STAT 145 (Notes)
Al Nosedal
anosedal@unm.edu
Department of Mathematics and Statistics
University of New Mexico
Fall 2013
.
.
.
.
.
.
CHAPTER 4
SCATTERPLOTS AND CORRELATION.
.
.
.
.
.
.
Definitions
I
A response variable measures an outcome of a study.
.
.
.
.
.
.
Definitions
I
A response variable measures an outcome of a study.
I
An explanatory variable may explain or influence changes in a
response variable.
.
.
.
.
.
.
Coral reefs
How sensitive to changes in water temperature are coral reefs? To
find out, scientists examined data on sea surface temperatures and
coral growth per year at locations in the Red Sea. What are the
explanatory and response variables? Are they categorical or
quantitative?
.
.
.
.
.
.
Solution
Sea-surface temperature is the explanatory variable; coral growth is
the response variable. Both are quantitative.
.
.
.
.
.
.
Scatterplot
A scatterplot shows the relationship between two quantitative
variables measured on the same individuals. The values of one
variable appear on the horizontal axis, and the values of the other
variable appear on the vertical axis. Each individual in the data
appears as the point in the plot fixed by the values of both
variables for that individual.
Always plot the explanatory variable, if there is one, on the
horizontal axis (the x axis) of a scatterplot. As a reminder, we
usually call the explanatory variable x and the response variable y .
If there is no explanatory-response distinction, either variable can
go on the horizontal axis.
.
.
.
.
.
.
Do heavier people burn more energy?
Metabolic rate, the rate at which the body consumes energy, is
important in studies of weight gain, dieting, and exercise. We have
data on the lean body mass and resting metabolic rate for 12
women who are subjects in a study of dieting. Lean body mass,
given in kilograms, is a person’s weight leaving out all fat.
Metabolic rate is measured in calories burned per 24 hours.
.
.
.
.
.
.
Data
Mass
36.1
54.6
48.5
42.0
50.6
42.0
Rate
995
1425
1396
1418
1502
1256
Mass
40.3
33.1
42.4
34.5
51.1
41.2
Rate
1189
913
1124
1052
1347
1204
The researchers believe that lean body mass is an important
influence on metabolic rate. Make a scatterplot to examine this
belief.
.
.
.
.
.
.
900
1000
1100
1200
1300
Metabolic Rate (calories/day)
1400
1500
Scatterplot
35
40
Lean Body Mass (kg)
45
50
.
.
55
.
.
.
.
Examining a Scatterplot
In any graph of data, look for the overall pattern and for striking
deviations from the pattern.
You can describe the overall pattern of a scatterplot by the
direction, form, and strength of the relationship.
An important kind of deviation is an outlier, an individual value
that falls outside the overall pattern of the relationship.
.
.
.
.
.
.
Positive Association, Negative Association
Two variables are positively associated when above-average values
of one tend to accompany above-average values of the other, and
below-average values also tend to occur together.
Two variables are negatively associated when above-average values
of one tend to accompany below-average values of the other, and
vice versa.
The strength of a relationship in a scatterplot is determined by
how closely the points follow a clear form.
.
.
.
.
.
.
10
20
30
40
y
50
60
70
80
Positive Association (scatterplot)
2
4
6
8
10
12
14
x
.
.
.
.
.
.
60
40
20
y
80
100
Negative Association (scatterplot)
2
4
6
8
10
12
14
x
.
.
.
.
.
.
-10
-5
0
y
5
10
15
NO Association (scatterplot)
2
4
6
8
10
12
14
x
.
.
.
.
.
.
Do heavier people burn more energy? (Again)
Describe the direction, form, and strength of the relationship
between lean body mass and metabolic rate, as displayed in your
plot.
.
.
.
.
.
.
Solution
The scatterplot shows a positive direction, linear form, and
moderately strong association.
.
.
.
.
.
.
Do heavier people burn more energy? (Again)
The study of dieting described earlier collected data on the lean
body mass (in kilograms) and metabolic rate (in calories) for both
female and male subjects.
Mass
36.1
54.6
48.5
42.0
50.6
42.0
Rate
995
1425
1396
1418
1502
1256
Sex
F
F
F
F
F
F
Mass
40.3
33.1
42.4
34.5
51.1
41.2
Rate
1189
913
1124
1052
1347
1204
.
Sex
F
F
F
F
F
F
.
.
.
.
.
More data
Mass
51.9
46.9
62.0
62.9
Rate
1867
1439
1792
1666
Sex
M
M
M
M
Mass
47.4
48.7
51.9
Rate
1322
1614
1460
.
Sex
M
M
M
.
.
.
.
.
a) Make a scatterplot of metabolic rate versus lean body mass for
all 19 subjects. Use separate symbols to distinguish women and
men. (This is a common method to compare two groups of
individuals in a scatterplot)
b) Does the same overall pattern hold for both women and men?
What is the most important difference between women and men?
.
.
.
.
.
.
1200
1400
1600
female
male
1000
Metabolic Rate (calories/day)
1800
Scatterplot
35
40
45
50
55
60
Lean Body Mass (kg)
.
.
.
.
.
.
b) Solution
For both men and women, the association is linear and positive.
The women’s points show a stronger association. As a group,
males typically have larger values for both variables (they tend to
have more mass, and tend to burn more calories per day).
.
.
.
.
.
.
Correlation
The correlation measures the direction and strength of the linear
relationship between two quantitative variables. Correlation is
usually written as r .
Suppose that we have data on variables x and y for n individuals.
The values for the first individual are x1 and y1 , the values for the
second individual are x2 and y2 , and so on.
