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AP Statistics: Section 9.1
Objectives
 DISTINGUISH between a parameter and a statistic

DEFINE sampling distribution

DISTINGUISH between population distribution, sampling distribution, and the distribution of sample data

DETERMINE whether a statistic is an unbiased estimator of a population parameter

DESCRIBE the relationship between sample size and the variability of an estimator
Introduction
The process of statistical inference involves using information from a sample to draw conclusions about a wider
population.
Population
Collect data from a representative Sample...
Sample
Make an Inference about the Population.
Different random samples yield different statistics. We need to be able to describe the sampling distribution of
possible statistic values in order to perform statistical inference.
We can think of a statistic as a random variable because it takes numerical values that describe the outcomes of
the random sampling process. Therefore, we can examine its probability distribution using what we learned in
previous chapters.
Parameters and Statistics
As we begin to use sample data to draw conclusions about a wider population, we must be clear about whether a
number describes a sample or a population.

A parameter is a number that describes some characteristic of the population. In statistical practice, the
value of a parameter is usually not known because we cannot examine the entire population.

A statistic is a number that describes some characteristic of a sample. The value of a statistic can be
computed directly from the sample data. We often use a statistic to estimate an unknown parameter.
What we have seen:
Mean
Proportion
Standard Deviation
Parameter
µ
p or π
σ
Statistic
𝑥̅
𝑝̂
s
Other parameters/statistics that we have seen: percentiles, Q1, Q3, median, IQR, variance, min, max
EXAMPLE 1: Identify the population, the parameter, the sample, and the statistic in each of the following settings.
a. A pediatrician wants to know the 75th percentile for
b. A Pew Research Center poll asked 1102 12- to 17the distribution of heights of 10-year-old boys so
year-olds in the United States if they have a cell
she takes a sample of 50 patients and calculates Q3
phone. Of the respondents, 71% said yes.
= 56 inches.
The population is all 12- to 17-year-olds in the US; the
The population is all 10-year-old boys; the parameter
parameter is the proportion with cell phones. The
of interest is the 75th percentile, or Q3. The sample is
sample is the 1102 12- to 17-year-olds in the
the 50 10-year-old boys included in the sample; the
sample; the statistic is the sample proportion with
statistic is the sample Q3 = 56 inches.
a cell phone, p̂ = 0.71.
Sampling Variability
How can
x.
x be an accurate estimator of μ?
After all, different random samples would produce different values of
This basic fact is called sampling variability: the value of a statistic varies in repeated random
sampling.
To make sense of sampling variability, we ask, “What would happen if we took many samples?”
EXAMPLE 2: (From http://www.rossmanchance.com/applets/Reeses/ReesesPieces.html.)
Background: Reese’s Pieces candies have three colors: orange, brown, and yellow. We want to estimate the
proportion that are Orange.
Sampling Distributions

The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible
samples of the same size from the same population.
In practice, it’s difficult to take all possible samples of size n to obtain the actual sampling distribution of a
statistic. Instead, we can use simulation to imitate the process of taking many, many samples.
There are actually three distinct distributions involved when we sample repeatedly and measure a variable of
interest.
1) The population distribution gives the values of the variable for all the individuals in the population.

In the Reeses example: The distribution of colors in the machine – use a bar graph to show
2) The distribution of sample data shows the values of the variable for all the individuals in the sample.

In the Reeses example: The distribution of colors in the selected candies – use a bar graph
3) The sampling distribution shows the statistic values from all the possible samples of the same size
from the population.

In the Reeses example: The dotplot that is to the right – it is an approx. sampling distribution
EXAMPLE 3: Consider the applet from last night and identify how each distribution was demonstrated:
Population distribution
Distribution of sample data
Sampling distribution
Graph 1 – the top distribution
Graph 2 when animate is turned on
Graphs 3 and 4
Judging Estimators
The fact that statistics from random samples have definite sampling distributions allows us to answer the
question, “How trustworthy is a statistic as an estimator of the parameter?” To get a complete answer, we
consider the center, spread, and shape.
 A statistic used to estimate a parameter is an unbiased estimator if the mean of its sampling
distribution is equal to the true value of the parameter being estimated.
To get a trustworthy estimate of an unknown population parameter, start by using a statistic that’s an
unbiased estimator. This ensures that you won’t tend to overestimate or underestimate.
Unfortunately, using an unbiased estimator doesn’t guarantee that the value of your statistic will be close
to the actual parameter value. Larger samples have a clear advantage over smaller samples. They are
much more likely to produce an estimate close to the true value of the parameter.

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 – less variable sampling distributions.

The spread of the sampling distribution does not depend on the size of the population, as long as the
population is at least 10 times larger than the sample – called the 10% condition.
EXAMPLE 4:
We can think of the true value of the population parameter as the bull’s- eye on a target and of the sample
statistic as an arrow fired at the target. Both bias and variability describe what happens when we take many
shots at the target.
Bias:
HIGH
Variability:
LOW
EXAMPLE 5:
The population is below:
The true variance is
approx. 38.69
The true range is 32.
Bias:
LOW
Variability:
HIGH
Bias:
HIGH
Variability:
HIGH
Bias:
LOW
Variability:
LOW
- This is the ideal!!
Here is the approx. sampling distribution
of the sample variance using 25 SRSs of
size 5. The mean is noted below.
Here is the approx. sampling distribution
of the sample range using 25 SRSs of
size 5. The mean is noted below.
mean = 40.96
mean = 12.20
Is this variance an unbiased estimator?
Why/why not?
Is this range an unbiased estimator?
Why/why not?
Probably, the mean of the approximate
sampling distribution is close to the true
variance.
This is an approx. sampling distribution.
If more samples had been taken, the
estimate would be closer.
Clearly. The sampling distribution is
NOT centered anywhere near the true
range. In order for the sample range to
be correct you would need the
population min and max in the sample.
And now for Mr. Igoe’s follow-up rant about the use of range in a statistics class . . . . IT IS BIASED
The lesson about center and spread should be clear: given a choice of statistics to estimate an unknown
parameter, choose one with no or low bias and minimum variability.
EXAMPLE 6: German Tank Problem
Here are several of the methods for estimating the total number of tanks (N) from last time:
 Copy the estimator
 Copy a sketch of the approximate sampling distribution
The total number of tanks was actually: 342
1.
2.
3.
4.
Biased/Unbiased? Explain
Biased/Unbiased? Explain
Biased/Unbiased? Explain
Biased/Unbiased? Explain
5.
6.
7.
8.
Biased/Unbiased? Explain
Biased/Unbiased? Explain
Biased/Unbiased? Explain
Biased/Unbiased? Explain
Of the unbiased estimators, which is best? Explain.
THOUGH ESTIMATES WILL VARY, FOCUS ON THE FOLLOWING:
The best estimators
 Should center themselves at the value 342 (the mean of the sampling distribution should be
this.)
 Should have a small spread.
Be aware, sometimes the best estimate is a biased estimate – this may or may not happen with the
samples and estimators selected.
The true best estimator is 𝒔𝒂𝒎𝒑𝒍𝒆 𝒎𝒂𝒙𝒊𝒎𝒖𝒎 +
𝒔𝒂𝒎𝒑𝒍𝒆 𝒎𝒂𝒙𝒊𝒎𝒖𝒎−𝒔𝒂𝒎𝒑𝒍𝒆 𝒔𝒊𝒛𝒆
𝒔𝒂𝒎𝒑𝒍𝒆 𝒔𝒊𝒛𝒆
this result at http://en.wikipedia.org/wiki/German_tank_problem.
. Wikipedia has a discussion of