Stats Ch 7 Probability and Samples: The Distribution of Sample Means (Gravetter & Wallnau)

Sampling error

The natural discrepancy, or amount of error, between a sample statistic and its corresponding population parameter

Distribution of sample means

Collection of the individual sample means for all of the possible random samples of a particular size (n) that can be obtained from the population.

Sampling distribution

Distribution of statistics obtained by selecting all of the possible samples of a specific size from a population

General characteristics of the distribution

Sample means should pile up around the population mean & represent the population.
Pile of sample means should tend to form a normal-shaped distribution.
The larger the sample size, the closer the sample means should be to the population mean.

Central limit theorem (don't need to know)

For any population with mean ? and standard deviation ?, the distribution of sample means for sample size n will have a mean of ? and a standard deviation of ?/?n and will approach a normal distribution as n approaches infinity
- Describes the distributio

Characteristics which describe the distribution

Shape.
Central tendency (mean, median & mode).
& Variability.

The shape of the distribution of sample means

Normal distribution if population is normal or sample is > 30.
= almost perfectly normal distribution

The mean of the distribution of sample means

Is Centered around population mean.
The mean of the distribution of sample means is always identical to the population mean; remember the expected value of M!

The standard error of M

The standard deviation of the distribution of sample means, ?M.
Provides a measure of how much distance is expected on average between a sample mean (M) and the population mean (?)

Law of large numbers

The larger the sample size (n), the more probable it is that the sample mean is closer to the population mean

The population standard deviation

As sample size (n) increases, the size of the standard error decreases.
When the sample consists of a single score (n = 1) the standard error is the same as the standard deviation. So if one DQ has a mean of $3.80 the SE would be $1.20 & since there's onl

A z-score for sample means

Can be used to describe the exact location of any specific sample mean within the distribution of sample means.
Which is, where the sample mean is located in relation to all of the other possible sample means that could have been obtained.

Standard error and reliability

Can be used as a measure of how reliable a sample mean is, e.g.consistency of different measurements of the same thing.
Influences on Standard Error & reliability are:
- The number of scores.
- The size of the population standard deviation

The distribution of sample means is almost perfectly normal in either of 2 conditions:

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

Variability of a distribution of scores is measured by the

standard deviation. i.e. how different they are from the mean.

Variability of a distribution of sample
means
is measured by

the standard deviation of the sample means, & is called the standard error of M and written as Om, SE or SEM

Mean of the distribution of sample means is called

the expected value of M.
-M is an
unbiased statistic
because the expected value of the distribution of sample means (Um) is the value of the population mean (u).

The standard error of M & its magnitude

Sample size! sample size! Sample size!

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

The standard error measures the standard amount of difference between a sample mean, M, and the population mean, u.
-Standard error of M = Om = standard distance between M & u.

The sample size

The law of large numbers states that the larger the sample size (n) the more probable it is that the sample mean will be close to the population mean.

The standard error of M formula

is used to incorporate into a z score value. Take the difference between the sample mean & population mean divided by the standard difference you can expect the two. (z=m-u/Om & Om=o/sq rt of n)