The means and standard deviations of the two variables are x̄ and
Sx for the x-values, and ȳ and Sy for the y -values. The correlation
r between x and y is
1 ∑
r=
n−1
n
i=1
(
xi − x̄
Sx
)(
yi − ȳ
Sy
.
)
.
.
.
.
.
Covariance
You can compute the covariance, Sxy using the following formula:
∑n
xi yi
nx̄ ȳ
Sxy = i=1
−
n−1
n−1
.
.
.
.
.
.
Correlation (alternative formula)
r=
Sxy
Sx Sy
where
r = correlation.
Sxy = covariance.
Sx = standard deviation of x.
Sy = standard deviation of y .
.
.
.
.
.
.
Example
Five observations taken for two variables follow
xi
4
6
11
3
16
yi
50
50
40
60
30
a) Develop a scatter diagram with x on the horizontal axis.
b) Compute the sample covariance.
c) Compute and interpret the sample correlation coefficient.
.
.
.
.
.
.
45
40
35
30
y
50
55
60
a) Scatterplot
4
6
8
10
12
14
16
x
.
.
.
.
.
.
x̄ and ȳ
First, let’s find x̄ and ȳ
x̄ =
ȳ =
4 + 6 + 11 + 3 + 16
=8
5
50 + 50 + 40 + 60 + 30
= 46
5
.
.
.
.
.
.
Sx
Now, let’s find Sx and Sy
Sx2 =
(4 − 8)2 + (6 − 8)2 + (11 − 8)2 + (3 − 8)2 + (16 − 8)2
4
Sx2 =
(−4)2 + (−2)2 + (3)2 + (−5)2 + (8)2
118
=
= 29.5
4
4
Sx = 5.4313
.
.
.
.
.
.
Sy
Sy2 =
(50 − 46)2 + (50 − 46)2 + (40 − 46)2 + (60 − 46)2 + (30 − 46)2
4
Sy2 =
(4)2 + (4)2 + (−6)2 + (14)2 + (−16)2
520
=
= 130
4
4
Sy = 11.4017
.
.
.
.
.
.
Covariance and Correlation
Finally, we find Sxy and r
5
∑
xi yi = (4)(50) + (6)(50) + (11)(40) + (3)(60) + (16)(30) = 1600
i=1
∑n
Sxy =
i=1 xi yi
n−1
r=
−
nx̄ ȳ
1600 (5)(8)(46)
=
−
= −60
n−1
4
4
Sxy
−60
=
= −0.9688
Sx Sy
(5.4313)(11.4017)
.
.
.
.
.
.
Coral reefs
Exercise 4.2 discusses a study in which scientists examined data on
mean sea surface temperatures (in degrees Celsius) and mean coral
growth (in millimeters per year) over a several-year period at
locations in the Red Sea. Here are the data:
Sea Surface Temperature
29.68
29.87
30.16
30.22
30.48
30.65
30.90
Growth
2.63
2.58
2.60
2.48
2.26
2.38
2.26
.
.
.
.
.
.
a) Make a scatterplot. Which is the explanatory variable?
b) Find the correlation r step-by-step. Explain how your value for r
matches your graph in a).
c) Enter these data into your calculator and use the correlation
function to find r .
.
.
.
.
.
.
Scatterplot
30.4
30.2
30.0
29.8
Coral Growth (mm)
30.6
30.8
Temperature is the explanatory variable.
2.3
2.4
2.5
2.6
Sea Surface Temperature
.
.
.
.
.
.
x̄ and ȳ
First, let’s find x̄ and ȳ
x̄ =
29.68 + 29.87 + 30.16 + 30.22 + 30.48 + 30.65 + 30.90
= 30.28
7
ȳ =
2.63 + 2.58 + 2.60 + 2.48 + 2.26 + 2.38 + 2.26
= 2.4557
7
.
.
.
.
.
.
Sx
Now, let’s find Sx and Sy
Sx2 =
(29.68 − 30.28)2 + ... + (30.65 − 30.28)2 + (30.90 − 30.28)2
6
Sx2 = 0.1845
Sx = 0.4296
.
.
.
.
.
.
Sy
Sy2 =
(2.63 − 2.4557)2 + ... + (2.38 − 2.4557)2 + (2.26 − 2.4557)2
6
Sy2 = 0.0249
Sy = 0.1578
.
.
.
.
.
.
Covariance
Finally, we find Sxy and r
7
∑
xi yi
= (29.68)(2.63) + ... + (30.65)(2.38) + (30.90)(2.26)
i=1
= 520.1504
∑n
Sxy
nx̄ ȳ
i=1 xi yi
−
n−1
n−1
520.1504 (7)(30.28)(2.4557)
=
−
6
6
= 86.6917 − 86.7516 = −0.0599
=
.
.
.
.
.
.
Correlation
r=
Sxy
−0.0599
=
= −0.8835
Sx Sy
(0.4296)(0.1578)
This is consistent with the strong, negative association depicted in
the scatterplot.
c) Ti-84 will give a value of r = −0.8914.
.
.
.
.
.
.
Example
Five observations taken for two variables follow
xi
1
2
3
4
5
yi
1
2
3
4
5
a) Develop a scatter diagram with x on the horizontal axis.
b) Compute the sample covariance.
c) Compute and interpret the sample correlation coefficient.
.
.
.
.
.
.
3
2
1
y
4
5
a) Scatterplot
1
2
3
4
5
x
.
.
.
.
.
.
Covariance and Correlation
x̄ = 3
ȳ∑= 3
5
i=1 xi yi
Sx = 1.58113883
Sy = 1.58113883
= 55
∑
Sxy =
nx̄ ȳ
xi yi
−
n−1
n−1
Sxy =
55 (5)(3)(3)
−
4
4
Sxy = 13.75 − 11.25 = 2.5
r=
Sxy
2.5
=
=1
Sx Sy
(1.58113883)(1.58113883)
.
