Statistics null hypothesis

Coverage factor Expanded uncertainty Method of least squares Multivariate statistics Null hypothesis Power of a test Probability levels Regression of j on jc... 

In attempting to reach decisions, it is useful to make assumptions or guesses about the populations involved. Such assumptions, which may or may not be true, are called statistical hypotheses and in general are statements about the probability distributions of the populations. A common procedure is to set up a null hypothesis, denoted by which states that there is no significant difference between two sets of data or that a variable exerts no significant effect. Any hypothesis which differs from a null hypothesis is called an alternative hypothesis, denoted by Tfj. 

Next, an equation for a test statistic is written, and the test statistic s critical value is found from an appropriate table. This critical value defines the breakpoint between values of the test statistic for which the null hypothesis will be retained or rejected. The test statistic is calculated from the data, compared with the critical value, and the null hypothesis is either rejected or retained. Finally, the result of the significance test is used to answer the original question. 

The test statistic for evaluating the null hypothesis is called an f-test, and is given as either... 

The variance for the sample of ten tablets is 4.3. A two-tailed significance test is used since the measurement process is considered out of statistical control if the sample s variance is either too good or too poor. The null hypothesis and alternative hypotheses are... 

If the null hypothesis is assumed to be true, say, in the case of a two-sided test, form 1, then the distribution of the test statistic t is known. Given a random sample, one can predict how far its sample value of t might be expected to deviate from zero (the midvalue of t) by chance alone. If the sample value oft does, in fact, deviate too far from zero, then this is defined to be sufficient evidence to refute the assumption of the null hypothesis. It is consequently rejected, and the converse or alternative hypothesis is accepted. 

The procedure for testing the significance of a sample proportion follows that for a sample mean. In this case, however, owing to the nature of the problem the appropriate test statistic is Z. This follows from the fact that the null hypothesis requires the specification of the goal or reference quantity po, and since the distribution is a binomial proportion, the associated variance is [pdl — po)]n under the null hypothesis. The primary requirement is that the sample size n satisfy normal approximation criteria for a binomial proportion, roughly np > 5 and n(l — p) > 5. 

I. Under the null hypothesis, it is assumed that the respective two samples have come from populations with equal proportions pi = po. Under this hypothesis, the sampling distribution of the corresponding Z statistic is known. On the basis of the observed data, if the resultant sample value of Z represents an unusual outcome, that is, if it falls within the critical region, this would cast doubt on the assumption of equal proportions. Therefore, it will have been demonstrated statistically that the population proportions are in fact not equal. The various hypotheses can be stated ... 

Suppose we have two methods of preparing some product and we wish to see which treatment is best. When there are only two treatments, then the sampling analysis discussed in the section Two-Population Test of Hypothesis for Means can be used to deduce if the means of the two treatments differ significantly. When there are more treatments, the analysis is more detailed. Suppose the experimental results are arranged as shown in the table several measurements for each treatment. The goal is to see if the treatments differ significantly from each other that is, whether their means are different when the samples have the same variance. The hypothesis is that the treatments are all the same, and the null hypothesis is that they are different. The statistical validity of the hypothesis is determined by an analysis of variance. 

The results of such multiple paired comparison tests are usually analyzed with Friedman s rank sum test [4] or with more sophisticated methods, e.g. the one using the Bradley-Terry model [5]. A good introduction to the theory and applications of paired comparison tests is David [6]. Since Friedman s rank sum test is based on less restrictive, ordering assumptions it is a robust alternative to two-way analysis of variance which rests upon the normality assumption. For each panellist (and presentation) the three products are scored, i.e. a product gets a score 1,2 or 3, when it is preferred twice, once or not at all, respectively. The rank scores are summed for each product i. One then tests the hypothesis that this result could be obtained under the null hypothesis that there is no difference between the three products and that the ranks were assigned randomly. Friedman s test statistic for this reads... 

This homogeneity value will become positive when the null hypothesis is not rejected by Fisher s E-test (F < Ei a)Vl)V2) and will be closer to 1 the more homogeneous the material is. If inhomogeneity is statistically proved by the test statistic F > Fi a>VuV2, the homogeneity value becomes negative. In the limiting case F = hom(A) becomes zero. 

