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ST 432 HW1 SP2017
2.3
The sampling design depends on a careful definition of the population of interest. As
it would be almost impossible to get a listing of all cars owned by residents of a city, a
better option would be to restrict the population of cars to something like "cars that
use city parking lots on a working day" or "cars that belong to people visiting the
malls on a weekend." Then, a listing of parking lots or sections of parking lots could
serve as frames for collections of cars.
2.23
(a)
One rating point represents one percent of the viewing households, or
95.1 million x 0.01 = 951,000 households
based on the fact that the sampled population is households.
(b)
As a percentage, a share is larger than a rating because the denominator of the
rating is the total number of sampled households, while the denominator of a
share is the total number of sampled households that actually have a TV set
turned on (viewing households).
3.4
(c)
95.1 million x 0.217 = 20.64 million households could have been viewing this
show
(d)
Much of the data collected by Nielsen depends upon people in the sampled
households either pushing a button on a People Meter or writing in a diary to
record what they are watching. This is far from a fool-proof system.
A sampling distribution is a distribution of all possible values of a statistic.
3.14
(a)
  E( x)   xp( x)
= 2(.443) + 3(.229) + 4(.200) + 5(.086) + 6(.028) + 7(.014) = 3.069
This approximation to the mean will be too large since the distribution of family size is
right-skewed.
(b)
(c)
(d)
 2  V ( x)   ( x   ) 2 p( x)  (2  3.069) 2 (.443) 
 1.458;   V ( x)  1.207
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
E ( x )  3.07 and standard deviation SD( x )  
3.15
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(a)
 (7  3.069) 2 (.014)
n  1.21 20  0.0605
The scatter plot shows that SAT and Percent are negatively correlated, with a
slightly curved pattern suggesting that the average score drops quickly as the
percentages begin to increase and then 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
ST 432 HW1 SP2017
exam.
(b) The correlation coefficient is -0.877, but this is not a good measure to use here
because of the curvature in the pattern. Correlation measures the strength of a
linear relationship between two variables.
Scatter plot between Average Score and Percent
Average State SAT vs Percent HS
Seniors Taking SAT
1250
A
v
e
r
a
g
e
S
A
T
1200
S 1150
c
1100
o
r
1050
e
1000
950
0
20
40
60
Percent Seniors Taking SAT
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80
100
ST 432 HW1 SP2017
3.20
A histogram of 200 sample means from samples of size 5 each are shown in the
second histogram below; the first histogram below is the histogram of the
population. The histogram of the sample means is somewhat skewed because the
population distribution of teachers per state is highly skewed. Even so, the mean of
the sampling distribution is 58,820, quite close to the population mean of 59,856.
The standard deviation of the sampling distribution is 25,643, quite close to the
theoretical value of 28,465.
Histogram of population: number of teachers in each of the 50 states.
Results for this problem will vary from student to student.
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ST 432 HW1 SP2017
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