Two-sided tests
In the example about randomly assigned red and blue uniforms during the 2004 Olympics, the contestant wearing red won 242 out of 441 times (or about 54.9% of the time). We conducted a hypothesis test to evaluate the claim by researchers Hill and Barton that red was likely to win more often than would be expected purely by chance. But think about this from the perspective of the Olympics officials: it's bad if red has an advantage, but it's also bad if blue has an advantage. We could adjust our alternative hypothesis to be:
HA: `p ne 0.50`
so that we would reject the null hypothesis (H0: p = 0.50) not only if we found evidence that the proportion of all matches won by contestants wearing red was above 50%, but also if we found evidence that it was below 50%. This is called a two-sided alternative hypothesis.
Computing the P-value for this revised alternative hypothesis on the calculator is very simple. All we need to do is change the form of the alternative hypothesis in the 1-PropZTest menu:
If we select the Draw option and press ENTER we get:
(Warning: you might need to turn off all of the STAT PLOTS and delete entries from the Y= menu if you get an error message.) Notice that the right side of this picture looks like the one we drew before (for the one-sided alternative hypothesis) but that there is now a similar area shaded in on the left side of the Normal model. The sample proportion we observed (`hat(p) = 0.549`) was just over 2 SDs above the hypothesized proportion (p = 0.50), and anything to the right of that value provides even stronger evidence that the true population proportion is higher than we hypothesized. But if we had observed a sample proportion 2 SDs below p = 0.50, that would provide evidence that the true population proportion was lower than the hypothesized value, and anything to the left of this would provide even stronger evidence. Because the alternative hypothesis is now that the population proportion is different from 0.50, any sample proportion significantly higher than 0.50 or significantly lower than 0.50 will provide evidence that the true population proportion is different from 0.50. So we need to add the probability from the right (upper) tail to the probability from the left (lower) tail. Because the Normal model is symmetric, the left probability will be the same as the right probability, so we can just double our original P-value. Note that the P-value given by the TI-84 above (P = 0.0406) is exactly twice the P-value we computed for the one-sided alternative hypothesis.