Population mean hypothesis testing

Under the null hypothesis of equal population means, the test statistic follows a t distribution with Kj 4- 2 - 2 degrees of freedom (df), assuming that the sample size in each group is large (that is, > 30) or the underlying distribution is at least mound shaped and somewhat symmetric. As the sample size in each group approaches 200, the shape of the t distribution becomes more like a standard normal distribution. Values of the test statistic that ate fat away from zero would contradict the null hypothesis and lead to its rejection. In particular, for a two-sided test of size a, the critical region (that is, those values of the test statistic that would lead to rejection of the null hypothesis) is defined by t[Pg.148]

To summarize Figure 18-1 in words, the top curve represents the characteristics of a population P0 with mean /x0. Also indicated in Figure 18-1 is the upper critical limit, marking the 95% point for a standard hypothesis test (//0) that the mean of a given sample is consistent with /x . A measured value above the critical value indicates that it would be too unlikely to have come from population P0, so we would conclude that such a reading came from a different population. Two such possible different, or alternate, populations are also shown in Figure 18-1, and labeled Pt and P2. Now, if in fact a random sample was taken from one of these alternate populations, there is a given probability, whose value depends on which population it came from, that it would fall above (or below) the upper critical limit indicated for H0. 

Analysis of Variance (ANOVA). Keeping in mind that the total variance is the sum of squares of deviations from the grand mean, this mathematical operation allows one to partition variance. ANOVA is therefore a statistical procedure that helps one to learn whether sample means of various factors vary significantly from one another and whether they interact significantly with each other. One-way analysis of variance is used to test the null hypothesis that multiple population means are aU equal. 

When two factors are being compared (e.g., two analysts, two methods, two certified reference materials) but the data are measurements of a series of independent test items that have been analyzed only once with each instance of the factor, it would not be sensible to calculate the mean of each set of data. However, the two measurements for each independent test item should be from the same population if the null hypothesis is accepted. Therefore the population mean of the differences between the measurements should be zero, and so the sample mean of the differences can be tested against this expected value (0). The arrangement of the data is shown in table 2.7. 

Alternatively, the dummy effect can be taken as the repeatability of the factor effects. Recall that a dummy experiment is one in which the factor is chosen to have no effect on the result (sing the first or second verse of the national anthem as the -1 and +1 levels), and so whatever estimate is made must be due to random effects in an experiment that is free of bias. Each factor effect is the mean of N/2 estimates (here 4), and so a Student s t test can be performed of each estimated factor effect against a null hypothesis of the population mean = 0, with standard deviation the dummy effect. Therefore the t value of the ith effect is ... 

For testing the hypothesis on the population mean we can use the following statistic ... 

This procedure is really equivalent to our earlier method of hypothesis testing, as an inspection of Eqs. (1.47)—(1.52) shows. In this part and the previous one, we have outlined the principles of statistical tests and estimates. In several examples, we have made tests on the mean, assuming that the population variance is known. This is rarely the case in experimental work. Usually we must use the sample variance, which we can calculate from the data. The resulting test statistic is not distributed normally, as we shall see in the next part of this chapter. 

If we desire to study the effects of two independent variables (factors) on one dependent factor, we will have to use a two-way analysis of variance. For this case the columns represent various values or levels of one independent factor and the rows represent levels or values of the other independent factor. Each entry in the matrix of data points then represents one of the possible combinations of the two independent factors and how it affects the dependent factor. Here, we will consider the case of only one observation per data point. We now have two hypotheses to test. First, we wish to determine whether variation in the column variable affects the column means. Secondly, we want to know whether variation in the row variable has an effect on the row means. To test the first hypothesis, we calculate a between columns sum of squares and to test the second hypothesis, we calculate a between rows sum of squares. The between-rows mean square is an estimate of the population variance, providing that the row means are equal. If they are not equal, then the expected value of the between-rows mean square is higher than the population variance. Therefore, if we compare the between-rows mean square with another unbiased estimate of the population variance, we can construct an F test to determine whether the row variable has an effect. Definitional and calculational formulas for these quantities are given in Table 1.19. 

We have met our first hypothesis test .The two-sample f-test is used to determine whether two samples have produced convincingly different mean values or whether the difference is small enough to be explained away as random sampling error. The data in each sample are assumed to be from populations that followed normal distributions and had equal SDs. 

There are many cases in which a scientist or an engineer needs to compare the mean of a data set with a known value. In some cases, the known value is the true or accepted value based on prior knowledge or experience. In other situations, the known value might be a value predicted from theory or it might be a threshold value that we use in making decisions about the presence or absence of a constituent. In all these cases, we use a statistical hypothesis test to draw conclusions about the population mean y. and its nearness to the known value, which we call p.Q. 

To obtain a variance that is an unbiased estimate of the population varianee so that valid confidence limits can be found for the mean, and various hypothesis tests can be applied. This goal can be reached only if every possible sample is equally likely to be drawn. 

Suppose our interest is in testing whether the population mean was equal to a particular hypothesized value, pg. A hypothesis testing process typically starts with a statement of the null and alternate hypotheses. The null hypothesis can be stated in the following manner ... 

If the null hypothesis is true - that is, the population mean is the hypothesized value, Pg - the value of the test statistic will be close to 0. The further the test statistic value is from 0 (either negative or positive) the less plausible is the hypothesized value, pg - that is, the null hypothesis should be rejected in favor of the alternate. 

If the value of the test statistic is in the critical region the null hypothesis is rejected and the conclusion is made that the population mean is not equal to p. When the null hypothesis is rejected, such a result is considered "statistically significant" at the a level, meaning that the result was unlikely (with probability no greater than a) to have been observed by chance alone. If the value of the test statistic is not in the critical region we fail to reject the null hypothesis. It is important to emphasize the fact that we cannot claim that the population mean is equal to Pq, but simply that the data were not sufficient to conclude that they were different. 

Under the null hypothesis of no difference in population means, and assuming somewhat symmetric distributions, the test statistic follows a t distribution with 298 (that is, 146 + 154 — 2) df. Therefore the critical region (values of the test statistic that lead to rejection) is defined as t < -1.968 and t > 1.968. Note that this particular entry is not in Appendix 2, but the closest is for 300 df. 

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. 

If the null hypothesis is rejected in this case, the conclusion would be that the population mean for the control treatment did not exceed that for the test group by more than The... 

The solution to this problem is to test the hypothesis that the two methods have equal population means. This is therefore a... 

Statistics such as the median and the trimmed mean are variously described as robust (i.e. suitable for use with a wide variety of population types) and/or resistant to outliers. Traditionally, robust and resistant statistics have been unpopular in classical statistics because it is often impossible to derive an analytical expression for the precision with which they can be estimated (i.e. formulae analogous to (4.7) above). This made it difficult to use the estimates in hypothesis tests. However, the advent of fast computers has radically altered the situation, since estimates of the precision of almost any ad hoc statistic can now be obtained by simulation tech-... 

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