Our previous two chapters based on references [1,2] describe how the use of the power concept for a hypothesis test allows us to determine a value for n at which we can state with both a- and / -% certainty that the given data either is or is not consistent with the stated null hypothesis H0. To recap those results briefly, as a lead-in for returning to our main topic [3], we showed that the concept of the power of a statistical hypothesis test allowed us to determine both the a and the j8 probabilities, and that these two known values allowed us to then determine, for every n, what was otherwise a floating quantity, D. 

The test to determine whether the bias is significant incorporates the Student s /-test. The method for calculating the t-test statistic is shown in equation 38-10 using MathCad symbolic notation. Equations 38-8 and 38-9 are used to calculate the standard deviation of the differences between the sums of X and Y for both analytical methods A and B, whereas equation 38-10 is used to calculate the standard deviation of the mean. The /-table statistic for comparison of the test statistic is given in equations 38-11 and 38-12. The F-statistic and f-statistic tables can be found in standard statistical texts such as references [1-3]. The null hypothesis (H0) states that there is no systematic difference between the two methods, whereas the alternate hypothesis (Hf) states that there is a significant systematic difference between the methods. It can be seen from these results that the bias is significant between these two methods and that METHOD B has results biased by 0.084 above the results obtained by METHOD A. The estimated bias is given by the Mean Difference calculation. 

The null hypothesis test for this problem is stated as follows are two correlation coefficients rx and r2 statistically the same (i.e., rx = r2)l The alternative hypothesis is then rj r2. If the absolute value of the test statistic Z(n) is greater than the absolute value of the z-statistic, then the null hypothesis is rejected and the alternative hypothesis accepted - there is a significant difference between rx and r2. If the absolute value of Z(n) is less than the z-statistic, then the null hypothesis is accepted and the alternative hypothesis is rejected, thus there is not a significant difference between rx and r2. Let us look at a standard example again (equation 60-22). 

And Z(n) = 0.89833, therefore Z(n), the test statistic, is less than 1.96, the z-statistic, and the null hypothesis is accepted - there is not a significant difference between the correlation coefficients. 

NOTE If Z(N) is greater than the absolute value of the z-statistic (Normal Curve one-tailed) we reject the null hypothesis and state that there is no significant difference in rl and r2 at the selected significance level. 

The most common techniques for detecting the presence of gross errors are based on so-called statistical hypothesis testing. This is based on the idea of testing the data set against alternative hypotheses (1) the null hypothesis, Ho, that no gross error is present, and (2) the alternative hypothesis, Hi, that gross errors are present. 

This statistic has chi-square distribution with / degrees of freedom under the null hypothesis, where / is the number of elements of 7. If T > xl /, Ho is rejected, otherwise Ho is accepted, a is the significance level of the test. 

Selection of the test statistic r, with a known distribution under the assumption that the null hypothesis holds. 

The statistic X2/T will be large when there is evidence of a dose-related increase or decrease in the tumor incidence rates, and small when there is little difference in the tumor incidence between groups or when group differences are not dose related. Under the null hypothesis of no differences between groups, X2lT has approximately a chi-squared distribution with one degree of freedom. 

For rather obscure reasons, x2 is known as the log-rank statistic. An approximate significance test of the null hypothesis of identical distributions of survival time in the two groups is obtained by x2 to a chi-square distribution on 1 degree of freedom. 

These can be used to calculate an F statistic to test the null hypothesis that all treatment effects are equal. 

Statistical hypothesis testing requires the formulation of a so-called null hypothesis H0 that should be tested, and an alternative hypothesis H which expresses the alternative. In most cases there are several alternatives, but the alternative to test has to be fixed. For example, if two distributions have to be tested for equality of the means, the alternative could be unequal means, or that one mean is smaller/larger than the other one. For simplicity we will only state the null hypothesis in this overview below but not the alternative hypothesis. For the example of testing for equality of the means of two random samples xl and x2 the R command for the two-sample f-test is... 

The above equations can directly be taken to construct statistical tests. For example, the null hypothesis that the intercept bo = 0 against the alternative b() / 0 is tested with the test statistic... 

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