Rejection region

Evaluation for the Phenyl Calibration and Test Data for a Model with Two PLS Components Applying a Rejection Region... 

Individual Reject Region Fig. 10 Weight control system. 

To carry out the statistical test, a test procedure must be implemented. The crucial elements of a test procedure are the formation of an appropriate test statistic and the identification of a rejection region. The test statistic is formulated from the data on which we will base the decision to accept or reject Hq. The rejection region consists of all the values of the test statistic for which Hq will be rejected. The null hypothesis is rejected if the test statistic lies within the rejection region. For tests concerning one or two means, the test statistic can be the z statistic if we have a large number of measurements or if we know a. Alternatively, we must use the t statistic for small numbers with unknown cr. When in doubt, the t statistic should be used. 

State the alternative hypothesis, H., and determine the rejection region ... 

The rejection regions are illustrated in Figure 7-2 for the 95% confidence level. Note that for p A pq, we can reject for either a positive value of z or for a negative value of z that exceeds the critical value. This is called a two-tailed test. For the 95% confidence level, the probability that z exceeds Zcn, is 0.025 in each tail or 0.05 total. Hence, there is only a 5% probability that random error will lead to a value of z 3 Zcnt or z s —Zcrit- The significance level overall is a = 0.05. From Table 7-1, the critical value is 1.96 for this case. 

Figure 1-1 Rejection regions for the 95% confidence level, (a) Two-tailed test for H,. Note the critical...

The choice of a rejection region for the null hypothesis is made so that we can readily understand the errors involved. At the 95% confidence level, for example, there is a 5% chance that we will reject the null hypothesis even though it is true. This could happen if an unusual result occurred that put our test statistic z or t into the rejection region. The error that results from rejecting when it is true is called a type I error. The significance level a gives the frequency of rejecting Hq when it is true. 

The one-sample t test is being used for a two-sided test of the null hypothesis, Hq. g = 0. For each of the following scenarios, define the rejection region for the test ... 

As the value of the test statistic is in the rejection region (only just, but still in it), the null hypothesis is rejected. The conclusion is that the distributions of the two populations from which the samples were selected differ in their location. The test treatment is associated with a greater reduction in SBP than placebo. 

Under the null hypothesis, this test statistic follows a standard normal distribution. The null hypothesis is rejected because the test statistic falls in the rejection region for a two-sided test of a = 0.05 based on the standard normal distribution (Z < - 1.96 ot Z > 1.96). 

This test statistic is not well defined in all cases, which means that a rejection region is not automatically defined from a known distribution. However, if some assumptions are made about the distribution of the random variable X, the distribution of the test statistic can be defined. The following assumptions are required for an appropriate use of ANOVA ... 

The F distribution with (k - 1) numerator df and (n - k) denominator df is used to define the rejection region for a test of size a. The critical region may be obtained from a table of values or provided by statistical software. Tabled F values... 

Construct the ANOVA table Having calculated the total sums of squares from all sources of variation, along with their degrees of freedom, we can now start to construct the ANOVA table. The only other calculations required are the mean squares for among-samples and within-samples (divide each sums of squares by its associated df) and the test statistic, F (divide among-samples mean square by within-samples mean square). All of this information is shown in the partial ANOVA table presented as Table 11.3. Determine if the test statistic is in the rejection region As always, we need to determine if the test statistic F falls in the rejection region. So far, we have not determined the... 

As the value of the test statistic, 11.67, is in the rejection region for this test of size a = 0.05 (that is, 11.67 > 3.89), the null hypothesis is rejected in favor of the alternate, which means that at least one pair of the population means is not equal. 

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