.
.
.
.
.
Facts about correlation
1. Correlation makes no distinction between explanatory and
response variables. It makes no difference which variable you call x
and which you call y in calculating the correlation.
2. Because r uses the standardized values of the observations, r
does not change when we change the units of measurement of x, y,
or both. The correlation r itself has no unit of measurement; it is
just a number.
3. Positive r indicates positive association between the variables,
and negative r indicates negative association.
4. The correlation r is always a number between −1 and 1. Values
of r near 0 indicate a very weak linear relationship. Perfect
correlation, r = 1 or r = −1, occurs only when the points on a
scatterplot lie exactly on a straight line.
.
.
.
.
.
.
Strong association but no correlation
The gas mileage of an automobile first increases and then
decreases as the speed increases. Suppose that this relationship is
very regular, as shown by the following data on speed (miles per
hour) and mileage (miles per gallon):
Speed
30
40
50
60
70
Mileage
24
28
30
28
24
.
.
.
.
.
.
Strong association but no correlation (cont.)
Make a scatterplot of mileage versus speed. Show that the
correlation between speed and mileage is r = 0. Explain why the
correlation is 0 even though there is a strong relationship between
speed and mileage.
.
.
.
.
.
.
27
24
25
26
Mpg
28
29
30
Scatterplot
30
40
50
60
70
Speed
Remember that correlation only measures the strength and
direction of a linear relationship between two variables.
.
.
.
.
.
.
Correlation
x̄ = 50
Sx = 15.8113
ȳ∑= 26.8
Sy = 2.6832
5
x
y
=
6700
i=1 i i
∑
nx̄ ȳ
xi yi
−
Sxy =
n−1
n−1
Sxy =
6700 (5)(50)(26.8)
−
4
4
Sxy = 1675 − 1675 = 0
r=
Sxy
0
=
=0
Sx Sy
(15.8113)(2.6832)
.
.
.
.
.
.
More facts about correlation
1. Correlation requires that both variables be quantitative, so that
it makes sense to do the arithmetic indicated by the formula for r.
2. Correlation measures the strength of only the linear relationship
between two variables. Correlation does not describe curved
relationships between variables, no matter how strong they are.
3. Like the mean and the standard deviation, the correlation is not
resistant: r is strongly affected by a few outlying observations.
4. Correlation is not a complete summary of two-variable data,
even when the relationship between the variables is linear. You
should give the means and standard deviations of both x and y
along with the correlation.
.
.
.
.
.
.
CHAPTER 5
REGRESSION
.
.
.
.
.
.
Regression Line
A regression line is a straight line that describes how a response
variable y changes as an explanatory variable x changes. We often
use a regression line to predict the value of y for a given value of x.
.
.
.
.
.
.
Review of Straight Lines
Suppose that y is a response variable (plotted on the vertical axis)
and 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
In this equation, b is the slope, the amount by which y changes
when x increases by one unit. The number a is the intercept, the
value of y when x = 0.
.
.
.
.
.
.
City mileage, highway mileage
We expect a car’s highway gas mileage (mpg) to be related to its
city gas mileage. Data for all 1040 vehicles in the government’s
2010 Fuel Economy Guide give the regression line
highway mpg = 6.554 + (1.016 x city mpg)
for predicting highway mileage from city mileage.
a) What is the slope of this line? Say in words what the numerical
value of the slope tells you.
b) What is the intercept? Explain why the value of the intercept is
not statistically meaningful.
c) Find the predicted highway mileage for a car that gets 16 miles
per gallon in the city. Do the same for a car with a city mileage of
28 mpg.
.
.
.
.
.
.
Solutions
a) The slope is 1.016. On average, highway mileage increases by
1.016 mpg for each additional 1 mpg change in city mileage.
b) The intercept is 6.554 mpg. This is the highway mileage for a
nonexistent car that gets 0 mpg in the city. Although this
interpretation is valid, such prediction would be invalid because it
involves considerable extrapolation.
c) For a car that gets 16 mpg in the city, we predict highway
mileage to be:
6.554 + (1.016)(16) = 22.81 mpg.
For a car that gets 28 mpg in the city, we predict highway mileage
to be:
6.554 + (1.016)(28) = 35.002 mpg.
.
.
.
.
.
.
What’s the line?
You use the same bar of soap to shower each morning. The bar
weighs 80 grams when it is new. Its weight goes down by 5 grams
per day on the average. What is the equation of the regression line
for predicting weight from days of use?
.
.
.
.
.
.
Solution
The equation is:
weight = 80 − 5 × days
The intercept is 80 grams (the initial weight), and the slope is −5
grams/day.
.
.
.
.
.
.
Least-Squares Regression Line
The least-squares regression line of y on x is the line that makes
the sum of the squares of the vertical distances of the data points
from the line as small as possible.
.
.
.
.
.
.
Equation of the Least-Squares Regression Line
We have data on an explanatory variable x and a response variable
y for n individuals. From the data, calculate the means x̄ and ȳ
and the standard deviations Sx and Sy of the two variables, and
their correlation r . The least-squares regression line is the line
ŷ = a + bx
with slope
b=r
Sy
Sx
and intercept
a = ȳ − bx̄
.
.
.
.
.
.
Coral reefs
Exercises 4.2 and 4.8 discuss a study in which scientists examined
data on mean sea surface temperatures (in degrees Celsius) and
mean coral growth (in millimeters per year) over a several-year
period at locations in the Red Sea. Here are the data:
Sea Surface Temperature
29.68
29.87
30.16
30.22
30.48
30.65
30.90
Growth
2.63
2.58
2.60
2.48
2.26
2.38
2.26
.
