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CHAPTER 5. A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING 5.1 Concepts When a number of animals or plots are exposed to a certain treatment, we usually estimate the effect of the treatment by the mean response of the experimental units. Statistically this is referred to as a point estimate of the parameter from the population of all possible treated animals or plots. Different samples, however, yield different estimates. It is desirable, therefore, to have a range of values so that we have a high degree of confidence that the true treatment mean will fall within this range. This is called confidence interval estimation. These ideas can be illustrated by the following example. Suppose we wish to know the mean heart rate of female horses (mares) after being treated by a drug. A random sample of 25 treated mares is taken and the heart rate of each is determined. The average heart rate is 40.5 beats per minute, and the standard deviation of the 25 determinations is 3. With 95% confidence, we would like to know a range of heart beats within which the population mean will fall. From t he standard deviation of our sample we estimate the standard error, S y = 3 / 25 = 0.6 , with df = 24. The 95% confidence interval is calculated as y ± t α S y = 40.5 + 2.06 (0.6) = 40.5 + 1.24, That is between 39.26 and 41.74. Based on the sample, our best single value estimate of μ is the sample mean, 40.5. Now, however, we are 95% confident that the μ is equal to or greater than 39.26 and smaller than or equal to 41.74. The idea of confidence intervals is related to the concept of hypothesis testing. A statistical hypothesis is an assumption made about some parameter of a population. The hypothesis being tested is often denoted by H0 and is called the null hypothesis since it implies that there is no real difference between the parameter and its hypothesized value. For example, it may have been hypothesized that the mean heart rate of the treated mare population (μ) is the same as that of the nontreatment population which is known to be 39 beats/minute. The null hypothesis to be tested is that μ = 39. A statistical procedure which provides a probablistic assessment of the truth or falsity of a hypothesis is called a statistical test. Figure 5-1 illustrates the process of hypothesis testing. The confidence interval procedure illustrated above can also be used for the purpose of hypothesis testing. For this example, the 95% confidence interval does not bracket the hypothesized value of 39, and therefore the sample data do not support the null hypothesis. Fig. 5-1. Procedure for testing a null hypothesis, H0: μ = μ*. The experiment is interested in a population with an unknown mean, μ. He wishes to test the hypothesis that the unknown mean is equal to a value μ*. The theoretical distribution of sample means drawn from the population centering on μ* is shown on the left. On the right the experimental information obtained from a sample is illustrated. The final stage of the testing process is to compare the experimental results with the hypothesized situation. An alterative hypothesis can be denoted by H1 and will be accepted if H0 is rejected. The nonrejection of H0 is a decision not to accept H1. Our alternative hypothesis for the mare heart rate study can be H1 : μ ≠ 39. Since the null hypothesis, μ = 39, was rejected, we accept this alternative. Note that the nonrejection of H0 is not always equivalent to accepting H0. It merely says that from the available information, there is not sufficient evidence to reject H0. It merely says that from the available information, there is not sufficient evidence to reject H0. In fact, H0 may be rejected if further evidence so indicates. However, in practice it is common to use the acceptance of H0 interchangeably with the nonrejection of H0. In stating the hypothesis that the mean heart rate of mares is 39, H0 : μ = 39, versus the alternative, H1 : μ ≠ 39, we reject H0 if the observed sample mean is significantly greater or smaller than 39. This is a two-tailed test. If, based on prior information, we know that the true mean heart rate cannot be less than 39, our alternative hypothesis would be, H1 : μ > 39. In this case, the test is called a one-tailed test. That is, we will reject the null hypothesis only if the observed sample mean is significantly greater than 39. Figure 5-2 illustrates these situations. Figure 5-2. One-tailed or two-tailed test: the mare heart rate example at α = 5%. One-tailed test is a more directed test; the experimenter must know the direction of the difference if it occurs. Therefore, it is more sensitive (smaller critical value) in detecting the differences than a two-tailed test at the same α level. Whenever a decision is made to reject or not to reject a null hypothesis, there is always a possibility that the conclusion is