invest_3ed.pdf

# To find the tail probability in the other direction

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distribution. To find the tail probability in the other direction, you can first determine the values that are as extreme in the other direction, 82.67 + 2.67 = 85.33, and then you would calculate P(X > 85.33) = P(X > 86) = 0.298. So then the two-sided p-value reported by the applet was P(X < 80) + P(X > 86) = 0.336 + 0.298 = 0.634. Another way to interpret “or more extreme” is to consider the outcomes that are even less likely to occur than the value observed. That is, we will include an observed value of x in our p-value calculation if the probability for that value is smaller than the probability of our observed value. In some situations, this will lead to slightly different results. Below is a portion of the binomial distribution with S = 2/3: x 79 80 81 82 83 84 85 86 87 P(X = x ) 0.0581 0.0655 0.0712 0.0748 0.0759 0.0742 0.0699 0.0635 0.0556 Here, P(X = 80) = 0.0655, P(X = 85) = 0.0699, P(X = 86) = 0.0635 so that 86 is the first value of x that has a smaller probability than 80. This alternative method then tells us to report the two-sided p-value as P(X < 80) + P(X > 86). However, if we had started with 86, then the two-sided p-value will be calculated from P(X > 86) + P(X < 79) as 79 is the first x value with a probability smaller than P(X = 86). When the sample size is large and the probability of success is near 0.5, these two methods should lead to very similar results. However, this latter approach (called the method of small p-values ) accounts for the non-symmetric shape that the binomial distribution often has with small n and S away from 0.5. This is the method used by R. Though we will not advocate one approach over the other here, you will often want to be aware of the algorithm used by your statistical package. In fact, Minitab uses a third approach altogether. Some other software packages find the (smaller) one-sided p-value and simply double it. We find this approach less satisfying when the binomial distribution is not symmetric. Practice Problem 1.4 (a) Do these data provide convincing evidence that the probability of leaning right is larger than 0.5? (State hypotheses, report the p-value, describing how you determined it and clarifying what it is in terms of the probability of P(X ….) , and interpret the strength of evidence against the null hypothesis.) (b) Do these data provide convincing evidence that the probability of leaning right differs from 0.5? (c) What two values are used in (b) for the cut-offs and how far is each from the expected value of X?

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Chance/Rossman, 2015 ISCAM III Investigation 1.5 49 Investigation 1.5: Kissing the Right Way (cont.) In the previous investigation, you learned how to decide whether a hypothesized value of the parameter is plausible based on a two-sided p-value. The two-sided p-value is used when you do not have a prior suspicion or interest in whether the hypothesized value is too large or too small. In fact, in many studies we may not even really have a hypothesized value, but are more interested in using the sample data to estimate the value of the parameter.
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