Alternative hypothesis test

A one-tailed test is required since the alternative hypothesis states that the population parameter is equal to or less than the hypothesized value. 

If the significance test is conducted at the 95% confidence level (a = 0.05), then the null hypothesis will be retained if a 95% confidence interval around X contains p,. If the alternative hypothesis is... 

Relationship between confidence intervals and results of a significance test, (a) The shaded area under the normal distribution curves shows the apparent confidence intervals for the sample based on fexp. The solid bars in (b) and (c) show the actual confidence intervals that can be explained by indeterminate error using the critical value of (a,v). In part (b) the null hypothesis is rejected and the alternative hypothesis is accepted. In part (c) the null hypothesis is retained. 

The value of fexp is then compared with a critical value, f(a, v), which is determined by the chosen significance level, a, the degrees of freedom for the sample, V, and whether the significance test is one-tailed or two-tailed. For paired data, the degrees of freedom is - 1. If fexp is greater than f(a, v), then the null hypothesis is rejected and the alternative hypothesis is accepted. If fexp is less than or equal to f(a, v), then the null hypothesis is retained, and a significant difference has not been demonstrated at the stated significance level. This is known as the paired f-test. 

On occasion, a data set appears to be skewed by the presence of one or more data points that are not consistent with the remaining data points. Such values are called outliers. The most commonly used significance test for identifying outliers is Dixon s Q-test. The null hypothesis is that the apparent outlier is taken from the same population as the remaining data. The alternative hypothesis is that the outlier comes from a different population, and, therefore, should be excluded from consideration. 

Individual comparisons using Fisher s least significant difference test are based on the following null hypothesis and one-tailed alternative hypothesis... 

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. 

Interpretation If the alternate hypothesis had been stated as //i Xmean is different from /r, a two-sided test is applied with 2.5% probability being provided for each possibility Xmean smaller than p" resp. Xmean larger than /i . Because 1.92 is smaller than 2.45, the test criterion is not exceeded, so Hi is rejected. On the other hand, if it was known beforehand that Xmean can only be smaller than p, the one-sided test is conducted under the alternate hypothesis H Xn,ean smaller than p in this case the result is elose, with 1.92 almost exceeding 1.94. 

Figure 1.34. Alternative hypothesis and the power of a t-test. Alpha (a) is the probability of rejecting an event that belongs to the population associated with it is normally in the range 0.05. .. 0.01. Beta (/3) is the probability that an event that is effectively to be associated with H is accepted as belonging to the population associated with Hq. Note that the power of the test to discriminate between hypotheses increases with the distance between ha and hb- >-a is fixed either by theory or by previous measurements, while hb can be adjusted (shifted along the x-axis), for examples see H - H4, Section 4.1. Compare with program HYPOTHESIS.

Hi = alternative hypothesis a = significance level, usually set at. 10,. 05, or. 01 t = tabled t value corresponding to the significance level a. For a two-tailed test, each corresponding tail would have an area of a/2, and for a one-tailed test, one tail area would be equal to a. If a2 3 4 is known, then z would be used rather than the t. t = (x- Po)/(s/Vn) = sample value of the test statistic. 

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. 

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). 

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. 

Two hypotheses are considered. Based on a random sample, the validity of the null hypothesis (Ho) is tested against the alternate hypothesis (Hi) in order to either reject or accept the first one. 

The first precise or calculable aspect of experimental design encountered is determining sufficient test and control group sizes to allow one to have an adequate level of confidence in the results of a study (that is, in the ability of the study design with the statistical tests used to detect a true difference, or effect, when it is present). The statistical test contributes a level of power to such a detection. Remember that the power of a statistical test is the probability that a test results in rejection of a hypothesis, H0 say, when some other hypothesis, H, say, is valid. This is termed the power of the test with respect to the (alternative) hypothesis H. ... 

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... 

II the difference approach, which typically utilises 2-sided statistical tests (Hartmann et al., 1998), using either the null hypothesis (H0) or the alternative hypothesis (Hi). The evaluation of the method s bias (trueness) is determined by assessing the 95% confidence intervals (Cl) of the overall average bias compared to the 0% relative bias value (or 100% recovery). If the Cl brackets the 0% bias then the trueness that the method generates acceptable data is accepted, otherwise it is rejected. For precision measurements, if the Cl brackets the maximum RSDp at each concentration level of the validation standards then the method is acceptable. Typically, RSDn> is set at <3% (Bouabidi et al., 2010),... 

However, failure to disprove the null hypothesis does not mean we can reject the alternative hypothesis and accept the null hypothesis. This is a subtle but extremely important point in hypothesis testing, especially when hypothesis testing is used to identify factors in research and development projects (see Section 1.2 and Table 1.1). 

Write a null hypothesis that might be useful for testing the hypothesis that = 13.62. What is the alternative hypothesis ... 

Suppose, however, that the alternative hypothesis is H, bp > 0. Values of bp significantly less than zero would not satisfy the alternative hypothesis. If we did disprove the null hypothesis and bp were greater than zero, then we should be twice as confident about accepting the alternative hypotheses (or we should have only half the risk of being wrong). If is obtained from a regular two-tailed /-table specifying a risk a, then the level of confidence in the test is 100(1 - a/2)%. 

First the hypotheses must be chosen. There are two (1) the null hypothesis denoted by H sub zero which Is assumed true until rejected, and (2) the alternative hypothesis denoted by H sub one or sub A for alternative which Is assumed false until the null hypothesis Is rejected. The logic of the test requires that the hypotheses be "mutually exclusive" and "jointly exhaustive." "Mutually exclusive" means that one and only one of the hypotheses can be true "jointly exhaustive" means that one or the other of the hypotheses must be true. Both cannot be false. The null hypothesis Is to reflect the status quo, which means that failure to reject It Is only continuation of a present loss. For the agricultural station, failure to Improve the status quo means that the old brand of seed, pesticide, or fertilizer Is used when. In fact, a new and better brand Is available. This Is a status quo loss of productivity (e.g. 

Significance testing can be divided into a small number of steps. It starts with the formulation of the Null hypothesis. This is the assumption, which is made about the properties of a population of data expressed mathematically, e.g. there is no bias in our measurements . The second step is the formulation of the alternative hypothesis, the opposite of the Null hypothesis, in the above example there is a bias . 

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