The Test Statistic

Hypothesis testing involves the use of distributions of known area, like the normal distribution, to estimate the probability of obtaining a certain value as a result of chance. The researcher is usually testing to see if the probability will be low because that means it is likely that the test result was not a mere coincidence but occurred because the researcher's theory is correct. It could mean, for example, that it is probably not just bad luck but faulty packaging equipment that caused you to get a box of raisin cereal with only five raisins in it.

Only two outcomes of a hypothesis test are possible: Either the null hypothesis is rejected, or it is not. You have seen that values from normally distributed populations can be converted to z-scores and their probabilities looked up in Table 2 in "Statistics Tables." The z‐score is one kind of test statistic that is used to determine the probability of obtaining a given value. In order to test hypotheses, you must decide in advance what number to use as a cutoff for whether the null hypothesis will be rejected. This number is sometimes called the critical or tabled value because it is looked up in a table. It represents the level of probability that you will use to test the hypothesis. If the computed test statistic has a smaller probability than that of the critical value, the null hypothesis will be rejected.

For example, suppose you want to test the theory that sunlight helps prevent depression. One hypothesis derived from this theory might be that hospital admission rates for depression in sunny regions of the country are lower than the national average. Suppose that you know the national annual admission rate for depression to be 17 per 10,000. You intend to take the mean of a sample of admission rates from hospitals in sunny parts of the country and compare it to the national average.

Your research hypothesis is:

The mean annual admission rate for depression from the hospitals in sunny areas is less than 17 per 10,000.

In notation: H a : μ 1 < 17 per 10,000

The null hypothesis is:

The mean annual admission rate for depression from the hospitals in sunny areas is equal to 17 per 10,000.

In notation: H 0: μ 1 = 17 per 10,000

Your next step is to choose a probability level for the test. You know that the sample mean must be lower than 17 per 10,000 in order to reject the null hypothesis, but how much lower? You settle on a probability level of 5 percent. That is, if the mean admission rate for the sample of sunny hospitals is so low that the chance of obtaining that rate from a sample selected at random from the national population is less than 5 percent, you will reject the null hypothesis and conclude that there is evidence to support the hypothesis that exposure to the sun reduces the incidence of depression.

Next, you look up the critical z‐score—the z‐score that corresponds to your chosen level of probability—in the standard normal table. It is important to remember which end of the distribution you are concerned with. Table 2 in "Statistics Tables" lists the probability of obtaining a given z‐score or lower. That is, it gives the area of the curve below the z‐score. Because a computed test statistic in the lower end of the distribution will allow you to reject your null hypothesis, you look up the z‐score for the probability (or area) of 0.05 and find that it is –1.65. If you were hypothesizing that the mean in sunny parts of the country is greater than the national average, you would have been concerned with the upper end of the distribution instead and would have looked up the z‐score associated with the probability (area) of 0.95, which is z = 1.65.

The critical z‐score allows you to define the region of acceptance and the region of rejection of the curve (see Figure 1). If the computed test statistic is below the critical z‐score, you can reject the null hypothesis and say that you have provided evidence in support of the alternative hypothesis. If it is above the critical value, you cannot reject the null hypothesis.

Figure 1.The z‐score defines the boundary of the zones of rejection and acceptance.

figure 

Suppose that the mean admission rate for the sample of hospitals in sunny regions is 13 per 10,000 and suppose also that the corresponding z‐score for that mean is –1.20. The test statistic falls in the region of acceptance; so you cannot reject the null hypothesis that the mean in sunny parts of the country is significantly lower than the mean in the national average. There is a greater than 5 percent chance of obtaining a mean admission rate of 13 per 10,000 or lower from a sample of hospitals chosen at random from the national population, so you cannot conclude that your sample mean could not have come from that population.