1. Probability & The Normal Distribution Statistics for the Social Sciences Psychology 340 Spring 2010

2. PSY 340 Statistics for the Social Sciences Reminders Quiz 2 due Thursday Homework 3 due Tues, Feb 2 Exam 1 Thurs Feb. 11

3. PSY 340 Statistics for the Social Sciences Basics of Probability Probability –Expected relative frequency of a particular outcome Outcome –The result of an experiment

4. PSY 340 Statistics for the Social Sciences Flipping a coin example What are the odds of getting a “heads”? One outcome classified as heads = 1 2 =0.5 Total of two outcomes n = 1 flip

5. PSY 340 Statistics for the Social Sciences Flipping a coin example What are the odds of getting two “heads”? Number of heads 2 1 1 0 One 2 “heads” outcome Four total outcomes =0.25 This situation is known as the binomial # of outcomes = 2 n n = 2

6. PSY 340 Statistics for the Social Sciences Flipping a coin example What are the odds of getting “at least one heads”? Number of heads 2 1 1 0 Four total outcomes =0.75 Three “at least one heads” outcome n = 2

7. PSY 340 Statistics for the Social Sciences Flipping a coin example HHH HHT HTH HTT THH THT TTH TTT Number of heads 3 2 1 0 2 2 1 1 2n2n = 2 3 = 8 total outcomes n = 3

8. PSY 340 Statistics for the Social Sciences Flipping a coin example Number of heads 3 2 1 0 2 2 1 1 Xfp 31.125 23.375 13 01.125 Number of heads 0123.1.2.3.4 probability.125.375 Distribution of possible outcomes (n = 3 flips)

9. PSY 340 Statistics for the Social Sciences Flipping a coin example Number of heads 0123.1.2.3.4 probability What’s the probability of flipping three heads in a row?.125.375 p = 0.125 Distribution of possible outcomes (n = 3 flips) Can make predictions about likelihood of outcomes based on this distribution.

10. PSY 340 Statistics for the Social Sciences Flipping a coin example Number of heads 0123.1.2.3.4 probability What’s the probability of flipping at least two heads in three tosses?.125.375 p = 0.375 + 0.125 = 0.50 Can make predictions about likelihood of outcomes based on this distribution. Distribution of possible outcomes (n = 3 flips)

11. PSY 340 Statistics for the Social Sciences Flipping a coin example Number of heads 0123.1.2.3.4 probability What’s the probability of flipping all heads or all tails in three tosses?.125.375 p = 0.125 + 0.125 = 0.25 Can make predictions about likelihood of outcomes based on this distribution. Distribution of possible outcomes (n = 3 flips)

12. PSY 340 Statistics for the Social Sciences Hypothesis testing Can make predictions about likelihood of outcomes based on this distribution. Distribution of possible outcomes (of a particular sample size, n) In hypothesis testing, we compare our observed samples with the distribution of possible samples (transformed into standardized distributions) This distribution of possible outcomes is often Normally Distributed

13. PSY 340 Statistics for the Social Sciences The Normal Distribution The distribution of days before and after due date (bin width = 4 days). 0 14 -14 Days before and after due date

14. PSY 340 Statistics for the Social Sciences The Normal Distribution Normal distribution

15. PSY 340 Statistics for the Social Sciences The Normal Distribution Normal distribution is a commonly found distribution that is symmetrical and unimodal. –Not all unimodal, symmetrical curves are Normal, so be careful with your descriptions It is defined by the following equation: 12-20 -33 Z-scores

16. PSY 340 Statistics for the Social Sciences Estimating Probabilities in a Normal Distribution 50%-34%-14% rule 12-20 50% -33 Number of heads 0123.1.1.2.2.3.3.4.4 probability.125.375 Same logic as before

17. PSY 340 Statistics for the Social Sciences Estimating Probabilities in a Normal Distribution 50%-34%-14% rule 50% 12-20 34.13% 13.59% -33

18. PSY 340 Statistics for the Social Sciences Estimating Probabilities in a Normal Distribution 12-20 Similar to the 68%-95%-99% rule 13.59% 2.28% 34.13% 13.59% 2.28% 34.13% 68% -33

19. PSY 340 Statistics for the Social Sciences Estimating Probabilities in a Normal Distribution 12-20 Similar to the 68%-95%-99% rule 13.59% 2.28% 34.13% 13.59% 2.28% 34.13% -33 95%

20. PSY 340 Statistics for the Social Sciences The Unit Normal Table z 0 : 1.00 : 2.31 2.32 Gives the precise proportion of scores (in z-scores) above or below a given score in a Normal distribution There are many ways that this table gets organized Learn to understand what is in the table What do the numbers represent? The normal distribution is often transformed into z-scores. Understand your table z 0

