Transcript Relationships Between Quantitative Variables

Chapter 10
Relationships
Between
Measurements
Variables
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.
Statistical Relationships
Correlation: measures the strength of a
certain type of relationship between
two measurement variables. (How
much does one variable explain the
other variable)
Regression: gives a numerical method
for trying to predict one measurement
variable from another.
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.
2
Statistical Relationships versus
Deterministic Relationships
Deterministic: if we know the value of one variable, we can
determine the value of the other exactly. e.g. relationship
between volume and weight of water.
Statistical: natural variability exists in both measurements.
Useful for describing what happens to a population or
aggregate. (Therefore, even when two variables are
strongly correlated we can still not predict one exactly from
the other, there will always be some variability due to
nature!)
*When researchers make claims about statistical relationships,
they are not claiming that the relationship will hold for
everyone… just that the relationship will hold for MOST
people
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.
3
Statistical Significance
A relationship is statistically significant if
the chances of observing the relationship
(between two measurement variables) in the
sample when actually nothing is going on in
the population are less than 5%.
* Less than 5% of the time we will see a sample that
tells us there is a relationship when in fact there is no
relationship between the variables.
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.
4
Measuring Strength
Through Correlation
A Linear Relationship: How close do the points in
a scatter plot follow a straight line?
Correlation (or the Pearson product-moment correlation
or the correlation coefficient) represented by the letter r:
• Indicator of how closely the values fall to a straight line.
• Measures linear relationships only; that is, it measures
how close the individual points in a scatterplot are to a
straight line.
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.
5
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.
6
Features of Correlations
Correlation is a number between –1 and 1
• –1 would correspond to a strong negative relationship (as one variable
increases the other decreases)
• 1 would correspond to a strong positive relationship (as one variable
increases the other increases)
• 0 would correspond to no relationship between the two
Strong Negative Corr
Strong Positive Corr
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.
variables
No relationship
7
Features of Correlations
The most important features of a correlation are:
•Magnitude: how close is the correlation to one or neg.
one (or how strong is the relationship)
•Sign is the correlation positive or negative (as one
variable increases do we expect the other to increase or
decrease)
*Correlations are unaffected if the units of measurement are changed.
For example, the correlation between weight and height remains
the same regardless of whether height is expressed in inches, feet
or millimeters (as long as it isn’t rounded off).
*Interpretation of Correlation: If the correlation between two
variables (say height and weight) is .6 then we know that 60% of
the variation in a persons height is explained by their weight.
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.
8
Computing the Correlation
To compute the correlation
coefficient we will need
the following formulas:
X
Y
55
100
60
120
38
80
18
65
Dev.in
X
Dev in
Y
Prod of
Dev
Dev in
X
squared
Dev in
Y
squared
Total:
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.
9
Specifying Linear
Relationships with Regression
Goal: Find a straight line that comes as close as
possible to the points in a scatterplot.
• Procedure to find the line is called regression.
• Resulting line is called the regression line.
• Formula that describes the line is called the
regression equation.
• Most common procedure used gives the least
squares regression line.
How do we find this least squares regression line?
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.
10
The Equation of the Line
y = a + bx
• a = intercept – where the line crosses the
vertical axis when x = 0.
• b = slope – how much of an increase there
is in y when x increases by one unit.
Ex. Deterministic Linear Relationship
y = temperature in Fahrenheit
x = temperature in Celsius
y = 32 + 1.8x
Intercept of 32 = temperature in F when C
temperature is zero. Slope of 1.8 = amount
by which F temperature increases when C
temperature increases by one unit.
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.
11
Ex. Linear Statistical Relationship:
Husbands’ and Wifes’ Ages
Scatterplot of British husbands’ and wives’ ages with
regression equation: y = 3.6 + 0.97x
Intercept: has no meaning.
Slope: as a wife’s age increases
by 1 year we would expect the
husbands age to increase by .97
of a year.
*Can use the regression line to
predict a husband’s age based
on a given wife’s age.
Ex. If wife is 20 we would
expect her husband to be
3.6 + .97(20) = 23 years old.
husband’s age = 3.6 + (.97)(wife’s age)
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.
12
Computing the Least Squares Line
To compute the least squares line we want to minimize the vertical
distances between the data points and a “best fit line.”
90
What would we expect y to be if we knew
that x=45?
70
y
110
Regression Analysis
20
30
40
x
50
60
If x increased by one unit how much would
we expect y to increase by?
If x was equal to zero what would we expect
y to be?
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.
13
Extrapolation
Not a good idea to use a regression equation to predict
values far outside the range where the original data fell.
No guarantee that the relationship will continue beyond the
range for which we have data.
Use the equation only for a minor extrapolation beyond the
range of the original data.
Ex. Can’t use a x-value outside of
the range of 10 to 60 to predict
the value of y. Thus, could not
use a x-value of say 100 to use
the regression line and predict
what y will be.
90
70
y
110
Regression Analysis
20
30
40
50
60
x
Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.
14