.
.
.
.
.
a) Use your calculator to find the mean and standard deviation of
both sea surface temperature x and growth y and the correlation r
between x and y . Use these basic measures to find the equation of
the least-squares line for predicting y from x.
b) Enter the data into your software or calculator and use the
regression function to find the least-squares line. The result should
agree with your work in a) up to roundoff error.
.
.
.
.
.
.
Solutions
a) x̄ = 30.28
ȳ = 2.4557
r = −0.8914.
Hence,
Sx = 0.4296
Sy = 0.1578
Sy
b=r
= (−0.8914)
Sx
(
0.1578
0.4296
)
= −0.3274
a = ȳ − bx̄ = 2.4557 − (−0.3274)(30.28) = 12.3693
b) Ti-84 gives: slope = - 0.3276 and intercept = 12.3758.
.
.
.
.
.
.
Do heavier people burn more energy?
We have data on the lean body mass and resting metabolic rate
for 12 women who are subjects in a study of dieting. Lean body
mass, given in kilograms, is a person’s weight leaving out all fat.
Metabolic rate, in calories burned per 24 hours, is the rate at
which the body consumes energy.
Mass
36.1
54.6
48.5
42.0
50.6
42.0
Rate
995
1425
1396
1418
1502
1256
Mass
40.3
33.1
42.4
34.5
51.1
41.2
Rate
1189
913
1124
1052
1347
1204
.
.
.
.
.
.
a) Make a scatterplot that shows how metabolic rate depends on
body mass. There is a quite strong linear relationship, with
correlation r = 0.876.
b) Find the least-squares regression line for predicting metabolic
rate from body mass. Add this line to your scatterplot.
c) Explain in words what the slope of the regression line tells us.
d) Another woman has a lean body mass of 45 kilograms. What is
her predicted metabolic rate?
.
.
.
.
.
.
900
1000
1100
1200
1300
Metabolic Rate (calories/day)
1400
1500
Scatterplot
35
40
Lean Body Mass (kg)
45
50
.
.
55
.
.
.
.
Solutions
b) the regression equation is
ŷ = 201.2 + 24.026x
where y = metabolic rate and x= body mass.
c) The slope tells us that on the average, metabolic rate increases
by about 24 calories per day for each additional kilogram of body
mass.
d) For x = 45 kg, the predicted metabolic rate is
ŷ = 1282.4 calories per day.
.
.
.
.
.
.
Facts about Least-Squares Regression
1. The distinction between explanatory and response variables is
essential in regression.
2. The least-squares regression line always passes through the
point (x̄, ȳ ) on the graph of y against x.
3. The square of the correlation, r 2 , is the fraction of the variation
in the values of y that is explained by the least-squares regression
of y on x.
.
.
.
.
.
.
What’s my grade?
In Professor Krugman’s economics course the correlation between
the student’s total scores prior to the final examination and their
final-examination scores is r = 0.5. The pre-exam totals for all
students in the course have mean 280 and standard deviation 40.
The final-exam scores have mean 75 and standard deviation 8.
Professor Krugman has lost Julie’s final exam but knows that her
total before the exam was 300. He decides to predict her
final-exam score from her pre-exam total.
a) What is the slope of the least-squares regression line of
final-exam scores on pre-exam total scores in this course? What is
the intercept?
b) Use the regression line to predict Julie’s final-exam score.
c) Julie doesn’t think this method accurately predicts how well she
did on the final exam. Use r 2 to argue that her actual score could
have been much higher (or much lower) than the predicted value.
.
.
.
.
.
.
Solutions
S
8
a) b = r Syy = (0.5) 40
= 0.1
a = ȳ − bx̄ = 75 − (0.1)(280) = 47.
Hence, the regression equation is ŷ = 47 + 0.1x.
b) Julie’s pre-final exam total was 300, so we would predict a final
exam score of
ŷ = 47 + (0.1)(300) = 77.
c) Julie is right ... with a correlation of r = 0.5, r 2 = 0.25, so the
regression line accounts for only 25% of the variability in student
final exam scores. That is, the regression line doesn’t predict final
exam scores very well. Julie’s score could, indeed, be much higher
or lower than the predicted 77.
.
.
.
.
.
.
Residuals
A residual is the difference between an observed value of the
response variable and the value predicted by the regression line.
That is, a residual is the prediction error that remains after we
have chosen the regression line:
residual = observed y - predicted y
residual = y − ŷ .
.
.
.
.
.
.
Residual Plots
A residual plot is a scatterplot of the regression residuals against
the explanatory variable. Residual plots help us assess how well a
regression line fits the data.
.
.
.
.
.
.
Residuals by hand
In Exercise 5.3 (page 115) you found the equation of the
least-squares line for predicting coral growth y from mean sea
surface temperature x.
a) Use the equation to obtain the 7 residuals step-by-step. That is,
find the prediction ŷ for each observation and then find the
residual y − ŷ .
b) Check that (up to roundoff error) the residuals add to 0.
c) The residuals are the part of the response y left over after the
straight-line tie between y and x is removed. Show that the
correlation between the residuals and x is 0 (up to roundoff error).
That this correlation is always 0 is another special property of
least-squares regression.
.
.
.
.
.
.
Solutions
a) The residuals are computed in the table below using
ŷ = −0.3276x + 12.3758.
xi
29.68
29.87
30.16
30.22
30.48
30.65
30.90
yi
2.63
2.58
2.60
2.48
2.26
2.38
2.26
yˆi
2.65632
2.59038
2.495384
2.475728
2.390552
2.33486
2.25296
yi − yˆi
- 0.022632
- 0.010388
0.104616
0.004272
- 0.130552
0.04514
0.00704
∑
b) (yi − yˆi ) = −0.002504 (they sum to zero, except for rounding
error.
c) From software, the correlation between xi and yi − yˆi is
−0.0000854, which is zero except for rounding.
