CH4 Question and answers verified to
pass
Review of Straight Lines - correct answers Suppose that Y is a Response
Variable (plotted on the vertical axis) & X is an Explanatory Variable (plotted
on the horizontal axis)
A straight line relating Y to X has an equation of the form:
- y = a + bx
- b = Slope
- a = Intercept
Slope (b) - correct answers The AMOUNT BY WHICH Y CHANGES WHEN X
INCREASES BY ONE UNIT
- b = r x (SDy/SDx)
The SIZE of it DEPENDS ON the UNITS in which WE MEASURE THE TWO
VARIABLES
- Ex: it is the change in fat gain in kilograms when NEA increases by 1 Cal
• There are 1000 grams in a kilogram
• If we measured fat gain in grams, this would be 1000 times larger, 3.44
• You CAN'T DETERMINE HOW IMPORTANT A RELATIONSHIP IS BY
LOOKING AT THE SIZE of this value in the Regression Line
Intercept (a) - correct answers The VALUE OF Y WHEN X EQUALS ZERO
- a = My - b(Mx)
,Although we need its value to complete the Regression Equation, it is
MEANINGFUL in context ONLY WHEN the EXPLANATORY VARIABLE CAN
actually TAKE VALUES CLOSE TO ZERO
- Ex: the relationship between time to finish a 1-mile race (X) pulse rate at the
finish line (Y)
• Because it is physiologically impossible to run 1 mile in 0 time, this value
would have no useful interpretation in this particular context
Regression Line - correct answers The LINE OF BEST FIT
A straight line that summarizes the linear relationship between 2 variables, but
only in a specific setting:
- When ONE OF THE VARIABLES IS THOUGHT TO HELP
EXPLAIN/PREDICT THE OTHER
- That is, REGRESSION DESCRIBES a RELATIONSHIP BETWEEN AN
EXPLANATORY VARIABLE (X) & a RESPONSE VARIABLE (Y)
It is a MATHEMATICAL MODEL that:
- Represents the UNDERLYING PATTERN in the data
- Provides an equation showing how the RESPONSE VARIABLE Y
CHANGES, ON AVERAGE, AS the EXPLANATORY VARIABLE X
CHANGES
• This equation may be used to PREDICT the EXPECTED VALUE OF Y FOR
A GIVEN VALUE OF X
Ex 5.1:
Figure 4.1 repeats Figure 3.2 (page 73), with the addition of a Regression
Line for predicting fat gain from NEA change
What does this line mean?
,Suppose that an individual's NEA increases by 400 Cal when she overeats
- From 400 Cal on the X axis, go up to the Regression Line & then over to the
Y axis
- The graph shows that the fat gain predicted by the linear model is a bit more
than 2 kg - correct answers Any straight line describing the NEA data has the
Form:
- fat gain = a + (b x NEA change)
- Where "fat gain" is Y & "NEA change" is X
The line in Figure 4.1 is the Regression Line with the equation:
- fat gain = 3.505 - (0.00344 x NEA change)
This equation makes it easy to predict fat gain
- If a person's NEA increases by 400 Cal when she overeats, substitute X =
400 in the equation
- The fat gain predicted by this model is:
• fat gain = 3.505 - (0.00344 x 400) = 2.13 kg
The slope, b = -0.00344
- Tells us that fat gained decreases by 0.00344 kg, on average, for each
added Calorie of NEA
- The Slope of a Regression Line is the expected rate of change in the
Response Variable as the Explanatory Variable changes
The intercept, a = 3.505 kg
- The estimated fat gain if NEA does not change (X = 0) when a person
overeats
, Residuals - correct answers Because we use the line to predict Y from X, the
prediction errors we make are errors in Y
- The vertical direction in the Scatterplot
The VERTICAL DISTANCES OF THE POINTS FROM THE LINE
-Y-Ŷ
A good Regression Line makes these AS SMALL AS POSSIBLE
- The most common way to make the COLLECTION OF VERTICAL
PREDICTION ERRORS "as small as possible" is the LEAST-SQUARES
method
Least-Squares Regression Line - correct answers The line that makes the
SUM OF the SQUARED VERTICAL DISTANCES of the data points from the
line AS SMALL AS POSSIBLE
Equation of the Least-Squares Regression Line - correct answers We have
data on an Explanatory Variable (X) & a Response Variable (Y) for n
individuals
From the data, we can calculate:
- The Means (Mx & My) & Standard Deviations (SDx & SDy) of the 2
Variables
- Their Correlation (r)
Least-Squares Regression Line:
- Ŷ = a + bx
- Slope = b = r x (SDy/SDx)
