1. Statistics and Quantitative Analysis U4320 Segment 4: Statistics and Quantitative Analysis Prof. Sharyn O’Halloran

2. Probability Distributions A. Distributions: How do simple probability tables relate to distributions? 1. What is the Probability of getting a head? ( 1 coin toss)

3. Probability Distributions (cont.) 2. Now say we flip the coin twice. The picture now looks like:

4. Probability Distributions (cont.) As number of coin tosses increases, the distribution looks like a bell-shaped curve:

5. Probability Distributions (cont.) 3. General: Normal Distribution Probability distributions are idealized bar graphs or histograms. As we get more and more tosses, the probability of any one observation falls to zero. Thus, the final result is a bell-shaped curve

6. Probability Distributions (cont.) B. Properties of a Normal Distribution 1. Formulas: Mean and Variance

7. Probability Distributions (cont.) 2. Note Difference with the Book The population mean is written as: Variance as: Example: Two tosses of a coin

8. Probability Distributions (cont.) 2. Note Difference with the Book (cont.) So the average, or expected, number of heads in two tosses of a coin is: 0*1/4 + 1*1/2 + 2*1/4 = 1.

9. Probability Distributions (cont.) 3. Expected Value

10. Probability Distributions (cont.) C. Standard Normal Distribution 1.Definition: a normal distribution with mean 0 and standard deviation 1. Z-values - Are points on the x-axis that show how many standard deviations that point is away from the mean m.

11. Probability Distributions (cont.) C. Standard Normal Distribution 2. Characteristics symmetric Unimodal. continuous distributions 3. Example: Height of people are normally distributed with mean 5'7" What is the proportion of people taller than 5'7"?

12. Probability Distributions (cont.) D. How to Calculate Z-scores Definition: Z-value is the number of standard deviations away from the mean Definition: Z-tables give the probability (score) of observing a particular z-value.

13. Probability Distributions (cont.) 1. What is the area under the curve that is greater than 1 ? Prob (Z>1) The entry in the table is 0.159, which is the total area to the right of 1.

14. Probability Distributions (cont.) 2. What is the area to the right of 1.64? Prob (Z > 1.64) The table gives 0.051, or about 5%.

15. Probability Distributions (cont.) 3. What is the area to the left of -1.64? Prob (Z < -1.64)

16. Probability Distributions (cont.) 4. What is the probability that an observation lies between 0 and 1? Prob (0 < Z < 1)

17. Probability Distributions (cont.) 5. How would you figure out the area between 1 and 1.5 on the graph? Prob (1 < Z < 1.5)

18. Probability Distributions (cont.) 6. What is the area between -1 and 2? Prob (-1 < Z < 2) P (-1<Z<0) =.341 P (0<z<2) =.50-.023 =.477  0.477 + 0.341 =.818

19. Probability Distributions (cont.) 7. What is the area between -2 and 2? Prob(-2 2) = 1 -.023 -.023 =.954

20. Probability Distributions (cont.) E. Standardization 1. Standard Normal Distribution -- is a very special case where the mean of distribution equals 0 and the standard deviation equals 1.

21. Probability Distributions (cont.) 2. Case 1: Standard deviation differs from 1 For a normal distribution with mean 0 and some standard deviation, you can convert any point x to the standard normal distribution by changing it to x/.

22. Probability Distributions (cont.) 3. Case 2: Mean differs from 0 So starting with any normal distribution with mean and standard deviation 1, you can convert to a standard normal by taking x- and using this as your Z-value. Now, what would be the area under the graph between 50 and 51? Prob (0<Z<1) =.341

23. Probability Distributions (cont.) 4. General Case: Mean not equal to 0 and SD not equal to 1 Say you have a normal distribution with mean & standard deviation. You can convert any point x in that distribution to the same point in the standard normal by computing This is called standardization. The Z-value corresponds to x. The Z-table lets you look up the Z-Score of any number.

24. Probability Distributions (cont.) 5. Trout Example: a. The lengths of trout caught in a lake are normally distributed with mean 9.5" and standard deviation 1.4". There is a law that you can't keep any fish below 12". What percent of the trout is this? (Can keep above 12) Step 1: Standardize Find the Z-score of 12: Prob (x>12) Step 2: Find z-score Find Prob (Z>1.79) Look up 1.79 in your table; only.037, or about 4% of the fish could be kept. 12 - 9.5 Z = --------- = 1.79. 1.4

25. Probability Distributions (cont.) 5. Trout Example (cont.): b. Now they're thinking of changing the standard to 10" instead of 12". What proportion of fish could be kept under the new limit? Standardize Prob (x>10) Step 2: Find z-score Prob (Z>.36) In your tables, this gives.359, or almost 36% of the fish could be kept under the new law. 10 - 9.5 Z = --------- = 0.36. 1.4

26. Joint Distributions A. Probability Tables 1. Example: Toss a coin 3 times. How many heads and how many runs do we observe? Def: A run is a sequence of one or more of the same event in a row Possible outcomes

27. Joint Distributions (cont.) 2. Joint Distribution Table

28. Joint Distributions (cont.) 3. Definition: The joint probability of x and y is the probability that both x and y occur. p(x,y) = Pr(X and Y) p(0, 1) = 1/8, p(1, 2) = 1/4, and p(3, 3) = 0.

29. Joint Distributions (cont.) B. Marginal Probabilities 1. Def: Marginal probability is the sum of the rows and columns. The overall probability of an event occurring. So the probability that there are just 1 head is the prob of 1 head and 1 runs + 1 head and 2 runs + 1 head and 3 runs = 0 + 1/4 + 1/8 = 3/8

30. Joint Distributions (cont.) C. Independence A and B are independent if P(A|B) = P(A).

31. Joint Distributions (cont.) Are the # of heads and the # of runs independent?

32. Correlation and Covariance A. Covariance 1.Definition of Covariance Which is defined as the expected value of the product of the differences from the means.

33. Correlation and Covariance 2.Graph

34. Correlation and Covariance B. Correlation 1.Definition of Correlation

35. Correlation and Covariance 2. Characteristics of Correlation

36. Correlation and Covariance 2. Characteristics of Correlation (cont.)

37. Correlation and Covariance 2. Characteristics of Correlation (cont.)

38. Correlation and Covariance 2. Characteristics of Correlation (cont.) Why?

39. Sample Homework GET /FILE 'gss91.sys'. The SPSS/PC+ system file is read from file gss91.sys The SPSS/PC+ system file contains 1517 cases, each consisting of 203 variables (including system variables). 203 variables will be used in this session. ------------------------------- COMPUTE AFFAIRS = XMARSEX. RECODE AFFAIRS (0,5,8,9 = SYSMIS) (1,2 = 0) (3,4 = 1). VALUE LABELS AFFAIRS 0 'BAD' 1 'OK'. The raw data or transformation pass is proceeding 1517 cases are written to the compressed active file. ***** Memory allows a total of 10345 Values, accumulated across all Variables. There also may be up to 1293 Value Labels for each Variable.

40. Sample Homework