Test of Hypothesis - Concise Formula Summary Ebenezer R.H.P Isaac Department of Computer Science and Engineering Anna University, College of Engineering, Guindy Email: ebeisaac@cs.annauniv.edu Abstract—This paper contains contracted material on the statistical tests of hypotheses focusing on salient formulas and notations. This includes tests concerning means, variances, pro- portions, and χ 2 tests of independence and goodness of fit. This paper is intended for reference and not a substitute for the main subject prescribed books. I. I NTRODUCTION In general terms, a statistical hypothesis a condition, state- ment or an assumption of one or more parameters of a population or a probability distribution [1]. In hypothesis testing, an initial statement about the system in question is taken a null hypothesis and is tested against an opposite statement framed as an alternative hypothesis based on a test statistic. The following are the general steps to be followed in a test of hypothesis. 1) Hypothesis: State the null hypothesis H 0 , and alternative hypothesis H 1 . The H 1 always contradicts H 0 . 2) Error Probability: It is usually the type I error known as the level of significance α. It may also seldom include the type II error β. 3) Criterion: State the test statistic (TS) and formulate the rejection region (RR) 1 that should be satisfied in order to reject H 0 . 4) Calculations: Calculate the value of the test statistic based on the sample data in the problem. 5) Decision: Make the decision whether to reject H o thus accepting H 1 , or not to reject H 0 Note that not rejecting H 0 does not affirm its acceptance. It may be only assumed accepted as per the given data sample. True acceptance can only be achieved if it holds for all samples of the population. In general, z and t-distributions are used to test hypothesis concerning mean and χ 2 (chi-square 2 ) and F distributions are used to test hypothesis concerning variances or standard deviations. A detailed explanation on all sections of this paper can be found in [2]. II. TERMS AND NOTATIONS There are a few terminology and notations to be clear before moving on to the statistical formulas. • Unit: a single item or entity • Population: a complete collection of units of a system • Sample: a subset of a population TABLE I COMMON NOTATIONS FOR TEST OF HYPOTHESIS Notation Description H 0 Null hypothesis H 1 Alternate hypothesis μ Population mean ¯ x Sample mean σ 2 Population variance s 2 Sample variance σ/ √ n Standard error s/ √ n Estimated standard error α Type I error (level of significance) β Type II error ν Degree of freedom (nu) The common notations used in the forthcoming sections are listed in Table I. The degree of freedom 3 can also be denoted as df. in some references and statistical tables. When discussing hypothesis of two or more means or variances, the concept of treatment should be understood. • Treatment: procedures, entities, or processes that are being compared • Experimental Unit: Basic unit exposed to one treatment or another • Response: the observed characteristic to be compared If N is the total number of items in the population, then the population variance is given by σ 2 = 1 N N  i=1 (x i − μ) 2 If n is the number of items in a sample set, then the sample variance of that set is given by s 2 = 1 n − 1 n  i=1 (x i − ¯ x) 2 The factor (n − 1) supposedly approximates the sample variance to its population variance. 1 also known as critical region 2 Greek letter ‘χ’ (chi), pronounced as kai 3 denoted as the Greek letter ‘ν ’, pronounced as ‘new’; shouldn’t be confused with the English letter ‘v’ 1