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CHAPTER 3
A REVIEW OF SOME BASIC CONCEPTS
3.1
Statistics has many definitions, but almost all involve the process of drawing conclusions
from data. Data is subject to variability, so some would say that statistics is the study of
variability with the objective of understanding its sources, measuring it, controlling
whatever is controllable, and drawing conclusions in the face of it. For sample survey
purposes, statistics involves a well-defined population, a sample selected according to an
appropriate probabilistic design, and a methodology for making inferences from the
sample to the population, usually in terms of estimation of population parameters.
3.2
A statistic is a function of (is calculated from) sample data whereas a parameter is a
numerical characteristic of a population. In a common opinion poll, a sample of 500
residents may be asked whether or no they favor a certain candidate for office. The
sample percentage is a statistic, but it is used to estimate the population percentage
favoring that candidate, an unknown parameter.
3.3
An estimator is a statistic used to estimate a population parameter, like the sample
proportion in Exercise 3.2.
3.4
A sampling distribution is a distribution of all possible values of a statistic.
3.5 The goodness of an estimator is usually measured by the standard deviation of its
sampling distribution. The margin of error refers to two standard deviations of the
sampling distribution of an estimator. Roughly speaking, the difference between an
estimator and the true value of the parameter being estimated will be less than the
margin of error with probability about .95.
3.6
An estimator should be unbiased (or nearly so) and have a small standard deviation of its
sampling distribution. In other words, in repeated usage, an estimatorโs values should
pile up close to the value of the parameter being estimated.
3.7
An unbiased estimator is one for which the sampling distribution centers at the true
value of the parameter being estimated.
7
3.8
3.10
The error of estimation refers to the difference between an estimator and the true value
of a parameter being estimated. It is measured by the standard deviation of the sampling
distribution of the estimator in question.
Summary Statistics
Calories
Cost in Dollars
w/Hydra w/o Hydra w/ Hydra w/o Hydra
Mean
64.78
64.62
.294
.27
Median 66.0
63.5
.260
.25
Stdev
8.51
9.09
.097
.05
60
60
.230
.225
Q1
70
70
.345
.320
Q3
Min
50
50
.220
.220
Max
80
80
.520
.350
Range
30
30
.300
.130
Scatterplot of cost vs. calories
0.50+
cost
0.40+
0.30+
0.20+
* Hydra
*
*
*
*
*
*
2
+———+———+———+———+———+—–calories
48.0
54.0
60.0
66.0
72.0
78.0
8
Scatterplot of cost vs. calories (without Hydra)
0.350+
*
*
cost
0.300+
*
*
0.250+
*
*
2
0.200+
+———+———+———+———+———+—–calories
48.0
54.0
60.0
66.0
72.0
78.0
(a)
The mean is a good summary number for typical calories per serving.
The standard deviation is a good summary number for the variation in the
calories.
Box plot of calories
———————————-I
+
I——————————–+———+———+———+———+———+—–48.0
54.0
60.0
66.0
72.0
78.0
(b)
Since there is an extreme value, the median is a good summary number for
typical cost per serving, and IQR (Q3 – Q1) is a good summary for the variation
in costs.
Box plot of cost
—————–Hydra
—I +
I*
———————–+———+———+———+———+———+
0.240
0.300
0.360
0.420
0.480
0.540
(c)
Because these drinks would not generally be combined by users, the totals have
little practical value here.
(d)
On the average calories per serving; not much impact
On the standard deviation: slight increase
On the average cost per serving: decrease
9
On the standard deviation of the cost per serving: decreased
Box plot of calories (without Hydra)
———————————-I
+
I——————————–+———+———+———+———+———+—–48.0
54.0
60.0
66.0
72.0
78.0
Box plot of cost (without Hydra)
——————————–I
+
I—————————————————+———+———+———+———+———+-0.225
0.250
0.275
0.300
0.325
0.350
3.11
(e)
There is no particularly influential drink on the average calories per saving, but
Snappple (80 calories) has the most influence as it is furthest from the mean.