.
.
.
.
.
.
Do heavier people burn more energy?
Return to the data of Exercise 5.4 (page 117) on lean body mass
and metabolic rate. We will use these data to illustrate influence.
a) Make a scatterplot of the data that is suitable for predicting
metabolic rate from body mass, with two new points added. Point
A: mass 42 kilograms, metabolic rate 1500 calories. Point B: mass
70 kilograms, metabolic rate 1400 calories. In which direction is
each of these points an outlier?
b) Add three least-squares regression lines to your plot: for the
original 12 women, for the original women plus Point A, and for
the original women plus Point B. Which new point is more
influential for the regression line? Explain in simple language why
each new point moves the line in the way your graph shows.
.
.
.
.
.
.
900
1000
1100
1200
1300
Metabolic Rate (calories/day)
1400
1500
Scatterplot
A
B
30
40
Lean Body Mass (kg)
50
60
.
70
.
.
.
.
.
1500
Least-squares regression lines
1000
1100
1200
1300
B
original
original + A
original + B
900
Metabolic Rate (calories/day)
1400
A
30
40
50
60
70
Lean Body Mass (kg)
.
.
.
.
.
.
Solutions
a) Point A lies above the other points; that is, the metabolic rate
is higher than we expect for the given body mass. Point B lies to
the right of the other points; that is, it is an outlier in the x (mass)
direction, and the metabolic rate is lower than we would expect.
b) In the plot, the dashed blue line is the regression line for the
original data. The dashed red line slightly that includes Point A; it
has a very similar slope to the original line, but a slightly higher
intercept, because Point A pulls the line up. The third line includes
Point B, the more influential point; because Point B is an outlier in
the x direction, it ”pulls” the line down so that it is less steep.
.
.
.
.
.
.
Influential observations
An observation is influential for a statistical calculation if removing
it would markedly change the result of the calculation.
The result of a statistical calculation may be of little practical use
if it depends strongly on a few influential observations.
Points that are outliers in either the x or the y direction of a
scatterplot are often influential for the correlation. Points that are
outliers in the x direction are often influential for the least-squares
regression line.
.
.
.
.
.
.
The endangered manatee
Table 4.1 gives 33 years of data on boats registered in Florida and
manatees killed by boats. If we made a scatterplot for this data
set, it would show a strong positive linear relationship. The
correlation is r = 0.951.
a) Find the equation of the least-squares line for predicting
manatees killed from thousands of boats registered. Because the
linear pattern is so strong, we expect predictions from this line to
be quite accurate - but only if conditions in Florida remain similar
to those of the past 33 years.
b) In 2009, experts predicted that the number of boats registered
in Florida would be 975,000 in 2010. How many manatees do you
predict would be killed by boats if there are 975,000 boats
registered? Explain why we can trust this prediction.
c) Predict manatee deaths if there were no boats registered in
Florida. Explain why the predicted count of deaths is impossible.
.
.
.
.
.
.
Table 4.1
Year
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
Boats
447
460
481
498
513
512
526
559
585
614
645
Manatees
13
21
24
16
24
20
15
34
33
33
39
Year
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
Boats
675
711
719
681
679
678
696
713
732
755
809
.
Manatees
43
50
47
53
38
35
49
42
60
54
66
.
.
.
.
.
Table 4.1 (cont.)
Year
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
Boats
830
880
944
962
978
983
1010
1024
1027
1010
982
Manatees
82
78
81
95
73
69
79
92
73
90
97
.
.
.
.
.
.
Solutions
a) The regression line is ŷ = −43.172 + 0.129x.
b) If 975, 000 boats are registered, then by our scale, x = 975, and
ŷ = −43.172 + (0.129)(975) = 82.6 manatees killed. The
prediction seems reasonable, as long as conditions remain the same,
because ”975” is within the space of observed values of x on which
the regression line was based. That is, this is not extrapolation.
c) If x = 0 (corresponding to no registered boats), then we would
”predict” −43.172 manatees to be killed by boats. This is absurd,
because it is clearly impossible for fewer than 0 manatees to be
killed. Note that x = 0 is well outside the range of observed values
of x on which the regression line was based.
.
.
.
.
.
.
Extrapolation
Extrapolation is the use of a regression line for prediction far
outside the range of values of the explanatory variable x that you
used to obtain the line. Such predictions are often not accurate.
.
.
.
.
.
.
Is math the key to success in college?
A College Board study of 15,941 high school graduates found a
strong correlation between how much math minority students took
in high school and their later success in college. New articles
quoted the head of the College Board as saying that ”Math is the
gatekeeper for success in college.” Maybe so, but we should also
think about lurking variables. What might lead minority students
to take more or fewer high school math courses? Would these
same factors influence success in college?
.
.
.
.
.
.
Solution
A student’s intelligence may be a lurking variable: stronger
students (who are more likely to succeed when they get to college)
are more likely to choose to take these math courses, while weaker
students may avoid them. Other possible answers may be
variations on this idea; for example, if we believe that success in
college depends on a student’s self-confidence, and perhaps
confident students are more likely to choose math courses.
.
.
.
.
.
.
Lurking Variable
A lurking variable is a variable that is not among the explanatory
or response variables in a study and yet may influence the
interpretation of relationships among those variables.
.
.
.
.
.
.
Chapter 8
PRODUCING DATA: SAMPLING.
.
.
.
.
.
.
Population, Sample, Sampling Design
The population in a statistical study is the entire group of
individuals about which we want information.
