ISCAM
Lab 2

1: Background 2: Files Needed 3: Data collection 4: Two-sided alternative 5: Two-sided p-values 6: Analysis 7: Another guess 8: Summary

Two-sided p-values

With a two-sided alternative hypothesis, sample proportions that are much larger than 0.74 OR much smaller than 0.74 will be considered evidence against the null hypothesis and in favor of the alternative hypothesis. In other words, we will look for values at least as extreme as the observed proportion in either direction. So how do we define "or more extreme" with a two-sided alternative....

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Probability Detour

(c) Suppose for the moment that the probability of turning right when kissing is = 0.90 and we observed n =50 couples. Use the One Proportion Inference applet to create the hypothesized (exact) binomial distribution for this scenario.

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Report the expected value (mean) and standard deviation of this distribution.

(d) Suppose we observe 42 couples leaning right. Report the Exact Binomial probability P(X ≤ 42).

(e) How far is 42 from the expected value? What value for the number of successes is the same distance above the expected value?

So one approach would be to find the probability of an outcome that far from the expected value in each direction, e.g., P(X ≤ 42 or X ≥ 48).

• In the applet above, check the Two-sided box.

(f) Does the applet use 48 for the right side? What value does it use instead?

Thought Question: Suggest another way of calculating a two-sided p-value than what you suggested in (e)? What do you think the applet is doing instead?

Click here when ready to proceed.

The “method of small p-value” determines a two-sided p-value by summing the probabilities for all outcomes with a probability equal to or smaller than the probability of the observed result. In this case, the applet calculates P(X ≤ 42) + P(X ≥ 49) because 49 is the first time the probability of an observation is less than 0.0643 (and any value larger than 49 will have an even smaller probability). That is, this is the first bar above the expected value that is shorter than the bar at 42 successes (first outcome with a smaller probability). Some software packages (e.g., R, applet) use this approach. Others (e.g., Minitab, JMP) use a different approach. Others find the (smaller) one-sided p-value and simply double it, but this doesn’t really account for the non-symmetric nature of the distribution when is close to zero or 1.

Definition: Two-sided p-values are used with two-sided alternative hypotheses (not equal to). A two-sided p-value considers outcomes in both tails that are at least as rare as the observed result, where at least as rare could imply a smaller probability of occurring or being as far or farther from the expected value. If the null distribution is symmetric these two approaches are equivalent and the two-sided pvalue will be double the one-sided p-value.