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rev12 - STAT1131 Statistics I/25.Nov.2011 Weekly Review 12...

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STAT1131 Statistics I/25.Nov.2011 Weekly Review 12 Last week we introduced the sign test to test the null hypothesis that a population median is equal to a particular value. On Monday I continued to explain how to calculate the p -value for the sign test, and the decision rule is that we reject the null hypothesis whenever the p -value is less than or equal to α . Suppose x is the number of “+” signs, out of n signs. If the alternative is “ > ”, then the larger the value of x , the more likely the alternative is true. The p -value is therefore equal to Pr( X x ), where X B( n, 0 . 5). If the alternative is “ < ”, then the p -value is Pr( X x ). If the alternative is “ ̸ =”, the p -value is equal to the smaller one of 2 × Pr( X x ) and 2 × Pr( X x ). Read Example 18.1 for an illustration. When n is large, we can approximate the binomial distribution by the normal distribution. See Section 18.2. The sign test is a nonparametric analogue to the one-sample t -test. We have another one, which is known as the signed-rank test . That is to say, the signed-rank test also tests the null hypothesis that a population median is equal to a particular value. The idea is to consider not only the sign but also the differences between the data and the hypothesized median. The procedure is as follows. 1. Calculate all differences between the data and the hypothesized median, 2. Rank the differences according to the absolute values. 3. We discard the data that are equal to the hypothesized median. 4. If two or more absolute values of the differences are equal, we assign each one the mean of the ranks which they jointly occupy. 5. Let T + be the sum of the ranks of all positive differences and T - be the sum of the ranks of all negative differences and T = min( T + , T - ). Using the same argument as the sign test above, we can see that 1
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if the alternative is “ < ”, the smaller the value of T + , the more likely the alternative is true; if the alternative is “ > ”, the smaller the value of T - , the more likely the alternative is true; if the alternative is “ ̸ =”, the smaller the value of T , the more likely the alternative is true. Therefore, what we need to decide is: how small is too small. The answer is: it is too small when it is less than or equal to T 2 α for a one-sided test and T α for a two-sided test.
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