(a)
Including the powdered drinks on the same list with the liquid drinks does not
have much effect on the average calories per serving, as their calorie figures
are within the range of the first data set. Including the powdered drinks lowers
the average cost per serving and increases the standard deviation of cost
because the new cost values are much lower than in the original set.
w/o powder
w powder
calories
mean
stdev
64.78
8.51
64.82
7.93
cost
mean
.294
.278
stdev
.097
.102
Parallel box plots of calories
w/o powder
———————————-I
+
I———————————
———————————-I
+
I——————————–+———+———+———+———+———+—–48.0
54.0
60.0
66.0
72.0
78.0
w/ powder
10
Parallel box plots of cost
—————I +
I————–
w/o powder
w/ powder
(b)
*
————————I +
I—-*
——————+———+———+———+———+——-0.160
0.240
0.320
0.400
0.480
Adding the light varieties to the list will not have much of an effect on the
average cost and standard deviation of cost.
Mean and standard deviation for two groups (cost)
w/o lites
w/ lites
mean
.294
.286
stdev
.097
.088
Parallel box plots of cost
w/o lites
w/ lites
(c)
——————–I +
I——————
*
————-I +
I——-O
—————-+———+———+———+———+———+
0.240
0.300
0.360
0.420
0.480
0.540
Adding the light varieties to the list will decrease the average calories per
serving and increase the standard deviation of calories because the new cost
figures are way below those of the original data set.
Mean and standard deviation for two groups (calories)
w/o lites
w/ lites
mean
64.78
55.45
stdev
8.51
22.69
11
Parallel box plots of calories
————–I
+ I————-
w/o lites
w/ lites
(d)
3.12
————-I +
I—————+———+———+———+———+———+—–0
15
30
45
60
75
O
*
Use the median because the median is not sensitive to extreme values.
Summary Statistics
Area
U.S
U.S.& Foreign
(a)
N
10
10
mean
25.10
4.80
median
12.50
1.00
stdev
Q1
22.02 7.50
7.18 0.00
Q3
51.25
10.00
Q3 – Q1
43.75
10.00
As shown on the stem plot, these data are split into two groups and neither the
mean nor the median are good measures of center. A more meaningful
summary statistic is the total number of endangered species, 251 for those
unique to the U.S. and 299 in the U.S. and foreign countries.
Box plot of U.S.
———————————————–I
+
I——————————————————–+———+———+———+———+——-10
20
30
40
50
Stem plot of U.S.
Stem-and-leaf of US
Leaf Unit = 1.0
3
(3)
4
4
3
3
(b)
0 368
1 023
2
3 7
4
5 057
Again, the total number of endangered species is a more meaningful statistic
than either the mean or median. For the world, this total is 791 species.
12
Stem plot of U.S & Foreign
Stem-and-leaf of US & Foreign
Leaf Unit = 1.0
5
5
3
3
3
2
2
2
2
1
0 00000
0 23
0
0
0 8
1
1
1
1 6
1 9
Box plot of U.S. & Foreign
———————–I +
I—————————————————–+———+———+———+———+———+—–0.0
3.5
7.0
10.5
14.0
17.5
3.13
(c)
No. See part (a).
(a)
58
1
(3 ร 16 + 2 ร 4 + 1 ร 2) =
= 2.32
25
25
(b)
ยต = โ xp( x ) = 3ร.64 + 2ร.16 + 1ร.08 + 0ร.12 = 2.32
(c)
V ( x ) = โ ( x โ ยต ) 2 p( x ) = โ x 2 p( x ) โ ยต 2
x
x
= 3 2 (.64) + 2 2 (.16) + 12 (.08) + 0 2 (.12) โ 2.32 2 = 6.48 โ 5.3824 = 10976
.
ฯ = V ( x ) = 105
.
3.14
(a)
ยต = E ( x ) = โ xp( x )
= 2(.443) + 3(.229) + 4(.200) + 5(.086) + 6(.028) + 7(.014) = 3.069
(b)
ฯ 2 = V( x) =
โ ( x โ ยต ) p(x) = 1.458
2
x
ฯ = V(x) = 1.207
13
(c)
(d)
3.15
The distribution of the sample data would reflect that of the population. Most
of the data values would pile up around 2 and 3 , with a few larger values. The
distribution of the sample would be skewed toward the larger values, with a
center at approximately 3.07 and a standard deviation of approximately 1.21.
The sample mean x has approximately a normal distribution with mean
ฯ
1.21
ยตx = ยตx = 3.07 and standard deviation ฯ x =
=
= 0.0605
n
20
(a)
The scatter plot shows that SAT and Percent are negatively correlated, with a
curved pattern suggesting that the average score drops quickly as the
percentages begin to increase and them levels off for higher percentages. The
decreasing scores with increasing percentage taking the exam makes practical
sense; in states with small percentages only the very best students are taking the
exam.
