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

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13 Sample Mean Take a sample of n independent observations from a distribution with an unknown . Data are n random variables x 1, … x n. 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” (H 0 ) asserts a particular value (  0 ) for the unknown parameter  of the distribution. Written as H 0 :  =  0 E.g., H 0 :  = 5 H 0 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 H 0 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. H 0 is accepted if the confidence interval includes  0. Otherwise H 0 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 x 1, … x n 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 — $25,000April — $20,000 February — $27,000May — $22,000 March — $29,000June — $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?