A sample is a part of the population from which we actually collect
information. We use a sample to draw conclusions about the entire
population.
A sampling design describes exactly how to choose a sample from
the population.
The first step in planning a sample survey is to say exactly what
population we want to describe. The second step is to say exactly
what we want to measure, that is, to give exact definitions of our
variables.
.
.
.
.
.
.
Customer satisfaction
A department store mails a customer satisfaction survey to people
who make credit card purchases at the store. This month, 45,000
people made credit card purchases. Surveys are mailed to 1000 of
these people, chosen at random, and 137 people return the survey
form.
a) What is the population of interest for this survey?
b) What is the sample? From what group is information actually
obtained?
.
.
.
.
.
.
Solutions
a) The population is all 45,000 people who made credit card
purchases.
b) The sample is the 137 people who returned the survey form.
.
.
.
.
.
.
How to sample badly
The final step in planning a sample survey is the sampling design.
A sampling design is a specific method for choosing a sample from
the population. The easiest- but not the best - design just chooses
individuals close at hand. A sample selected by taking the members
of the population that are easiest to reach is called a convenience
sample. Convenience samples often produce unrepresentative data.
.
.
.
.
.
.
Bias
The design of a statistical study is biased if it systematically favors
certain outcomes.
.
.
.
.
.
.
Voluntary response sample
A voluntary response sample consists of people who choose
themselves by responding to a broad appeal. Voluntary response
samples are biased because people with strong opinions are most
likely to respond.
.
.
.
.
.
.
Sampling on campus
You see a woman student standing in front of the student center,
now and then stopping other students to ask them questions. She
says that she is collecting student opinions for a class assignment.
Explain why this sampling method is almost certainly biased.
.
.
.
.
.
.
Solution
It is a convenience sample; she is only getting opinions from
students who are at the student center at a certain time of day.
This might underrepresent some group: commuters, graduate
students, or nontraditional students, for example.
.
.
.
.
.
.
Simple Random Sampling
A simple random sample (SRS) of size n consists of n individuals
from the population chosen in such a way that every set of n
individuals has an equal chance to be the sample actually selected.
.
.
.
.
.
.
Random digits
A table of random digits is a long string of the digits 0, 1, 2, 3, 4,
5, 6, 7, 8, 9 with these two properties:
1. Each entry in the table is equally likely to be any of the 10
digits 0 through 9.
2. The entries are independent of each other. That is, knowledge
of one part of the table gives no information about any other part.
.
.
.
.
.
.
Using Table B to choose an SRS
Label: Give each member of the population a numerical label of
the same length.
Table: To choose a SRS, read from Table B successive groups of
digits of the length you used as labels. Your sample contains the
individuals whose labels you find in the table.
.
.
.
.
.
.
Apartment living
You are planning a report on apartment living in a college town.
You decide to select three apartment complexes at random for
in-depth interviews with residents. Use software or Table B to
select a simple random sample of 4 of the following apartment
complexes. If you use Table B, start at line 122.
.
.
.
.
.
.
Ashley Oaks
Bay Pointe
Beau Jardin
Bluffs
Brandon Place
Briarwood
Brownstone
Burberry Place
Cambridge
Chauncey Village
Country View
Country Villa
Crestview
Del-Lynn
Fairington
Fairway Knolls
Fowler
Franklin Park
Georgetown
Greenacres
Mayfair Village
Nobb Hill
Pemberly Courts
Peppermill
Pheasant Run
River Walk
Sagamore Ridge
Salem Courthouse
Village Square
Waterford Court
.
.
.
.
.
.
Solution
01
02
03
04
05
06
07
08
09
10
Ashley Oaks
Bay Pointe
Beau Jardin
Bluffs
Brandon Place
Briarwood
Brownstone
Burberry Place
Cambridge
Chauncey Village
11
12
13
14
15
16
17
18
19
20
Country View
Country Villa
Crestview
Del-Lynn
Fairington
Fairway Knolls
Fowler
Franklin Park
Georgetown
Greenacres
.
21
22
23
24
25
26
27
28
29
30
.
Mayfair Village
Nobb Hill
Pemberly Courts
Peppermill
Pheasant Run
River Walk
Sagamore Ridge
Salem Courthouse
Village Square
Waterford Court
.
.
.
.
Solution (cont.)
With Table B, enter at line 122 and choose 13 = Crestview, 15 =
Fairington, 05 = Brandon Place, and 29 = Village Square.
.
.
.
.
.
.
Inference about the Population
The purpose of a sample is to give us information about a larger
population. The process of drawing conclusions about a population
on the basis of sample data is called inference because we infer
information about the population from what we know about the
sample.
Inference from convenience samples or voluntary response samples
would be misleading because these methods of choosing a sample
are biased. We are almost certain that the sample does not fairly
represent the population. The first reason to rely on random
sampling is to eliminate bias in selecting samples from the list of
available individuals.
.
.
.
.
.
.
Sampling frame
The list of individuals from which a sample is actually selected is
called the sampling frame. Ideally, the frame should list every
individual in the population, but in practice this is often difficult. A
frame that leaves out part of the population is a common source of
undercoverage.
Suppose that a sample of households in a community is selected at
random from the telephone directory. What households are
omitted from this frame? What types of people do you think are
likely to live in these households? These people will probably be
underrepresented in the sample.
.
.
.
.
.
.
Solution
This design would omit households without telephones or with
unlisted numbers. Such households would likely be made up of
poor individuals (who cannot afford a phone), those who choose
not to have phones, and those who do not wish to have their
phone number published.
.
.
.
.
.
.
Cautions about sample surveys
Random selection eliminates bias in the choice of a sample from a
list of the population. When the population consists of human
beings, however, accurate information from a sample requires more
than a good sampling design.
