Define the
sampling distribution of a statistic
.
Define an
unbiased statistic
and an
unbiased estimator
.
Describe what is meant by the
variability of a statistic
.
Construction Objectives:
Students will be able to:
Explain how to
describe
a sampling distribution.
Explain how
bias
and
variability
are related to estimating with a sample.
Vocabulary:
Parameter – a number that describes the population
Statistic – a number that can be computed from the sample data without making use of any unknown parameters
μ (Greek letter mu) – symbol used for the mean of a population
x
̄
(x-bar) – symbol used for the mean of the sample
Sampling Distribution (of a statistic) – the distribution of values taken by the statistic in all possible samples of the same
size from the same population
Bias – the level of trustworthiness of a statistic
Unbiased Statistic – a statistic whose sampling distribution mean is equal to the true value of the parameter being
estimated; also known as an
unbiased estimator
Variability (of a statistic) – a description of the spread of the statistic’s sampling distribution
Key Concepts:
•
Population Parameters
–
Usually unknown and are estimated by sample statistics using techniques we will learn
–
Mean:
μ
–
Standard Deviation:
σ
–
Proportion:
p
•
Sample Statistics
–
Used to estimate population parameters
–
Mean:
x
̄
–
Standard Deviation:
s
–
Proportion:
p
̂
Sampling Distribution
In other words:
a sampling distribution of proportions is using
the proportion of an individual sample as the data point of the
samples of
p
̂
– the “bigger” sample.
Population of passengers going through the airport
Daily
sample
of 100
Daily
sample
of 100
Daily
sample
of 100
Daily
sample
of 100
Daily
sample
of 100
Daily
sample
of 100
Sampling Distribution of p
̂

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Chapter 9:
Sampling Distributions
Example 1:
Upon entry to an airport’s customs area each passenger presses a button and either a green arrow comes on
(directing the passenger on through) or a red arrow comes on (directing them to a customs agent) and they have the bags
searched.
Homeland Security sets the “search” parameter at 30%.
a)
What type of probability distribution applies here?
b)
What are the mean and standard deviation of this distribution?
c)
Each of you represents a day, 8 in total, that we are going to simulate a simple random sampling of 100 passengers
passing through the airport.
We want to know what your individual average proportion of those who got the green
arrow.
This we will refer to as p-hat or p.
To do this we will use our calculator.
̂
d)
We can also use our calculator to simulate this and just get the total number, which represents p-hat or p.
̂
e)
Describe the distribution below
Example 2:
Which of these sampling distributions displays large or small bias and large or small variability?

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