Parameters and Statistics
What is the average income of American households? Each March,
the government’s Current Population Survey (CPS) asks detailed
questions about income. The random sample of 54,100 households
contacted in March 2009 had a mean “total money income” of
$68,424 in 2008.That $68,424 describes the sample, but we use it to
estimate the mean income of all households.
Population
Parameter
the mean income μ of all households
Sample
Statistics
x  $68,424
  population mean
x  sample mean
p  population proportion
pˆ  sample proportion
From Ghosts to Cold Cabins
Identify the population, the parameter, the sample, and the statistic
in each of the following settings
(b) During the winter months, the
temperatures
outside
thea random
Starneses’
(a)
The Gallup Poll
asked
cabin inofColorado
staywhether
well below
sample
515 U.S.can
adults
or
freezing
(32°F, orin0°C)
for weeks
not
they believe
ghosts.
Of theat a
time. To prevent
respondents,
160 the
saidpipes
“Yes.”from
freezing, Mrs. Starnes sets the
thermostatAll
at US
50°F.
She wants to know
Population:
adults
how low the temperature actually gets
Parameter:
p, A
the
proportion
of all U.S. adults who believe in ghosts
in the cabin.
digital
thermometer
recordsthe
the515
indoor
temperature
20
Sample:
people
who wereatinterviewed
randomly chosen times during a given
160
day. The minimum
reading
pˆ 

0.31 is 38°F.
Statistic:
515
CHECK YOUR UNDERSTANDING
Each boldface number below is the value of either a parameter
or a statistic. In each case, state which it is and use appropriate
notation to describe the number.
2.On
OnTuesday,
a New York–to–Denver
flight, 8%
ofTea
the filled
125 in a plant
1.
the bottles of Arizona
Iced
passengers
were
foraverage
randomof
security
screening
were
supposed
toselected
contain an
20 ounces
of iced
before
boarding.
According
to sampled
the Transportation
tea.
Quality
control
inspectors
50 bottles Security
at random
Administration,
10% of passengers
at thiscontained
airport are
from
the day’s production.
These bottles
an
chosen for
random
screening
average
of 19.6
ounces
of iced tea.
The
or 10%ofoficed
passengers
Theparameter
parameterisispμ==0.10,
20 ounces
tea
.08
8% of of
The statistic is pˆx019
theiced
sample
tea of passengers
.6orounces
Sampling Cards
Describe what you see: shape, center,
spread and any unusual values
Is this a
Sampling
Distribution?
Sample Median
We used Fathom to simulate choosing 500 SRSs of size 5 from
the deck of cards. The graph below shows the distribution
of thefrom
sample
median
for
Dot
Plot these 500
Measures
Sample
of Collecti...
2
3
samples.
4 5 6 7 8 9 10
SampleMedian
(c)
another
student prepared
a different
deck of
(a) Suppose
Is this thethat
sampling
distribution
of the sample
median?
cards
Justifyand
yourclaimed
answer.that it was exactly the same as the one used
in the activity. However, when you took an SRS of size 5, the
(b)
Describe
theDoes
distribution.
Are convincing
there any obvious
outliers?
median
was 4.
this provide
evidence
that the
student’s deck is different?
Biased vs. Unbiased estimators
Sampling sleep
Calculate the sample IQR of sleep
hours and the sample maximum of
sleep hours and plot the values of
these statistics on the dot plots
below:
Average IQR:
Average Max:
Make a graph of the population of sleep hours and calculate
the true values of the IQR and maximum.
Sleep
Based on these values and the approximate sampling
distributions, do either of these statistics appear to be
unbiased estimators?
Who Watches Survivor?
Why sample size matters
Television executives and
companies who advertise on TV
are interested in how many
viewers watch particular shows.
According to Nielsen ratings,
Survivor was one of the mostwatched television shows in the
United States during every week
that it aired. Suppose that the
true proportion of U.S. adults who
have watched Survivor is p = 0.37.
Variability of a Statistic
Sample size of 100
Sample size of 1000
Variability of a Statistic
The variability of a statistic is described by the spread
of its sampling distribution.
•This spread is determined primarily by the size of the
random sample.
• Larger samples give smaller spread.
•The population should be at least 10 times larger
than the sample.
Exercises on page 428,
#1-13 odds,
# 17-20 all
Read 7.2
We used Fathom software to simulate
choosing 500 SRSs of size n = 20 from
a population of 200 chips, 100 red
and 100 blue. Below is a dot plot of
the values of p̂ , the sample
proportion of red chips, from these
500 samples.
(a)
Is this theyour
sampling
distribution of p̂ ?
(c) Suppose
teacher
prepares
a bag
200 chipsAre
(b) Describe
thewith
distribution.
and
that halfoutliers?
of them are
thereclaims
any obvious
red. A classmate takes an SRS of
20 chips; 17 of them are red.
What would you conclude about
your teacher’s claim? Explain.