PSY 307 – Statistics for the Behavioral Sciences

1. PSY 307 – Statistics for the Behavioral Sciences Chapter 8 – The Normal Curve, Sample vs Population, and Probability

2. Demos • How normal distributions are generated: • http://www.ms.uky.edu/~mai/java/stat/GaltonMachine.html • How changes in the mean and std deviation affect the shape of the normal distribution: • http://onlinestatbook.com/chapter6/varieties_demo.html • Finding the proportion for a given z score: • http://onlinestatbook.com/java/normal.html • Finding the z-score for a given portion of the distribution: • http://onlinestatbook.com/java/normalshade.html

3. A Family of Normal Curves • A normal curve has a symmetrical, bell-like shape. • The lower half (below the mean) is the mirror image of the upper half. • Values for the mean, median and mode are always the same number. • The mean and SD specify the location and shape (steepness) of the normal curve.

4. Different Normal Curves Same SD but different Means Same Mean but different SDs

5. Z-Score • Indicates how many SDs an observation is above or below the mean of the normal distribution. • Formula for converting any score to a z-score: Z =X – ms

6. Properties of Z-Scores • A z-score expresses a specific value in terms of the standard deviation of the distribution it is drawn from. • The z-score no longer has units of measure (lbs, inches). • Z-scores can be negative or positive, indicating whether the score is above or below the mean.

7. Standard Normal Curve • By definition has a mean of 0 and an SD of 1. • Standard normal table gives proportions for z-scores using the standard normal curve. • Proportions on either side of the mean equal .50 (50%) and both sides add up to 1.00 (100%).

8. Finding Proportions Actually +/-1.96

9. Finding Exact Proportions • http://davidmlane.com/hyperstat/z_table.html • http://www.sfu.ca/personal/archives/richards/Table/Pages/Table1.htm

10. Other Distributions • Any distribution can be converted to z-scores, giving it a mean of 0 and a standard deviation of 1. • The distribution keeps its original shape, even though the scores are now z-scores. • A skewed distribution stays skewed. • The standard normal table cannot be used to find its proportions.

11. Why Samples? • Population – any complete set of observations or potential observations. • Sample – any subset of observations from a population. • Usually of small size relative to a population. • Optimal size depends on variability and amount of error acceptable.

12. A Sample comes from a Population

13. Random Samples • To be random, all observations must have an equal chance of being included in the sample. • The selection process must guarantee this. • Random selection must occur at each stage of sampling. • Casual or haphazard is not the same as “random.”

14. Techniques for Random Selection • Fishbowl method – all observations represented on slips of paper drawn from a fishbowl. • Depends on thoroughness of stirring. • Random number tables – enter the table at a random point then read in a consistent direction. • Random digit dialing during polling.

15. Hypothetical Populations • Cannot be truly randomly sampled because all observations are not available for sampling. • Treated as real populations and sampled using random procedures. • Inferential statistics are applied to samples from hypothetical populations as if they were random samples.

16. Random Assignment • Random assignment ensures that, except for random differences, groups are similar. • When a variable cannot be controlled, random assignment distributes its effect across groups. • Any remaining difference can be attributed to effect, not uncontrolled variables.

17. How to Assign Subjects • Flip a coin. • Choose even/odd numbers from a random number table. • Assign equal numbers of subjects to each group by pairs: • When one subject goes to one group, the next goes to the other group. • Extend the same process to larger numbers of groups.

18. Probability • The proportion or fraction of times a particular outcome is likely to occur. • Probabilities permit speculation based on observations. • Relative frequency of heights also suggests the likelihood of a particular height occurring. • Probabilities of simple outcomes are combined to find complex outcomes

19. Addition Rule • Used to predict combinations of events. • Mutually exclusive events are events that cannot happen together. • Add the separate probabilities to find out the probability of any one of the outcomes occurring. • Pr(A or B) = Pr(A) + Pr(B)

20. Addition Rule (Cont.) • When events can occur together, addition must be adjusted for the overlap between outcomes. • Add the probabilities then subtract the amount that is shared (counted twice): • Drunk drivers = .40 • Drivers on drugs = .20 • Both = .12

21. Multiplication Rule • Used to calculate joint probabilities – events that both occur at the same time. • Birthday coincidence • http://www.cut-the-knot.org/do_you_know/coincidence.shtml • Pr(A and B) = [Pr(A)][Pr(B)] • The events combined must be independent of each other. • One event does not influence the other.

22. Dependent Outcomes • Dependent – when one outcome influences the likelihood of the other outcome. • The probability of the dependent outcome is adjusted to reflect its dependency on the first outcome. • The resulting probability is called a conditional probability. • Drunk drivers & drug takers example.

23. 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).