PSY292 - Chapter 7

Describe the distribution of sample means (shape, expected value, and standard error) for samples of n = 36 selected from a population with a mean of μ = 100 and a standard deviation of σ = 12.

Because n > 30, the distribution will be normal. Expected value: μ = 100 SE: 12/(√36) = 2

Distribution of Sample Means

Consists of the sample means for all the possible random samples of a specific size (n) from a specific population.

T/F: As sample size increases, the value of expected value also increases.

False. The expected value does not depend on sample size.

Central Limit Theorem

For any population with a mean *μ and standard deviation of *σ, the distribution of sample means (M) for sample size n will have a mean of *μ and a standard deviation of *σ/√n and will approach a normal distribution as n approaches infinity Provides a precise description of the distribution that would be obtained if you selected every possible sample, calculated every sample mean, and constructed the distribution of the sample mean.

The standard error tells you....

How much error to expect if you are using a sample mean to represent a population mean (specifies precisely how well a sample mean estimates its population mean) It provides a measure of how much difference is expected from one sample to another. Measures how well an individual sample mean represents the entire distribution. It provides a measure of how much distance is reasonable to expect between a sample mean and the overall mean for the distribution of sample means

If the sample size (n) is greater than 30....

The distribution of sample means is a normal distribution.

The distribution of sample means is not always a normal distribution. Under what circumstances is the distribution of sample means not normal?

The distribution of sample means will NOT be normal when it is based on small samples (n < 30) selected from a population that's not normal.

Central Limit Theorem - Shape

The distribution of sample means will be almost perfectly normal if a sample is n ≥ 30 OR if the population is normal

Law of Large Numbers

The larger the sample size (n), the more probable it is that the sample mean is close to the population mean. Also note that the standard error decreases in relation to the square root of the sample size

Central Tendency

The mean of the distribution of sample means is identical to the mean of the population from which the samples are selected. The mean of the distribution of sample means is called the expected value of M.

Variability

The standard deviation of the distribution of sample means is called the *standard error of M*

Standard error (σM) measures........

The standard distance between a sample mean (M) and the population mean (μ). SE is the standard deviation of the distribution of sample means

When the standard error is small....

Then all of the sample means are close together and have similar values If the standard error is large, then the sample means are scattered over a wide range and there are big differences from one sample to another. Bigger samples have smaller error, and smaller samples have bigger error.

When the sample consists of a single score (n = 1), the standard error is.....

same as the standard deviation (σM = σ).

*Shape - The distribution of sample means is almost perfectly normal if either of the following two conditions is satisfied:

1. The population from which the samples are selected is a normal distribution. 2. The number of scores (n) in each sample is relatively large, around 30 or more. As n gets larger, the distribution of sample means more closely approximates a normal distribution

Benefits of the Central Limit Theorem

It describes the distribution of sample means for any population, no matter what shape, mean, or standard deviation The distribution of sample means "approaches" a normal distribution very rapidly Describes the distribution of sample means by identifying the three basic characteristics that describe any distribution: *shape, *central *tendency, and *variability.

Expected Value of M

Mean of the distribution of sample means is equal to the mean of the population of scores (μ)