Sampling Distribution of the Mean

Probability distribution of means for all possible random samples of a given size from some population.

What does ̅X represent?

Sample Mean

Samples = normal alphabet

What does µ represent?

Population mean

Populations = Greek Alphabet

What does μ(petit ̅X) represent?

mu sub X-bar

Sampling distribution of the mean

What does s represent?

Sample standard deviation

What does σ represent?

Population standard deviation

What does σ(petit ̅X) represent?

sigma sub X-bar

Standard of error of the mean, or simply the standard error.

Without peeking, list the special symbols for the:

- mean of the population
- mean of the sampling distribution of the mean
- mean of the sample
- standard error of the mean
- standard deviation of the sample
- standard deviation of the population.

- µ
- mu sub X-bar
- ̅X
- sigma sub X-bar
- s
- σ

Mean of the Sampling Distribution of the Mean (µ sub X-bar)

The mean of all samples means always equals the population mean.

True or False?

The mean of all sample means, µ sub X-bar, always equals the value of a particular sample mean.

False. It always equals the value of the population mean.

True or False?

The mean of all sample means, µ sub X-bar, equals 100 if, in fact, the population mean equals 100.

True.

True or False?

The mean of all sample means, µ sub X-bar, usually equals the value of a particular sample mean.

False. Because of chance, most sample means tend to be either larger or smaller than the mean of all sample means.

True or False?

The mean of all sample means, µ sub X-bar, is interchangeable with the population mean.

True.

Standard Error of the Mean (σ sub X-bar)

A rough measure of the average amount by which sample means deviate from the mean of the sampling distribution or from the population mean.

σ sub X-bar = σ / √n

True or False?

The standard error of the mean (σ sub X-bar) roughly measures the average amount by which sample means deviate from the population mean.

True.

True or False?

The standard error of the mean (σ sub X-bar) measures varibility in a particular sample.

False. It measures variability among sample means.

True or False?

The standard error of the mean (σ sub X-bar) increases in value with larger sample sizes.

False. It decreases in value with larger sample sizes.

True or False?

The standard error of the mean (σ sub X-bar) equals 5, givent that σ = 40 and n = 64

True.

Central Limit Theorem

Regardelss of the population shape, the shape of the sampling distribution of the mean approximates a normal curve if the sample size is sufficiently large.

True or False?

The central limit theorem states that, with sufficiently large sample sizes, the shape of the population is normal.

False. The shape of the population remains the same regardless of sample size.

True or False?

The central limit theorem states that, regardeless of sample size, the shape of the sampling distributions of the mean is normal.

False. It requires that the sample size be sufficiently large - usually between 25 and 100.

True or False?

The central limit theorem ensures that the shape of the sampling distribution of the mean equals the shape of the population.

False. It ensures that the shape of the sampling distribution approximates a normal curve, regardless of the shape of the population (which remains intact).

True or False?

The central limit theorem applies to the shape of the sampling distribution - not to the shape of the population and not to the shape of the sample.

True.