PSY 307 – Statistics for the Behavioral Sciences

Chapter 9 – Sampling Distribution of the Mean

Random Sampling

Population

Sample 1

Mean =

Mean = x1

Repeated Random Sampling

Population

Sample 1

Sample 2Sample 3

Sample 1

Sample 4

Mean = x1

Mean = x2

Mean = x3

Mean = x4

All Possible Random Samples

Sample 1Sample 3

Sample 3Sample 3

Sample 3Sample 3

Sample 3Sample 3

Sample n

Population

Mean = x

Mean =

Sampling Distribution of the Mean

Probability distribution of means for all possible random samples of a given size from some population. Used to develop a more accurate

generalization about the population. All possible samples of a given size

– not the same as completely surveying the population.

Mean of the Sampling Distribution

Notation: x = sample mean = population mean x = mean of all sample means

The mean of all of the sample means equals the population mean.

Most sample means are either larger or smaller than the population mean.

Standard Error of the Mean

A special type of standard deviation that measures variability in the sampling distribution.

It tells you how much the sample means deviate from the mean of the sampling distribution ().

Variability in the sampling distribution is less than in the population:

x < .

Central Limit Theorem

The shape of the sampling distribution approximates a normal curve. Larger sample sizes are closer to

normal. This happens even if the original

distribution is not normal itself.

Demo

Central Limit Theorem: http://onlinestatbook.com/stat_sim/sampl

ing_dist/index.html

Why the Distribution is Normal

With a large enough sample size, the sample contains the full range of small, medium & large values. Extreme values are diluted when

calculating the mean. When a large number of extreme

values are found, the mean may be more extreme itself. The more extreme the mean, the less

likely such a sample will occur.

Probability and Statistics

Probability tells us whether an outcome is common (likely) or rare (unlikely).

The proportions of cases under the normal curve (p) can be thought of as probabilities of occurrence for each value.

Values in the tails of the curve are very rare (uncommon or unlikely).

Z-Test for Means

Because the sampling distribution of the mean is normal, z-scores can be used to test sample means.

To convert a sample mean to a z-score, use the z-score formula, but replace the parts with sample statistics: Use the sample mean in place of x Use the hypothesized population mean in place

of the mean Use the standard error of the mean in place of

the standard deviation

Z-Test

To convert any score to z:z = x –

Formula for testing a sample

mean:z = x –

x

Formula

Aleks refers to x or M. This is the standard error of the mean.

It is easiest to calculate the standard error of the mean using the following formula:

Step-by-Step Process

State the research problem. State the statistical hypotheses

using symbols: H0: = 500, H1: ≠ 500.

State the decision rule: e.g., p<.05 Do the calculations using formula. Make a decision: accept or reject H0

Interpret the results.

Decision Rule

The decision rule specifies precisely when the null hypothesis can be rejected (assumed to be untrue).

For the z-test, it specifies exact z-scores that are the boundaries for common and rare outcomes: Retain the null if z ≥ -1.96 or z ≤ 1.96 Another way to say this is retain H0

when: -1.96 ≤ z ≤ 1.96

Compare Your Sample’s z to the Critical Values

-1.96 1.96

.025.025COMMON

= .05

Assumptions of the z-test

A z-test produces valid results only when the following assumptions are met: The population is normally distributed

or the sample size is large (N > 30). The population standard deviation is

known. When these assumptions are not

met, use a different test.