80 90
Percent of Students Taking Test
Percent of Students
6