2_Probability and Statistics

Hence an important benefit of a large sample size is

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Unformatted text preview: Y ) . The associated random variable, SY / n , is called the standard error of Y Now, we define the standard error of y as se(y ) = sY / n . Then, the confidence interval can be written as [y ca/2 ⋅ se(y )] o Because the standard error se(y ) = sY / n shrinks to zero as the sample size grows – all else being equal – a larger sample size means a smaller confidence interval. Hence, an important benefit of a large sample size is that it results in smaller confidence intervals VER. 9/11/2012. © P. KOLM 54 Remarks (2/2): Note that for a = 0.05 , ca/2 (n ) > 1.96 . But ca/2 (n ) 1.96 as n ¥ o This means that for large samples we can construct confidence intervals just using the normal distribution (cf. CLT) o For smaller samples, a rule of thumb for an approximate 95% confidence interval is [y 2 ⋅ se(y )] VER. 9/11/2012. © P. KOLM 55 Example (1/2): Tony Gwynn Before a strike prematurely ended the 1994 major league baseball season, Tony Gwynn of the San Diego Padres had 165 hits in 419 at bats, for a 0.394 batting average. There was discussion about whether Gwynn was a potential 0.400 hitter that year. Was he? VER. 9/11/2012. © P. KOLM 56 Example (2/2): Solution This question can be stated in terms of Gwynn’s probability of getting a hit on a particular at bat, call it q . Let Yi be the Bernoulli(q) indicator equal to unity if Gwynn gets a hit during his i-th at bat, and zero otherwise. Then, Y1,...,Yn is a random sample from a Bernoulli(q) distribution, where q is the probability of success, and n=419. A point estimate of q is Gwynn’s batting average, the proportion of successes, is y = 0.394 Using that se(y ) = y (1 - y ) / n » 0.024 we obtain the standard normal approximation for the 95% confidence interval as 0.394 1.96(.024), or about [0.347, 0.441] Therefore, based on Gwynn’s average up to strike, there is not very strong evidence against q = 0.400 , as this value is well within the 95% confidence interval VER. 9/11/2012. © P. KOLM 57 Hypothesis Testing Hypothesis is a belief concerning a parameter typically stated in terms of a question that has a definite yes or no answer Parameter may be population mean, proportion, correlation coefficient, etc., but can also be regression coefficients and performance measures Examples: o Is the amount of overtime work equal for males and females? o Did the (passive) portfolio manager track the benchmark? o Did the (active) portfolio manager beat the benchmark? o Is a trading strategy better than another? o Is there a correlation between the five-year interest rate and spot gold prices? VER. 9/11/2012. © P. KOLM 58 Hypothesis Testing: Null vs. Alternative Hypothesis Null hypothesis, denoted by H 0 , is prevalent opinion, previous knowledge, basic assumption, prevailing theory o It is assumed to be true as long as we find evidence against it Alternative hypothesis, denoted by H 1 , is the “rival opinion” o If a sample gives strong enough evidence against null hypothesis then alternative hy...
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This document was uploaded on 02/17/2014 for the course COURANT G63.2751.0 at NYU.

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