Ho, null hypothesis

Pi = Xi/ui, the sample proportion from population 1 P2 = X2/ri2, the sample proportion from population 2 a = significance level Ho = null hypothesis Hi = alternative hypothesis... 

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. 

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. 

For example, we wish set up the null hypothesis (Ho) and claim that there is no difference (5) between the control or placebo (/Tc) and the drug being trialed (/Td). This is set against the alternative hypothesis (Ha), which states that indeed there is a difference (5) between the control and drug under trial. Mathematically, the representation is given as follows ... 

The smaller the P-value the less likely it is that, under the null hypothesis, i.e. HO is true, we would have obtained the observed value of the test statistic. So, a small value of P is likely to mean that HO is untrue, in which case we should prefer HI. However, it is possible that HO is true but a rather unlikely event has taken place. Thus accepting the 5% level of significance (P<0.05) in rejecting HO means that 95 times out of 100 we are probably correct in our decision, but 5 times out of 100 we run the risk of rejecting HO when in fact it is true. Rejecting the null hypothesis when it is actually true is referred to as a Type 1 error. Accepting HO when it is not true is a Type 2 error. The probability of a Type 1 error is given the symbol a the probability of a Type 2 error the symbol 3 (Table 21.3). 

Table 21.3 Relationship between Type 1 (a) and Type 2 (P) errors and accepting the null hypothesis (HO) and rejecting the alternative hypothesis (HI)...

Weak control of the FWER, also referred to as the experimentwise error rate in some traditional statistics books, controls the maximum probability of rejecting at least one null hypothesis H0i when all Ho, = 1,..., g, are true. Weak control of the FWER is inadequate in practice because, if there exists at least one differentially expressed gene, then there is no guarantee that the probability of incorrectly inferring the nondifferentially expressed genes as differentially expressed is controlled. 

Table V lists the estimates of the intercept and the coefficients of Equation 3. The entries labelled T FOR HO PARAMETER = 0 are the t-values for testing the null hypothesis that any parameter equals zero. The value of each entry under PR > T answers the question, "If the parameter is really equal to zero, what is the probability of getting a larger value of t " A small value for this probability indicates that it is unlikely that the parameter is actually equal to zero. For example, the probability-significance value for the coefficient of X1X2X3 is 0.0002, consequently, the hypothesis that ai23 = 0 acceptable. The finding that all the parameters of Equation 3 are significant confirms the results obtained from the RSQUARE procedure.

When two samples are veiy similar, t approaches zero when they are different, t approaches infinity. The value of f is used to calculate the P value using Student s f-test tables, given in the appendix of this book. The P value is tte probability that the two distribution means are the same that is, Aj = Ag. When the P value is greater than a critical accepted value (typically 5% [21] or the experimental error due to both sampling and size determination if it is lai ger) then the null hypothesis (Ho Aj = A2) is accepted (i.e., the two populations are considered to be the same). Ceramic powder size distributions are often represented by log-normal distributions and not by normal distributions. For this reason the t statistic must be augmented for use with lognormal distributions. Equation (2.59) can be modified for this purpose to... 

Ho = assumption or null hypothesis regarding the population proportion Hi = alternative hypothesis a= significance level, usually set at. 10,. 05, or. 01 z = Tabled Z value corresponding to the significance level a. The sample sizes required for the z approximation according to tne magnitude of po are given in Table 3-5. 

To test the null hypothesis Ho = fxi = fX2 against the alternative hypothesis Hi = fxi fX2, assuming normal distributions and equality of variances (af = a ), the... 

If normality of the data cannot be accepted, Spearman s correlation coefficient and its corresponding nonparametric test can be used for the null hypothesis Ho =... 

Standard deviation Degrees of freedom Null hypothesis Ho... 

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