In this tutorial, you're going to learn about the difference between a one-tailed and a two-tailed test in a hypothesis test. You will specifically focus on:
Suppose you have your favorite pop, Liter O'Cola and it's come out with new Diet Liter O'Cola. They think that it's indistinguishable so they obtain 120 individuals to do the taste test. If the claim is true, you would expect about 50%, or 60 people, to guess correctly just based on the fact that they were guessing and they guessed it right if the taste was indistinguishable.
What if some people can taste the difference? What would you expect the proportion of people correctly selecting the diet cola to be? You would say it's some number over 50%. At least half of the people will be able to correctly identify which cup is the diet cola.
These are your null and alternative hypotheses. Your null says that p, the true proportion of people who can correctly identify the diet cola is 1/2, half the people. Your alternative hypothesis suspects that maybe more than half of people will be able to select the diet cola correctly.
This is called a one-tailed test. You're only interested in testing whether or not the true proportion of people who can guess correctly or identify which one is the diet cola is over half. You don't care if it's under half. If it's under half, that actually works in Liter O'Cola's favor.
A test for when you have reason to believe the population parameter is higher or lower than the assumed parameter value of the null hypothesis
Next suppose you are presented with a different scenario. You suspect that Liter O'Cola is under-filling their bottles. Unsurprisingly, the bottles are supposed to contain 1 liter of cola.
State the null and alternative hypothesis for this. Do you think this is a one-tailed test or two-tailed test?
This is another example of a one-tailed test. The null hypothesis says that the average amount of cola in the bottle is 1 liter over all the bottles that Liter O'Cola makes. The alternative is that maybe we think that it's less than 1. They're under-filling the bottles. The average amount is less than 1 liter.
If the average amount, mu, was greater than 1 liter, you wouldn't really have a claim against Liter O'Cola because you're actually getting more pop than they say that they're giving us.
You're only going to give them trouble if they're under-filling their bottles.
Let's take a look at a third example. Liter O'Cola also claims 35 grams of sugar in its bottles of cola.
Anything over that, and the pop will taste too sweet. Anything under that, and the pop won't taste quite sweet enough. They won't get the refreshing Liter O'Cola taste that people have come to expect. So we think that Liter O'Cola might have altered their formula recently because it tastes different. So what do you think the null and alternative hypotheses will be here with respect to sugar?
Here, the null hypothesis is that the mean grams of sugar will be the same as it was before, 35. What about the alternative hypothesis? Well, if they've changed their formula, you don't know if they added more sugar or put in less sugar. But they're only going to be in trouble if they put in a different amount of sugar than before. The mean, a number of grams of sugar in the bottle, is different than 35. And this is a two-tailed test. They're going to be in trouble if they put in significantly more than 35 or significantly less.
A test for when you have reason to believe the population parameter is different from the assumed parameter value of the null hypothesis
One-tailed tests are preferred to two-tailed tests because they're more powerful. Statistical power means that they have a higher likelihood of actually detecting a difference if one is present. Let's take one last look visually at what a one-tailed test and a two-tailed test look like. This is what a one-tailed test with a p-value of 5% would look like.
This would be under the alternative hypothesis that you have something less than a particular number, like a mean is less than 1, like you had in the one example. You end up with one tail area here of about 5%.
Whereas, this is what a p-value of 5% would look like on a two-tailed test.
You are interested in what's the probability that you would get at least as extreme on either side of a value as you ended up with from our sample. Itt could either be extremely low or extremely high. Something that is extremely different from what you would have expected. Whereas, on this side, you're only going to get them in trouble if it's extremely lower than what you would have expected.
One-tailed tests only can test whether or not there's evidence of a statistic being significantly higher or significantly lower than a particular parameter, like mu or p. Whereas two-tailed tests will tell whether or not the statistic, x bar or p hat, is significantly different on the high or the low side from the claimed parameter. You learned about one-tailed tests, which have two versions, a left- tailed test, where you say in the alternative hypothesis that it's less than this claimed parameter, a right- tailed test, which means that it's larger than the claimed parameter, or there can be a two-sided test, where we claim that simply the true value is different than the claimed parameter, not equal to.
Source: This work adapted from Sophia Author Jonathan Osters.
A hypothesis test where the alternative hypothesis only states that the parameter is higher than the stated value from the null hypothesis.
A hypothesis test where the alternative hypothesis only states that the parameter is lower than the stated value from the null hypothesis.
A hypothesis test where the alternative hypothesis states that the parameter is different from the stated value from the null hypothesis; that is, the parameter's value is either higher or lower than the value from the null hypothesis.
A hypothesis test where the alternative hypothesis only states that the parameter is higher (or lower) than the stated value from the null hypothesis.