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University: Temple University

Course: Government in American Society (POLS 8101)

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AP Statistics – Chapter 7 Notes: Sampling Distributions

7.1 – What is a Sampling Distribution?

Parameter – A parameter is a number that describes some characteristic of the population

Statistic – A statistic is a number that describes some characteristic of a sample

Symbols

used

Sample

Statistic

Population

Parameter

Proportions

ˆ

p

p

Means

x

Sampling Distribution – the distribution of all values taken by a statistic in all possible samples of the same

size from the same population

A statistic is called an unbiased estimator of a parameter if the mean of its sampling distribution is equal to

the parameter being estimated

Important Concepts for unbiased estimators

The mean of a sampling distribution will always equal the mean of the population for any sample size

The spread of a sampling distribution is affected by the sample size, not the population size.

Specifically, larger sample sizes result in smaller spread or variability.

7.2 – Sample Proportions

Choose an SRS of size n from a large population with

population proportion p having some characteristic of

interest.

Let 𝑝 be the proportion of the sample having that

characteristic. Then the mean and standard deviation

of the sampling distribution of 𝑝 are

Mean: 𝜇𝑝

=𝑝 Std. Dev.: 𝜎𝑝

=√𝑝(1−𝑝)

𝑛

With the Z-Statistic: 𝑧 = 𝑝

−𝑝

√𝑝(1−𝑝)

𝑛

CONDITIONS FOR NORMALITY

The 10% Condition

Use the formula for the standard deviation of

ˆ

p

only

when the size of the sample is no more than 10% of

the population size (𝑛 ≤ 1

10𝑁).

The Large Counts Condition

We will use the normal approximation to the

sampling distribution of

ˆ

p

for values of n and p that

satisfy

10np

and

(1 ) 10np

.

7.3 – Sample Means

Suppose that

x

is the mean of a sample from a large

population with mean

and standard deviation

.

Then the mean and standard deviation of the

sampling distribution of

x

are

Mean: 𝜇𝑥=𝜇 Std. Dev.: 𝜎𝑥=𝜎

√𝑛

With the Z-Statistic: 𝑧 = 𝑥−𝜇

𝜎√𝑛

CONDITIONS FOR NORMALITY

If an SRS is drawn from a population that has the

normal distribution with mean

and standard

deviation

, then the sample mean

x

will have the

normal distribution

( , )Nn

for any sample size.

Central Limit Theorem

If an SRS is drawn from any population with mean

and standard deviation

, when n is large

( 30)n

, the sampling distribution of the sample

mean

x

will have the normal distribution

( , )Nn

.

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