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Unformatted text preview: WHAT PRACTITIONERS NEED TO KNOW ... About iHlypothesis Testing Mark Kritzman Financial analysts work with noisy data. As a consequence, it is often difficult to determine whether observed results are due to a real effect or reside within the realm of noise. This column reviews the methodology referred to as h3^othesis testing, used to distingmsh real effects from noise. Comparing Proportions from a Small Sample Suppose we wish to test whether or not a coin is fair. We begin by defining the null hypothesis and the alternative hypothesis. In this example, the null hypothesis, denoted by HQ, is that the coin is fair. The alternative hypothesis, denoted by H^, is that the coin is biased. Next we need to compute a test statistic. We do so by repeatedly tossing the coin and observing the number of heads or tails. Finally, we need to compute a P value, which is the probability that the test statistic would occur if the null hj^othesis were true. The estimation of the P value depends on the notion of a Bernoulli trial. A Bernoulli trial has three properties. (1) Its result must be characterized by a success or a failure. (2) The probability of a success must be the same for all of the trials. (3) The outcome of each trial must be independent of the outcomes of the other trials. The toss of a coin clearly satisfies the conditions of a Bernoulli trial. The fraction of successes from a sequence of Bernoulli trials is called a binomial random variable and serves as the test statistic in this example. The P value, which is the probability of observing a particular test statistic from a binomial distribu- tion, is given by Equation 1: P(X) = n! X! (n - X)!' (1 ,\n-X (1) where p (X) = probability of X heads in n tosses, n = number of tosses in the sample, p = expected proportion of heads resulting from tossing a fair coin. Mark Kritzman is a Partner of Windham Capital Management. X = number of heads observed in the sample and ! = a factorial (for example, 5! = 5 x 4 x 3 x 2 X 1). Suppose we toss the coin 10 times and observe eight heads. In our example, n equals 10, p equals 0.50 and X equals 8. If we substitute these values into Equation 1, we find that there is only a 4.39% probability of observing eight heads in 10 tosses. This probability equals the P value. If we are willing to tolerate a 5% chance of error, we would reject the null hypothesis that the coin is fair in favor of the alternative hypothesis that the coin is biased. It is important to note that we can never accept the null hypothesis. We can reject the null hypoth- esis or we can fail to reject the null hypothesis. This is the only term with which I am familiar that constitutes a triple negative. In the event we fail to reject the null hypothesis, we conclude that the test lacks sufficient power to accept the alternative hypothesis....
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- Three '10
- Normal Distribution, Null hypothesis, Portfolio Manager, Mark Kritzman