PSY 307 – Statistics for the Behavioral Sciences Chapter 9 – Sampling Distribution of the Mean

Random Sampling Population Sample 1 Mean =  Mean = x 1

Repeated Random Sampling Population Sample 1 Sample 2 Sample 3 Sample 1 Sample 4 Mean = x 1 Mean = x 2 Mean = x 3 Mean = x 4

All Possible Random Samples Sample 1 Sample 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: pling_dist/index.html pling_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: H 0 :  = 500, H 1 :  ≠ 500.  State the decision rule: e.g., p<.05  Do the calculations using formula.  Make a decision: accept or reject H 0  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 ≥ or z ≤ 1.96 Another way to say this is retain H 0 when: ≤ z ≤ 1.96

Compare Your Sample’s z to the Critical Values COMMON  =.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.