Choosing the Sample Size for Testing M

Suppose that after obtaining the sample, we compute ounces. The computed value of the test statistic is Because the rejection region is z ⬍ ⫺1.645, the computed value of z does not fall in the rejection region. What is our conclusion? In similar situations in previous sections, we would have concluded that there is insufficient evidence to reject H . Now, however, knowing that bm ⱕ .01 when m ⱕ 16.27, we would feel safe in our conclusion to accept H : m ⱖ 16.37. Thus, the manufacturer is somewhat secure in concluding that the mean fill from the examined machine is at least 16.37 ounces. With a slight modification of the sample size formula for the one-tailed tests, we can test H : m ⫽ m H a : m ⫽ m for a specified a, b, and ⌬, where b m ⱕ b, whenever 兩m ⫺ m 兩 ⱖ ⌬ Thus, the probability of Type II error is at most b whenever the actual mean differs from m by at least ⌬. A formula for an approximate sample size n when testing a two-sided hypothesis for m is presented here: z ⫽ y ⫺ 16.37 s 兾 1n ⫽ 16.35 ⫺ 16.37 .225 兾 180 ⫽ ⫺ .795 y ⫽ 16.35 Approximate Sample Size for a Two-Sided Test of H : ␮ ⴝ ␮ Note: If s 2 is unknown, substitute an estimated value to get an approximate sample size. n ⬵ s 2 ∆ 2 z a 兾2 ⫹ z b 2

5.6 The Level of Significance of a Statistical Test

In Section 5.4, we introduced hypothesis testing along rather traditional lines: we defined the parts of a statistical test along with the two types of errors and their as- sociated probabilities a and bm a . The problem with this approach is that if other researchers want to apply the results of your study using a different value for a then they must compute a new rejection region before reaching a decision concerning H and H a . An alternative approach to hypothesis testing follows the following steps: specify the null and alternative hypotheses, specify a value for a, collect the sample data, and determine the weight of evidence for rejecting the null hypothe- sis. This weight, given in terms of a probability, is called the level of significance or p-value of the statistical test. More formally, the level of significance is defined as follows: the probability of obtaining a value of the test statistic that is as likely or more likely to reject H as the actual observed value of the test statistic, assuming that the null hypothesis is true. Thus, if the level of significance is a small value, then the sample data fail to support H and our decision is to reject H . On the other hand, if the level of significance is a large value, then we fail to reject H . We must next decide what is a large or small value for the level of significance. The following decision rule yields results that will always agree with the testing procedures we introduced in Section 5.5. level of significance p-value We illustrate the calculation of a level of significance with several examples. EXAMPLE 5.12 Refer to Example 5.7. a. Determine the level of significance p-value for the statistical test and reach a decision concerning the research hypothesis using a ⫽ .01. b. If the preset value of a is .05 instead of .01, does your decision concern- ing H a change? Solution a. The null and alternative hypotheses are H : m ⱕ 380 H a : m ⬎ 380 From the sample data, with s replacing s, the computed value of the test statistic is The level of significance for this test i.e., the weight of evidence for reject- ing H is the probability of observing a value of greater than or equal to 390 assuming that the null hypothesis is true; that is, m ⫽ 380. This value can be computed by using the z-value of the test statistic, 2.01, because p-value ⫽ P ⱖ 390, assuming m ⫽ 380 ⫽ Pz ⱖ 2.01 Referring to Table 1 in the Appendix, Pz ⱖ 2.01 ⫽ 1 ⫺ Pz ⬍ 2.01 ⫽ 1 ⫺ .9778 ⫽ .0222. This value is shown by the shaded area in Figure 5.13. Because the p-value is greater than a .0222 ⬎ .01, we fail to reject H and conclude that the data do not support the research hypothesis. y y z ⫽ y ⫺ 380 s 兾 1n ⫽ 390 ⫺ 380 35.2 兾 150 ⫽ 2.01 Decision Rule for Hypothesis Testing Using the p -Value 1. If the p-value ⱕ a, then reject H . 2. If the p-value ⬎ a, then fail to reject H . FIGURE 5.13 Level of significance for Example 5.12 z = 0 f z z 2.01 p = .0222 b. Another person examines the same data but with a preset value for a ⫽ .05. This person is willing to support a higher risk of a Type I error, and hence the decision is to reject H because the p-value is less than a .0222 ⱕ .05. It is important to emphasize that the value of a used in the decision rule is preset and not selected after calculating the p-value.