21. PSY 340 Statistics for the Social Sciences The Unit Normal Table –Contains the proportions of a Normal distribution –Proportion between the z-score and left side of the distribution –Proportion in the tail to the right of corresponding z-scores –Proportion between the z-score and the mean Note: This means that this table lists only positive Z scores The normal distribution is often transformed into z-scores. In tail Understand your table ZProp in Body Prop in tail Prop btwn mean and z 0.00 0.01 0.02 : 1.0 : 1.3 : 4.00.5000.5040.5080 :.8413 :.9032 :.99997.5000.4960.4920 :.1587 :.0968 :.00003.0000.0040.0080 :.3413 :.4032 :.49997 From the left side of the dist. z 0

22. PSY 340 Statistics for the Social Sciences The Unit Normal Table z.00.01 0 : 1.0 : 2.3 2.4 : 0.5000 : 0.1587 : 0.0107 0.0082 : 0.4960 : 0.1562 : 0.0104 0.0080 : –Contains the proportions in the tail to the left of corresponding z-scores of a Normal distribution This means that the table lists only positive Z scores The different columns give the second decimal place of the z-score The normal distribution is often transformed into z-scores. In tail The unit normal table I have provided on- line (see ‘statistical tables’ link at top of labs) Understand your table z 0

23. PSY 340 Statistics for the Social Sciences The Unit Normal Table z Mean to ZIn tail 0 : 1.00 : 2.31 2.32 : 0.0000 : 0.3413 : 0.4896 0.4898 : 0.5000 : 0.1587 : 0.0104 0.0102 : –Contains the proportions –Proportion between the z-score and the mean –Proportion in the tail to the left of corresponding z-scores of a Normal distribution Note: This means that this table lists only positive Z scores The normal distribution is often transformed into z-scores. Mean to Z In tail Understand your table z 0

24. PSY 340 Statistics for the Social Sciences The Unit Normal Table z.00.01 -3.4 -3.3 : 0 : 1.0 : 3.3 3.4 0.0003 0.0005 : 0.5000 : 0.8413 : 0.9995 0.9997 0.0003 0.0005 : 0.5040 : 0.8438 : 0.9995 0.9997 –Contains the proportions to the left of corresponding z-scores of a Normal distribution This table lists both positive and negative Z scores The normal distribution is often transformed into z-scores. From the left side of the dist. Another common way the unit normal table is presented in textbooks Understand your table z 0

25. PSY 340 Statistics for the Social Sciences Using the Unit Normal Table 1.Convert raw score to Z score (if necessary) 2. Draw normal curve, where the Z score falls on it, shade in the area for which you are finding the percentage 3. Make rough estimate of shaded area’s percentage (using 50%-34%-14% rule) Steps for figuring the percentage below a particular raw or Z score: ZProp in Body Prop in tail Prop btwn mean and z 0.00 0.01 0.02 : 1.0 : 1.3 : 4.00.5000.5040.5080 :.8413 :.9032 :.99997.5000.4960.4920 :.1587 :.0968 :.00003.0000.0040.0080 :.3413 :.4032 :.49997

26. PSY 340 Statistics for the Social Sciences Using the Unit Normal Table 4. Find exact percentage using unit normal table –Use your sketch and understanding of the table 5. Check the exact percentage is within the range of the estimate from Step 3 Steps for figuring the percentage below a particular raw or Z score: ZProp in Body Prop in tail Prop btwn mean and z 0.00 0.01 0.02 : 1.0 : 1.3 : 4.00.5000.5040.5080 :.8413 :.9032 :.99997.5000.4960.4920 :.1587 :.0968 :.00003.0000.0040.0080 :.3413 :.4032 :.49997

27. PSY 340 Statistics for the Social Sciences Suppose that you got a 630 on the SAT. What percent of the people who take the SAT get your score or worse? SAT Example problems The population parameters for the SAT are: μ = 500, σ = 100, and it is Normally distributed From the table: z(1.3) =.0968 That’s 9.68% above this score So 90.32% got your score or worse

28. PSY 340 Statistics for the Social Sciences The Normal Distribution You can go in the other direction too –Steps for figuring Z scores and raw scores from percentages (or proportions): 1. Draw normal curve, shade in approximate area for the percentage (using the 50%-34%-14% rule) 2. Make rough estimate of the Z score where the shaded area starts 3. Find the exact Z score using the unit normal table - So now you’re looking for a percentage/proportion in the body of the table, and then looking to see what z-score it corresponds to 4. Check that your Z score is similar to the rough estimate from Step 2 5. If you want to find a raw score, change it from the Z score

29. PSY 340 Statistics for the Social Sciences Testing Hypotheses Looking ahead: –Core logic of hypothesis testing Considers the probability that the result of a study could have come about if the experimental procedure had no effect How do we determine this? Based on standard error or an estimate of the standard error “Studies” typically look not at single scores, but rather samples of scores. So we need to think about the probability of getting samples with particular characteristics (means). Next time: –The distribution of sample means