Understanding Sampling Distribution in Behavioral Sciences

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

2. Random Sampling Mean = m Population Sample 1 Mean = x1

3. Repeated Random Sampling Population Sample 4 Mean = x4 Sample 3 Sample 2 Mean = x2 Sample 1 Sample 1 Mean = x3 Mean = x1

4. All Possible Random Samples Population Mean = m Sample 1 Sample 3 Sample 3 Sample 3 Sample 3 Sample 3 Sample 3 Sample 3 Sample n Mean = mx

5. 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.

6. Mean of the Sampling Distribution • Notation: • x = sample mean • m = population mean • mx = 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.

7. 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 (m). • Variability in the sampling distribution is less than in the population: • sx < s.

8. 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.

9. Demo • Central Limit Theorem: • http://onlinestatbook.com/stat_sim/sampling_dist/index.html

10. 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.

11. 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).

12. 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

13. Z-Test • To convert any score to z: z = x – m s • Formula for testing a sample mean: z = x – m sx

14. Formula • Aleks refers to sx or sM. • This is the standard error of the mean. • It is easiest to calculate the standard error of the mean using the following formula:

15. Step-by-Step Process • State the research problem. • State the statistical hypotheses using symbols: H0: m = 500, H1: m ≠ 500. • State the decision rule: e.g., p<.05 • Do the calculations using formula. • Make a decision: accept or reject H0 • Interpret the results.

16. 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

17. Compare Your Sample’s z to the Critical Values a = .05 .025 .025 COMMON -1.96 1.96

18. 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 s is known. • When these assumptions are not met, use a different test.