Hypothesis Testing - 8, Sections 8.3 through 8.8

Two-tailed test (also called a "Nondirectional test")

A test that rejects extreme outcomes in either tail of the distribution

One-tailed test (also called a "Directional test")

A test that rejects extreme outcomes in one specified tail of the distribution

In general, two-tailed tests are far more common than one-tailed tests for several reasons. One reason for this is that the investigator may have no idea what the data will look like and therefore has to be prepared for any eventuality.

Another common reason for preferring two-tailed tests is that the investigators are reasonably sure the data will come out one way but want to cover them- selves in the event that they are wrong. This type of situation arises more often than you might think. (Carefully formed hypotheses have an annoying habit of being phrased in the wrong direction, for reasons that seem so obvious after the event.)

Finger Tap Test computerized finger-tapping test that produces scores from nonclinical participants with a mean of 59 and a standard deviation of 7 (rounded) over a 10-second trial. *Finally, suppose that an individual has just been sent to us who taps at a rate of 45 taps in 10 seconds. Is his score sufficiently below the mean for us to assume that he did not come from a population of neurologically healthy people* We begin by assuming that the individual's score does come from the population of healthy scores. This is the null hypothesis 1H02. If H0 is true, we auto- matically know the mean and the standard deviation of the population from which he was supposedly drawn (59 and 7, respectively). With this information we are in a position to calculate the probability that a score as low as his would be obtained from this population. If the probability is very low, we can reject H0 and conclude that he did not come from the healthy population. Conversely, if the probability is not particularly low, then the data represent a reasonable result under H0, and we would have no reason to doubt its validity and thus no reason to doubt that the person is healthy.

Because we have specified that we will reject H0 if the probability of the data under H0 is less than .05, we will conclude that we have a person whose score does not come from a population of healthy people. More specifically, we conclude that a finger-tapping rate of 45 could not reasonably have come from a population of scores with a mean equal to 59 and a standard deviation equal to 7.

*Sample statistics*

Statistics calculated from a sample and used primarily to describe a sample

Type II error

The error of not rejecting H₀ when it is false

Type I error

The error of rejecting H0 when it is true

Alternative hypothesis (H1)

The hypothesis that is adopted when H0 is rejected; usually the same as the research hypothesis.

α (alpha)

The probability of a Type I error

β (beta)

The probability of a Type II error

Power

The probability of correctly rejecting a false null hypothesis The power of a test is the probability of rejecting H₀ when it is actually false. Because the probability of failing to reject a false H₀ is β, then the power must equal 1 - β

Rejection level

The probability with which we are willing to reject H₀ when it is, in fact, correct

Significance level

The probability with which we are willing to reject H₀ when it is, in fact, correct.

*Test statistics*

The results of a statistical test

Rejection region

The set of outcomes of an experiment that will lead to rejection of H₀.

Critical value

The value of a test statistic at or beyond which we will reject H0.

t test

it is sufficient to say that the t test is often used, among other things, to determine whether two samples were drawn from populations with the same means

What does the null hypothesis tell us about two population means?

the difference between the two population means is zero, or null