To begin, we need an accurate and complete list of the population.
Because such a list is rarely available, most samples suffer from
some degree of undercoverage. A sample survey of households, for
example, will miss not only homeless people but prison inmates
and students in dormitories. An opinion poll conducted by calling
landline telephone numbers will miss households that have only cell
phones as well as households without a phone. The results of
national sample surveys therefore have some bias if the people not
covered differ from the rest of the population.
A more serious source of bias in most sample surveys is
nonresponse, which occurs when a selected individual cannot be
contacted or refuses to cooperate.
.
.
.
.
.
.
Nonresponse
Academic sample surveys, unlike commercial polls, often discuss
nonresponse. A survey of drivers began by randomly sampling all
listed residential telephone numbers in the United States.
Of 45,956 calls to these numbers, 5029 were completed. What was
the rate of nonresponse for this sample? (Only one call was made
to each number. Nonresponse would be lower if more calls were
made.)
.
.
.
.
.
.
Solution
5029
The response rate was 45956
= 0.1094, so the nonresponse rate
was 1 − 0.1094 = 0.8906
.
.
.
.
.
.
Undercoverage and nonresponse
Undercoverage occurs when some groups in the population are left
out of the process of choosing the sample.
Nonresponse occurs when an individual chosen for the sample
can’t be contacted or refuses to participate.
.
.
.
.
.
.
CHAPTER 9
Producing Data: Experiments
.
.
.
.
.
.
Observation vs Experiment
An observational study observes individuals and measures variables
of interest but does not attempt to influence the responses. The
purpose of an observational study is to describe some group or
situation.
An experiment, on the other hand, deliberately imposes some
treatment on individuals in order to observe their responses. The
purpose of an experiment is to study whether the treatment causes
a change in the response.
.
.
.
.
.
.
Subjects, Factors, Treatments
The individuals studied in an experiment are often called subjects,
particularly when they are people.
The explanatory variables in an experiment are often called factors.
A treatment is any specific experimental condition applied to the
subjects. If an experiment has more than one factor, a treatment is
a combination of specific values of each factor.
.
.
.
.
.
.
Sealing food packages
A manufacturer of food products uses package liners that are
sealed at the top by applying heated jaws after the package is
filled. The customer peels the sealed pieces apart to open the
package. What effect does the temperature of the jaws have on
the force needed to peel the liner? To answer this question,
engineers obtain 20 pairs of pieces of package liner. They seal five
pairs at each of 250◦ F, 275◦ F, 300◦ F, and 335◦ F. Then they
measure the force needed to peel each seal.
a) What are the individuals?
b) There is one factor (explanatory variable). What is it, and what
are its values?
c) What is the response variable?
.
.
.
.
.
.
Solutions
a) The individuals are pairs of pieces of package liner.
b) The explanatory variable is temperature of jaws, with four
values: 250◦ F, 275◦ F, 300◦ F, and 335◦ F.
c) The response variable is the force needed to peel each seal.
.
.
.
.
.
.
An industrial experiment
A chemical engineer is designing the production process for a new
product. The chemical reaction that produces the product may
have higher or lower yield, depending on the temperature and
stirring rate in the vessel in which the reaction takes place. The
engineer decides to investigate the effects of combinations of two
temperatures (50◦ C and 60◦ C) and three stirring rates (60 rpm,
90 rpm, and 120 rpm) on the yield of the process. She will process
two batches of the product at each combination of temperature
and stirring rate.
a) What are the individuals and the response variable in this
experiment?
b) How many factors are there? How many treatments?
.
.
.
.
.
.
Solutions
a) The individuals are batches of the product; the response variable
is the yield of a batch.
b) There are two factors (temperature and stirring rate) and six
treatments (temperature-stirring rate combinations).
60 rpm
90 rpm
120 rpm
50◦
Treatment 1
Treatment 3
Treatment 5
60◦
Treatment 2
Treatment 4
Treatment 6
.
.
.
.
.
.
Randomized Comparative Experiment
An experiment that uses both comparison of two or more
treatments and random assignment of subjects to treatments is a
randomized comparative experiment.
.
.
.
.
.
.
Completely Randomized Design
In a completely randomized experimental design, all the subjects
are allocated at random among all the treatments.
.
.
.
.
.
.
Adolescent obesity
Adolescent obesity is a serious health risk affecting more than 5
million young people in the United States alone. Laparoscopic
adjustable gastric banding has the potential to provide a safe and
effective treatment. Fifty adolescents between 14 and 18 years old
with a body mass index higher than 35 were recruited from the
Melbourne, Australia, community for the study. Twenty-five were
randomly selected to undergo gastric banding, and the remaining
25 were assigned to a supervised lifestyle intervention program
involving diet, exercise, and behavior modification. All subjects
were followed for two years and their weight loss was recorded.
a) Outline the design of this experiment. What is the response
variable?
b) Carry out the random assignment of 25 adolescents to the
gastric-banding group using software or Table B, starting at line
130.
.
.
.
.
.
.
Solutions
a) The diagram is provided. The response variable is weight loss.
b) Label the students from 01 to 50, and pick 25 to receive
Treatment 1 (the rest will receive Treatment 2). If using Table B,
line 130, the sample is 05, 16, 48, 17, 40, 20, 19, 45, 32, 41, 04,
25, 29, 43, 37, 39, 31, 50, 18, 07, 13, 33, 30, 21, 36.
.
.
.
.
.
.
Diagram
Group 1
25 Subjects
Treatment 1
Gastric banding
Random
Assignment
Compare
weight loss
Group 2
25 Subjects
Treatment 2
Lifestyle intervention
.
.
