Section 8.1
Statistics and
Sampling Variability
Introduction
 Suppose I want to know the average GPA of seniors at
Glacier Peak.
 I could track down every senior at GP and ask them their
GPA, this would give me the true mean of the population
 OR
 I could take a sample
The Foundation of Chapter 8
 Suppose that x is a discrete random variable that is equal to an
individual’s score on the 2010 AP Statistics Exam.
 As a first year AP stats teacher, I am interested in the mean score
of all students who took the exam in 2010 so that I may set
realistic expectations for myself and my students for this year.
 What is the population that I am interested
in?_____________________
 What statistic am I interested in? __________________
The Foundation of Chapter 8
 Let let µ denote the true mean score of all students who took the
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AP Stats exam in 2010.
To learn something about µ, I might obtain a sample of 50
students and determine the mean score from the sample. This
sample may produce a mean of 3.01. So, x = 3.01.
How close is this mean to the true population mean, µ?
If I selected another sample of 50 students and computed the
mean score, would this second x be near 3.01 or would it be
quite different?
If I repeated this process many, many times and plotted the
resulting means, I would create a sampling distribution of x. This
would give me an idea about the long run behavior of the sample
mean and could help me determine the true population mean, µ.
The Foundation of Chapter 8
 Questions regarding the repeated sampling can be addressed
by studying what is called the sampling distribution of
 Just as the distribution of a numerical variable describes its
long-run behavior, the sampling distribution
of provides
x
information about the long-run behavior of when sample
after sample is selected.
x
x
 I can obtain information about a population characteristic by
selecting a sample.
 Sample Mean:
 Population Mean:
 Often different from one another, and rarely actual values from
the data set
Basic Terms
 Any quantity computed from values in a sample is called a
statistic (x, s, p, etc.)
 Any quantity computed from values in a population is called
a population characteristic or parameter (, , )
Sampling Variability
 The observed value of a statistic depends on the particular
sample selected from the population.
 Typically, the value of the statistic varies from sample to
sample.
 This variability is called sampling variability.
Constructing a Sampling Distribution
 Suppose I want to take a random sample of size n = 50
from the population of all students who took the 2009
AP Stats exam.
 There are many, many different possible samples that
might result.
 We now define a hypothetical population, which consists
of all the different possible samples of size n = 50.
 This is called a population of samples. The population of
samples is viewed as a population because it consists of
every different sample; it is a complete collection of all
possible samples.
Constructing a Sampling Distribution
 Just as a variable associates a value with every individual in
the population and can be described by its distribution, a
statistic associates a value with each individual sample in the
population of samples.
 Therefore, a statistic can also be described by a distribution.
 The distribution of a statistic is called its sampling
distribution.
Ex: For 4 students, the number of siblings they each have is 0, 1, 3,
4. Select samples of size n=2 and find the average number of
siblings in each sample. Obtain the sampling distribution of x.
Sample of Size 2
x
Probability (x )
Ex: From the previous example, determine the x , the
mean value of x
.
Ex: Consider a population consisting of the following five values,
which represent the number of DVD rentals from the Red Box at
Fred Meyer for a given month of five families.
 The values are: 8, 14, 16, 10, 11.
 Compute the mean of this population.
 Notice that the 5 slips of paper you created for the warm up
correspond to these values.
 Turn your paper over and randomly select a sample of size n
= 2 and compute the mean of your sample.
 Come up and record your sample and mean on the
following table. Copy the values in the table.
Ex: Record your sample and mean in the table below.
Sample Mean Sample Mean Sample Mean
Ex: Construct a histogram for the sample mean
data in the table.
Ex: Consider the density histogram for the
sample mean data in the previous slide.
 Are most of the values of x near the population mean?
 Do the x values differ a lot from sample to sample or
do they tend to be similar?
Homework
Page 409 1,2,3,4a,7,10
Read pages 411-420