### Chapter 5. Sampling: Sampling Distributions

### Sampling Distribution of the Sample Proportion

A type of variable which is commonly studied in statistics is the *binary *variable.

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Binary Variable

**Definition**

A **binary **or **dichotomous **variable is a categorical variable that can only take on #2# possible values.

**Examples**

- True/false
- Success/failure
- Yes/no
- On/off

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The mean of a binary variable is mathematically equivalent to the *proportion*.

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Proportion

**Definition**

In statistics, a **proportion** refers to the fraction of a group that possesses a particular characteristic.

The population and sample proportion are denoted #\pi# and #\hat{p}#, respectively.

**Formula**

#\text{proportion}=\cfrac{\text{# of individuals with characterstic}}{\text{total number of individuals}}#

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When using a sample proportion to estimate a population proportion, the *sampling **distribution of the sample proportion *can be used to determine how much estimation error is reasonable to expect.

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Sampling Distribution of the Sample Proportion

The **sampling distribution of the sample proportion** is the probability distribution of the sample proportions of every possible sample of a particular size #n# that can be drawn from a population.

The *mean* of the distribution of sample proportions is called the **expected value of the sample proportion **and is denoted #\mu_{\hat{p}}#.

The *standard deviation* of the distribution of sample proportions is called the **standard error of the sample proportion **and is denoted #\sigma_{\hat{p}}#. The standard error is a measure of how much discrepancy to expect between a sample proportion #\hat{p}# and the population proportion #\pi#.

Conditions for Normality

For any population of which a proportion #\pi# possesses a particular characteristic, the *sampling distribution of the sample proportion* for samples of size #n# may be considered approximately normal if both of the following conditions are satisfied:

- We expect there to be at least #10#
*positive*cases: #n \cdot \pi \geq 10# - We expect there to be at least #10#
*negative*cases: #n \cdot (1-\pi) \geq 10#

If both these conditions are satisfied, the *sampling distribution of the sample proportions *may be considered approximately normal with parameters:

- #\mu_{\hat{p}}=\pi#
- #\sigma_{\hat{p}}=\sqrt{\cfrac{\pi \cdot(1-\pi)}{n}}#

\[\hat{p} \sim N(\pi, \sqrt{\cfrac{\pi \cdot (1-\pi)}{n}})\]

#\mu_{\hat{p}} = 0.50#

Investigate whether the *sampling distribution of the sample proportion *may be considered approximately normal:

- #n\cdot \pi = 160 \cdot 0.50 = 80 \geq 10#
- #n\cdot (1-\pi) = 160 \cdot (1-0.50) = 80 \geq 10#

Since both conditions are satisfied, the *expected value of the sample proportion*, #\mu_{\hat{p}}#, is equal to the population proportion #\pi#:

\[\mu_{\hat{p}} = \pi= 0.50\]

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