2_Probability and Statistics

Its level is called the p value or observed

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Unformatted text preview: pothesis comes into force Sometimes it happens that an innocent person is proved guilty o Same may happen in hypothesis testing: We may reject the null hypothesis although it is true (Type I error) We may fail to reject null hypothesis when it is actually false (Type II error) o There is always a risk of drawing the wrong conclusion. This risk is due to sampling error. Its level is called the p-value or observed significance level11 o Typical significant levels include 1%, 5% VER. 9/11/2012. © P. KOLM 59 Hypothesis: Examples H 0 : The amount of overtime work is equal for males and females H 1 : The amount of overtime work is not equal for males and females H 0 : The portfolio manager’s tracking error relative to the benchmark was less than or equal to 50bps (annually) H 1 : The portfolio manager’s tracking error relative to the benchmark was greater than 50bps (annually) H 0 : The Sharpe ratio of trading strategy A is greater than or equal the Sharpe ratio of trading strategy B H 1 : The Sharpe ratio of trading strategy A is less than the Sharpe ratio of trading strategy B H 0 : There is no correlation between interest rate and gold price H 1 : There is correlation between interest rate and gold price VER. 9/11/2012. © P. KOLM 60 Testing the Mean from a Population That Is Normally Distributed Let us assume Y1,...,Yn N (m, s 2 ) Null hypothesis: H 0 : m = m0 Alternative hypotheses: H 1 : m ¹ m0 This is called a two-sided alternative VER. 9/11/2012. © P. KOLM 61 Basic idea: We should reject H 0 in favor of H 1 when the value of the sample average, y , is “sufficiently” different from m0 , How should we determine when y is large enough for H 0 to be rejected at a chosen significance level? The answer to this question requires knowing the probability of rejecting the null hypothesis when it is true (i.e. the probability of a Type I error, or a = P (reject H 0 | H 0 )) We calculate the t statistic from the data y = (y1,..., yn ) , t= y - m0 sy / n = y - m0 se(y ) We know that under the null hypothesis the corresponding random variable Tº Y - m0 S/ n follows a t distribution with n - 1 degrees of freedom Our question becomes: So when is T “large”? VER. 9/11/2012. © P. KOLM 62 Yes, we need to choose a significance level ( a ) to determine this!12 Then we can proceed in a similar fashion to what we did when we constructed confidence intervals. Let us choose significance level a = 5% Mathematically, we want to determine the critical value c such that the probability of a Type I error is 5%, that is P (| T |> c | H 0 ) = 5% Decision criteria: We reject H 0 if | t |> c Area = 0.95 Area = 0.025 Area = 0.025 -c Rejection region VER. 9/11/2012. © P. KOLM c Rejection region 63 Example (see also GoogleRet.xlsx) Recall t statistic: y - m0 y - m0 t= = se(y ) sy / n Let n = 22 and choose a = 0.05 Critical value corresponds to the 97.5th percentile in the t21 distribution (recall tn -1 where n = 22) This means that H 0 is rejected only 5% of the time when it is true How do we find c? o Statistical tables of the t distribution o Statistical software Excel: TINV(0.05,21) Matlab: tinv(0.975,21) o c = 2.08 Our decision criteria: The absolute value of the t statistic must exceed 2.08 in order to rejec...
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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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