(b)
The correlation coefficient is -0.877, but this is not a good measure to use here
because of the curvature in the patter. Correlation measures the strength of a
linear relationship between two variables.
Scatter plot between Average Score and Percent
x
1200
xx x xx
xx
x
x x x
x
x
x
x
x x
x x
Aver age
x
xx
x
1050
x
x
x
x
x
x
x
x xx
x
x
x
x
x
2
x
xxx
x
x
x
x
x
x
0
3.16
20
40
60
80
Tabled below are the new probabilities for samples of size 2 and estimates of the
population total for the unequal probabilities of selection that favor the smaller
population values.
14
Per cent
Sample
Probability
ฯห pps
{1.2}
{1.3}
{1.4}
{2,3}
{2,4}
{3,4}
{1,1}
{2.2}
{3,3}
{4,4}
0.32
0.08
0.08
0.08
0.08
0.02
0.16
0.16
0.01
0.01
3.75
16.25
21.25
17.50
22.50
35.00
2.50
5.00
30.00
40.00
Calculation of expectations yields:
E ฯห pps = 10
( )
V (ฯห pps) = 81.25
3.17
The weights given in Section 3.3 for the four population values are w1 = 4.0916, w2 =
4.0916, w3 = 1.3236 and w4=1.3236. The sum of the weights for each of the six
possible samples, along with the probabilities of selecting each of these samples, are
shown in the accompanying table. The expected value of the sum of the weights turns
out to be 4.00, the number of values in the population.
Sample
3.18
Sum of weights
{1,2}
8.1832
{1,3}
{1,4}
{2,3}
{2,4}
{3,4}
5.4152
5.4152
5.4152
5.4152
2.6472
Probability of sample,
unequal weights
.0222
.1111
.1111
.1111
.1111
.5333
For samples of size n=2 taken with probabilities proportional to the populations of the
states, the pertinent data and the probabilities of selection with probabilities
proportional to the population, both with and without replacement, are given in the
first table that follows. With replacement probabilities of selection (ฮด) are directly
proportional to the population sizes. Without replacement probabilities of selection
(ฯ) are found by first finding the probability for each possible sample, given on the
15
second table. Note that there are 21 with replacement samples of size 2, but only 15
without replacement samples. (Also, note that the ฯโs sum to 2, the sample size.)
Students
Teachers
Population
570
206
973
207
158
101
42
17
69
15
11
8
35
13
64
13
11
6
With
Replacement
Probability of
selection, ฮด
0.25
0.09
0.45
0.09
0.08
0.04
Without
Replacement
Probability of
selection,ฯ
0.536150
0.214110
0.746890
0.214110
0.191280
0.097451
The estimates of the total number of teachers are found by using the formulas of
Section 3.3. The expected value of each set of estimates, with their appropriate
probability distributions, is 162, the total number of teachers for New England.
3.19
Sample
Probability
(with
replacement)
Estimate
(with
replacement)
Probability
(without
replacement)
Estimate
(without
replacement)
{1.2}
{1,3}
{1,4}
{1,5}
{1,6}
{2,3}
{2,4}
{2,5}
{2,6}
{3.4}
{3,5}
{3,6}
(4,5}
{4,6}
{5,6}
{1,1}
{2,2}
{3,3}
{4,4}
{5,5}
{6,6}
0.0450
0.2250
0.0450
0.0400
0.0200
0.0810
0.0162
0.0144
0.0072
0.0810
0.0720
0.0360
0.0144
0.0072
0.0064
0.0625
0.0081
0.2025
0.0081
0.0064
0.0016
178.444
160.667
167.333
152.750
184.000
171.111
177.778
163.195
194.445
160.000
145.417
176.667
152.083
183.333
168.750
168.000
188.889
153.333
166.667
137.500
200.000
0.054725
0.354545
0.054725
0.048406
0.023750
0.118142
0.017802
0.015738
0.007706
0.118142
0.104585
0.051477
0.015738
0.007706
0.006812
157.735
170.719
148.394
135.844
160.429
171.781
149.456
136.906
161.491
162.441
149.890
174.476
127.565
152.150
139.600
No; the proportions of students in the various states are about the same as the
proportions of the total population.
