Lecture 5: Hypothesis Testing
A hypothesis is a claim about a parameter that you’re interested in. The simplest hypotheses are about the parameters of a single variable, such as the mean of a population. But there are more complicated hypotheses, as we’ll see when we get to regression analysis; these hypotheses are about the parameters that control the relationship between two or more variables.
Some simple hypotheses:
• The average number of customers in this store per day is greater than 10.
• Condoms from this production line will break less than 1% of the time.
• The average number of years it takes to graduate from CSUN is 6.5.
If we wanted to test the claim, we could state the null and alternative hypotheses like so:
H0: μ = 6.5
H1: μ ≠ 6.5
Here we’re taking the administration’s claim as the null; we are giving them the benefit of a doubt, and will only reject their claim with sufficient evidence to the contrary.
On the other hand, what if the CSUN administration claims the average number of years to graduate from CSUN is no more than 6.5? Then there is only one way they could be wrong: if the average is really higher. We could state the null and alternative hypotheses like so:
H0: μ < 6.5
H1: μ > 6.5
Again, we’re giving the administration the benefit of a doubt by putting their claim as the null.
These two kinds of test are different. The first is called a two-tail test, because there are two ways we could reject the null. The second is called a one-tail test, because there is only one way we could reject the null.
We can see the difference by looking at the distribution of sample means around the population mean. [Draw the bell curve with mean centered 6.5. Show regions to both the left and right of 6.5, indicating two ways to reject the hypothesis that the mean really is 6.5: because the sample mean is especially small or especially large.] [Then draw the same bell curve, but with a somewhat larger region on the right, showing a rejection of the null because the sample mean is especially large. Have no similar region on the left.]
We could have done a different one-tail test. What if CSUN’s administration claimed the average graduation time was at least 6.5? Then we would say:
H0: μ > 6.5
H1: μ < 6.5
Again, this gives the benefit of a doubt to the administration.
III. Significance Levels and Type I and Type II Errors
Remember from the lecture on CI’s that we had to choose a significance level, designated α. This was the probability that a CI generated from a sample would not include the true mean.
Now, we’ll use the same significance level, or α, for the probability that a hypothesis test will reject the null hypothesis even though it’s true. This probability corresponds to the shaded area in the distributions just examined. If the true mean is 6.5, we could still (by chance) get a sample far enough from 6.5 that we reject the null hypothesis. In the two-tail test, this could happen with an especially large or especially small sample mean. In the one-tail test, this could happen only with an especially large sample mean (for the claim that graduation rates are no more than 6.5).
The level of significance is often set at 0.10, or 10%. For the two-tail test, we need to split this between the tails, for 0.05 or 5% each. For the one-tail test, we put all the weight in a single tail.
But we could choose a different significance level. In general, for significance level α, put α/2 in each tail for a two-tail test, α in the appropriate tail for a one-tail test.
We usually make the...