Understanding Probability Distributions and Hypothesis Testing in Econometrics

1. MF-852 Financial Econometrics Lecture 4 Probability Distributions and Intro. to Hypothesis Tests Roy J. Epstein Fall 2003

2. Distribution of a Random Variable • A random variable takes on different values according to its probability distribution. • Certain distributions are especially important because they describe a wide variety of random variables. • Binomial, Normal, student’s t

3. Binomial Distribution • Random variable has two outcomes, 1 (“success”) and 0 (“failure”) • Coin flip: heads = 1, tails = 0 • P(success) = p • P(failure) = q = (1 – p) • Binomial distribution yields probability of x successes in n outcomes. • Excel will do the calculations.

4. Tails and Body of a Distribution

5. Binomial Example (RR p. 20) • Medical treatment has p = .25. • n = 40 patients • What is probability of at least 15 successes (cures) • I.e, P(x 15)?

6. Normal Distribution • A normally distributed random variable: • Is symmetrically distributed around its mean • Can take on any value from – to + • Has a finite variance • Has the famous “bell” shape • “Standard normal:” mean 0, variance 1.

7. Standard Normal Distribution

8. N(0, .5) Distribution

9. N(0,1) Probabilities • Suppose z has a standard normal distribution. What is: • P(z  1.645)? • P(z  –1.96)? • Excel will tell us!

10. N(0,1) and Standardized Variables • Suppose x is N(12,10). • What is P(x  24.8) ?

11. Key Properties of Normal Distribution • Sum of 2 normally distributed random variables is also normally distributed. • The distribution of the average of independent and identically distributed NON-NORMAL random variables approaches normality. • Known as the Central Limit Theorem • Explains why normality is so pervasive in data

13. Sample Mean • Take a sample of n independent observations from a distribution with an unknown . • Data are n random variables x1, … xn. • We estimate the unknown population mean with the sample mean “xbar”:

14. Properties of Sample Mean • Sample mean is unbiased!

15. Properties of Sample Mean • Sample mean has variance. But the variance is reduced with more data.

16. Null Hypothesis • “Null hypothesis” (H0) asserts a particular value (0) for the unknown parameter  of the distribution. • Written as H0 :  = 0 • E.g., H0 :  = 5 • H0 usually concerns a value of particular interest (e.g., given by a theory)

17. Null Hypothesis • xbar is unlikely to equal 0 exactly. • Samples have sampling error, by definition. • Is xbar still consistent with H0 being a true statement? • This involves a hypothesis test.

18. Hypothesis Testing • Hypothesis testing finds a range for  called the confidence interval. • The confidence interval is the set of acceptable hypotheses for , given the available data. • H0 is accepted if the confidence interval includes 0. • Otherwise H0 is rejected.

19. Confidence Interval • confidence interval = xbar  allowable sampling error • How wide should the interval be around xbar? • Customary to use a 95% confidence interval. • The interval will include the true  95% of the time • Each tail probability is 2.5%.

20. Construction of Confidence Interval • If x1, … xn are normally distributed then xbar is normally distributed. Then: • The 95% confidence interval is

21. Confidence Interval Example • You are a restaurant manager. Burgers are supposed to weigh 5 ounces on average. The night shift makes burgers with a standard deviation of 0.75 ounces. • You eat 12 burgers from the night shift and xbar is 5.4 ounces. What is a 95% confidence interval for the weight of the night shift burgers? • You eat 8 more burgers that have an average weight of 5.25 ounces. What is a 95% confidence interval for this sample? • What is a 95% confidence interval based on all 20 burgers?

22. Sample Variance • Usually the population variance, as well as the mean, is unknown. • Estimate 2 with the sample variance: • We divide by n-1, not n. • What is the sample variance of xbar?

23. Sample Variance • Usually the population variance, as well as the mean, is unknown. • Estimate 2 with the sample variance: • We divide by n-1, not n. • What is the sample variance of xbar?

24. t-distribution • Confidence intervals use the t-distribution instead of the normal when the variance is estimated from the sample. • T-distribution has fatter tails than the normal. • Confidence intervals are wider because we have less information.

25. t distribution (3 dof)

26. Confidence Interval with t-distribution • You hired Leslie, a new salesperson. Leslie made the following sales each month in the first half: • January — $25,000 April — $20,000 • February — $27,000 May — $22,000 • March — $29,000 June — $35,000 • What is a 95% confidence interval for Leslie’s monthly sales? (assume monthly sales are normally distributed) • Suppose you knew that the standard deviation of sales was $1,500. How would your conclusion change?

27. Significance Levels • Assuming H0, what is the probability that the sample value would be as extreme as the value we actually observed? • Alternative to confidence interval • Equal to

28. Type 1 and Type 2 Error • Accept or reject H0 based on the confidence interval. • Type 1 error: reject H0 when it is true. • What is probability of this? • Type 2 error: accept H0 when it is false. • How important is this?