16
3.20
A histogram of 200 sample means from samples of size 5 each are shown in the
histogram. This distribution is somewhat skewed because the population distribution
of teachers per state is highly skewed. Even so, the mean of the sampling distribution
is 61, 147, quite close to the population mean of 59,856. The standard deviation of
the sampling distribution is 26,772, quite close to the theoretical value of 28,645.
Histogram
Means f rom samples of 5
40
35
30
25
20
15
10
5
0
40000
80000
120000
160000
Mean
3.21
p(u1 ) = p(u 2 ) = โ
โ
โ
= p(u N ) = 1 / N
ฯ 2 = V ( y ) = E ( y โ ยต ) 2 = โ ( y โ ยต ) 2 p( y ) =
y
3.22
1 N
โ ( ui โ ยต ) 2
N i =1
Let ai denote the number of times a particular yi value from the population appears in
the sample. This number could be greater than 1 because the sampling is with
replacement. Then,
N
n
y
yi 1
1
ai i
=
ฯห =
ฮดi
ฮดi n
n
โ
โ
i =1
i=1
where n is the sample size and N the population size. Since E(ai ) = nฮดi , it follows
that E(ฯห )= ฯ .
Now, ฯห is the mean of n independent variables, and so its variance is given by
V (ฯห )=
1
n
N
โ
๏ฃซy
๏ฃถ
ฮดi ๏ฃฌ i โ ฯ ๏ฃท
๏ฃญฮดi
๏ฃธ
i =1
17
2
The estimate of this variance can be rewritten as follows:
n
โ
y
ห (ฯห) = 1 โ
1
V
( i โ ฯห ) 2
n n โ 1 i =1 ฮด i
n
=
โ
1 1
y
โ
[( i โ ฯ ) โ (ฯห โ ฯ )] 2
n n โ 1 i =1 ฮดi
n
โ
1
1
y
= โ
[ ( i โ ฯ)2 โ
n n โ 1 i =1 ฮดi
N
=
n
โ (ฯห โ ฯ ) ]
2
i=1
N
1
1
y
โ
[ ai ( i โ ฯ )2 โ
ai (ฯห โ ฯ ) 2 ]
n n โ 1 i =1 ฮด i
โ
โ
i =1
N
=
โ
1
y
1
โ
[ ai ( i โ ฯ )2 โ n(ฯห โ ฯ ) 2 ]
n n โ 1 i =1 ฮด i
Taking expected values shows that:
ห (ฯห )] =
E[ V
=
3.23
N
1 1
y
โ
[ nฮด i ( i โ ฯ ) 2 โ nE(ฯห โ ฯ ) 2 ]
n n โ 1 i =1
ฮดi
โ
1
1
โ
[n 2 V (ฯห) โ nV (ฯห )] = V (ฯห )
n n โ1
Aspirin
Placebo
Exercise Vigorously
Yes
No
Total
7 910
2997
10907
7861
3060
10921
Cigarette smoking
Never
Past
Current
Total
5431
4373
1213
11017
5488
4301
1225
11014
(a)
Compare the two columns we see that the counts are nearly the same across all
categories. The randomization scheme did a good job in balancing these
variables between the two groups.
(b)
No.
(c)
5431
=.49,
11017
2997
No.
=.27,
10907
5488
=.50 are nearly the same.
11014
3060
=.28 are nearly the same.
10921
18
3.24
Heart Attack
Yes
No
Total
Yes,
3.25
No,
Aspirin
139
10861
11000
139
=.012636
11000
Placebo
239
10761
11000
239
=.021727 are not close.
10682
Stroke
Yes
No
Total
Aspirin
119
10881
11000
Placebo
98
10902
11000
119
=.0108
11000
98
=.0089 are close.
11000
.021727
.0089
= 172
=.82 , we can find Aspirin is more
. ,
.012636
.0108
effective as a possible prevention for heart attacks than for strokes.
Comparing the two ratios
3.26
The rate of heart attacks for the smokers (21/1213 = 0.0173) is greater than the rate
for those who never smoked (55/5431 = 0.0101), but the effectiveness of the aspirin is
about the same for both groups. One way to demonstrate the latter is to look at the
ratio of the heart attack rates for aspirin and placebo treatments, as shown here.
21 / 1213
= .573,
37 / 1225
55 / 5431
= .579
96 / 5488
19

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