- Intercept = a = My - b(Mx)
pass
Review of Straight Lines - correct answers Suppose that Y is a Response
Variable (plotted on the vertical axis) & X is an Explanatory Variable (plotted
on the horizontal axis)
A straight line relating Y to X has an equation of the form:
- y = a + bx
- b = Slope
- a = Intercept
Slope (b) - correct answers The AMOUNT BY WHICH Y CHANGES WHEN X
INCREASES BY ONE UNIT
- b = r x (SDy/SDx)
The SIZE of it DEPENDS ON the UNITS in which WE MEASURE THE TWO
VARIABLES
- Ex: it is the change in fat gain in kilograms when NEA increases by 1 Cal
• There are 1000 grams in a kilogram
• If we measured fat gain in grams, this would be 1000 times larger, 3.44
• You CAN'T DETERMINE HOW IMPORTANT A RELATIONSHIP IS BY
LOOKING AT THE SIZE of this value in the Regression Line
Intercept (a) - correct answers The VALUE OF Y WHEN X EQUALS ZERO
- a = My - b(Mx)
,Although we need its value to complete the Regression Equation, it is
MEANINGFUL in context ONLY WHEN the EXPLANATORY VARIABLE CAN
actually TAKE VALUES CLOSE TO ZERO
- Ex: the relationship between time to finish a 1-mile race (X) pulse rate at the
finish line (Y)
• Because it is physiologically impossible to run 1 mile in 0 time, this value
would have no useful interpretation in this particular context
Regression Line - correct answers The LINE OF BEST FIT
A straight line that summarizes the linear relationship between 2 variables, but
only in a specific setting:
- When ONE OF THE VARIABLES IS THOUGHT TO HELP
EXPLAIN/PREDICT THE OTHER
- That is, REGRESSION DESCRIBES a RELATIONSHIP BETWEEN AN
EXPLANATORY VARIABLE (X) & a RESPONSE VARIABLE (Y)
It is a MATHEMATICAL MODEL that:
- Represents the UNDERLYING PATTERN in the data
- Provides an equation showing how the RESPONSE VARIABLE Y
CHANGES, ON AVERAGE, AS the EXPLANATORY VARIABLE X
CHANGES
• This equation may be used to PREDICT the EXPECTED VALUE OF Y FOR
A GIVEN VALUE OF X
Ex 5.1:
Figure 4.1 repeats Figure 3.2 (page 73), with the addition of a Regression
Line for predicting fat gain from NEA change
What does this line mean?
,Suppose that an individual's NEA increases by 400 Cal when she overeats
- From 400 Cal on the X axis, go up to the Regression Line & then over to the
Y axis
- The graph shows that the fat gain predicted by the linear model is a bit more
than 2 kg - correct answers Any straight line describing the NEA data has the
Form:
- fat gain = a + (b x NEA change)
- Where "fat gain" is Y & "NEA change" is X
The line in Figure 4.1 is the Regression Line with the equation:
- fat gain = 3.505 - (0.00344 x NEA change)
This equation makes it easy to predict fat gain
- If a person's NEA increases by 400 Cal when she overeats, substitute X =
400 in the equation
- The fat gain predicted by this model is:
• fat gain = 3.505 - (0.00344 x 400) = 2.13 kg
The slope, b = -0.00344
- Tells us that fat gained decreases by 0.00344 kg, on average, for each
added Calorie of NEA
- The Slope of a Regression Line is the expected rate of change in the
Response Variable as the Explanatory Variable changes
The intercept, a = 3.505 kg
- The estimated fat gain if NEA does not change (X = 0) when a person
overeats
, Residuals - correct answers Because we use the line to predict Y from X, the
prediction errors we make are errors in Y
- The vertical direction in the Scatterplot
The VERTICAL DISTANCES OF THE POINTS FROM THE LINE
-Y-Ŷ
A good Regression Line makes these AS SMALL AS POSSIBLE
- The most common way to make the COLLECTION OF VERTICAL
PREDICTION ERRORS "as small as possible" is the LEAST-SQUARES
method
Least-Squares Regression Line - correct answers The line that makes the
SUM OF the SQUARED VERTICAL DISTANCES of the data points from the
line AS SMALL AS POSSIBLE
Equation of the Least-Squares Regression Line - correct answers We have
data on an Explanatory Variable (X) & a Response Variable (Y) for n
individuals
From the data, we can calculate:
- The Means (Mx & My) & Standard Deviations (SDx & SDy) of the 2
Variables
- Their Correlation (r)
Least-Squares Regression Line:
- Ŷ = a + bx
- Slope = b = r x (SDy/SDx)
- Intercept = a = My - b(Mx)
