AP Statistics: Section 9.1Sampling Distributions

What is the usual way to gain information about some

characteristic of a population?

sample. a By taking

We must note, however, that the sample information we gather may

differ from the true population characteristic we are trying to

measure. Furthermore, the sample information may differ from sample to

sample.

This sample-to-sample variability, called

____________________________ poses a problem when we try to

generalize our findings to the population. We need to gain an understanding of this variability.

y variabiltsampling

A parameter is

A statistic is

.population a describest number tha a

data. sample from computednumber a

In statistical practice, the value of a parameter is unknown since we

cannot examine the entire population. In practice, we often

use a statistic to estimate an unknown parameter.

The population mean is represented by the symbol ___

(Greek: Mu), the population standard deviation by ___(Greek:

Sigma) and the population proportion by ___.

p

The sample mean is represented by the symbol ____ (x bar), the sample standard deviation by ____ and the sample population by ____ (p hat).

xxsp̂

Example: Identify the number that appears in boldface type as a

parameter or a statistic, and then write an equation using the proper

symbol from above and the number from the statement

A department store reports that 84% of all customers who use the

store’s credit plan pay their bills on time.

paramter.84p

A consumer group, after testing 100 batteries of a certain brand, reported an average of 63 hr of

use.

statistichours 63x

We can view a sample statistic as a random variable, because we have no way of predicting exactly what statistic value we will get from a sample, BUT,

given a population parameter, we know how these sample statistics will

behave in repeated sampling.

Before we continue, we need to discuss two important definitions:

The population distribution of a variable is the distribution of

values of the variable among all individuals in the population.

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 population.

Careful: The population distribution describes the

individuals that make up the population. A sampling distribution describes how a statistic varies in many samples of size n from the

population.

Consider flipping a coin 10 times. We would expect to get 5 heads out of the 10, but we realize that we could also get 4 or 6 or 7 or

even 10. Let’s simulate this using our graphing calculators.

:MATH/PRB/7

535. 135. 6.

1Yscl10Ymax0Ymin

.1Xscl1Xmin0Xmax

WINDOW

How many different samples of size 10 are possible in this situation?

1024210

Let’s increase our sample size to 25.

52. 120. 54.

1L,.5,20)/25randBin(25

Hopefully, most of us found that as we increased the sample size from 10 to 25, the mean and the median of our sample proportions became closer together and

both became closer to .5. Also we should find that the standard deviation grows

smaller and our distribution of the sample proportions became closer to being a

normal distribution.

Since a sampling distribution is a distribution, we can use the tools of data analysis to describe the

distribution: ________, ________, __________ and __________.

shape centerspread outliers

CUSS oRemember t

Example: According to 2005 Nielsen ratings, Survivor:

Guatemala was one of the most-watched TV shows in the US during every week that it aired. Suppose

that the true proportion of US adults who watched Survivor: Guatemala was p = 0.37.

Describe the distribution of sample proportions at the right for samples of size n = 100 of people

who watched Survivor: Guatemala.

outliers no.3 is range

.37 approx. iscenter Normal approx.

Describe the distribution of sample proportions for samples of size n = 1000 of people who

watched Survivor: Guatemala.

outliers no.12 is range

.37 approx. iscenter Normal approx.

A statistic used to estimate a parameter is unbiased if the mean of

its sampling distribution equals the true value of the population

parameter.

The statistic is called an unbiased estimator of the parameter.

An unbiased statistic will sometimes fall above the true

value of the parameter and sometimes below. There is no tendency to overestimate or

underestimate the parameter, hence the “unbiased.”

We will see in sections 9.2 & 9.3 that are both unbiased

estimators of population parameters.

The variability of a statistic is described by the spread of its

sampling distribution.

This spread is determined by the sampling design and the sample

size.

Larger samples give a ________ spread.

smaller

As long as the population is larger than the sample by at least a factor of 10, the spread of the

sampling distribution is approximately the same for any

population size.

This means that a statistic from an SRS of size 2500 from the more

than 300 million residents of the US is just as precise as an SRS of

size 2500 from the 750,000 inhabitants of San Francisco.

For a better understanding of bias and variability, think of the center of a target as the true population

parameter and an arrow shot at the target as a sample statistic.

high bias low bias high bias low bias low variability high variability high variability low variability

ty. variabililow and bias low have willsize sufficient of samples random from computed statisticschosen Properly