.
.
.
.
The Logic of Randomized Comparative Experiments
Randomized comparative experiments are designed to give good
evidence that differences in the treatments actually cause the
differences we see in the response. The logic is as follows:
I
Random assignment of subjects forms groups that should be
similar in all respects before the treatments are applied.
.
.
.
.
.
.
The Logic of Randomized Comparative Experiments
Randomized comparative experiments are designed to give good
evidence that differences in the treatments actually cause the
differences we see in the response. The logic is as follows:
I
Random assignment of subjects forms groups that should be
similar in all respects before the treatments are applied.
I
Comparative design ensures that influences other than the
experimental treatments operate equally on all groups.
.
.
.
.
.
.
The Logic of Randomized Comparative Experiments
Randomized comparative experiments are designed to give good
evidence that differences in the treatments actually cause the
differences we see in the response. The logic is as follows:
I
Random assignment of subjects forms groups that should be
similar in all respects before the treatments are applied.
I
Comparative design ensures that influences other than the
experimental treatments operate equally on all groups.
I
Therefore, differences in average response must be due either
to the treatments or to the play of chance in the random
assignment of subjects to the treatments.
.
.
.
.
.
.
Principles of Experimental Design
The basic principles of statistical design of experiments are:
I
Control the effects of lurking variables on the response, most
simply by comparing two or more treatments.
.
.
.
.
.
.
Principles of Experimental Design
The basic principles of statistical design of experiments are:
I
Control the effects of lurking variables on the response, most
simply by comparing two or more treatments.
I
Randomize - use chance to assign subjects to treatments.
.
.
.
.
.
.
Principles of Experimental Design
The basic principles of statistical design of experiments are:
I
Control the effects of lurking variables on the response, most
simply by comparing two or more treatments.
I
Randomize - use chance to assign subjects to treatments.
I
Use enough subjects in each group to reduce chance variation
in the results.
.
.
.
.
.
.
Statistical Significance
An observed effect so large that it would rarely occur by chance is
called statistically significant.
.
.
.
.
.
.
The Placebo and Double-Blind Methods
A placebo is a dummy treatment. Experiments in medicine and
psychology often give a placebo to a control group because just
being in an experiment can affect responses.
In a double-blind experiment, neither the subjects nor the people
who interact with them know which treatment each subject is
receiving.
.
.
.
.
.
.
Testosterone for older men
As men age, their testosterone levels gradually decrease. This may
cause a reduction in lean body mass, an increase in fat, and other
undesirable changes. Do testosterone supplements reverse some of
these effects? A study in the Netherlands assigned 237 men aged
60 to 80 with low or low-normal testosterone levels to either a
testosterone supplement or a placebo. The report in the Journal of
the American Medical Association described the study as a
”double-blind, randomized, placebo-controlled trial”. Explain each
of these terms to someone who knows no statistics.
.
.
.
.
.
.
Solution
”Double-blind” means that the treatment (testosterone or
placebo) assigned to a subject was unknown to both the subject
and those responsible for assessing the effectiveness of that
treatment. ”Randomized” means that the patients were randomly
assigned to receive either the testosterone supplement or a
placebo. ”Placebo-controlled” means that some of the subjects
were given placebos. Even though these possess no medical
properties, some subjects may show improvement or benefits just
as a result of participating in the experiment; the placebos allow
those doing the study to observe this effect.
.
.
.
.
.
.
Matched Pairs Design
A matched pairs design compares two treatments. Choose pairs of
subjects that are as closely matched as possible. Use chance to
decide which subject in a pair gets the first treatment. The other
subject in that pair gets the other treatment.
Sometimes each ”pair” in a matched pairs design consists of just
one subject, who gets both treatments one after the other. Use
chance to decide the order in which subjects receive the
treatments.
.
.
.
.
.
.
Comparing breathing frequencies in swimming
Researchers from the United Kingdom studied the effect of two
breathing frequencies on performance times and on several
physiological parameters in front crawl swimming. The breathing
frequencies were one breath every second stroke (B2) and one
breath every fourth stroke (B4). Subjects were 10 male collegiate
swimmers. Each subject swam 200 meters, once with breathing
frequency B2 and once on a different day with breathing frequency
B4.
Describe the design of this matched pairs experiment, including the
randomization required by this design.
.
.
.
.
.
.
Solution
Each swimmer swims one time using each breathing technique (B2
and B4). A coin is tossed to determine the order in which these
techniques are used.
.
.
.
.
.
.
Alcohol and heart attacks
Many studies have found that people who drink alcohol in
moderation have lower risk of heart attacks than either nondrinkers
or heavy drinkers. Does alcohol consumption also improve survival
after a heart attack? One study followed 1913 people who were
hospitalized after severe heart attacks. In the year before their
heart attacks, 47% of these people did not drink, 36% drank
moderately, and 17% drank heavily. After four years, fewer of the
moderate drinkers had died.
a) Is this an observational study or an experiment? Why? What
are the explanatory and response variables?
b) Suggest some lurking variables that may be confounded with the
drinking habits of the subjects. The possible confounding makes it
difficult to conclude that drinking habits explain death rates.
.
.
.
.
.
.
Solution
a) This is an observational study; the subjects chose their own
”treatments” (how much to drink). The explanatory variable is
alcohol consumption, and the response variable is whether or not a
subject dies.
b) Many answers are possible. For example, some nondrinkers
might avoid drinking because of other health concerns. We do not
know what kind of alcohol (beer? wine? whiskey?) the subjects
were drinking.
.
.
.
.
.
.
Confounding
Two variables (explanatory variables or lurking variables) are
confounded when their effects on a response variable cannot be
distinguished from each other.
.
.
.
.
.
.