Chapter 8 Tests of Hypotheses Based on a Single Sample 8 1 Hypotheses and Procedures Test Hypotheses The null hypothesis denoted H0 is the claim that is initially assumed to be true The alternative hypothesis denoted by Ha is the assertion that is contrary to H0 Possible conclusions from hypothesistesting analysis are reject H0 or fail to reject H0 Hypotheses H0 may usually be considered the skeptic s hypothesis Nothing new or interesting happening here And anything interesting observed is due to chance alone Ha may usually be considered the researcher s hypothesis Rules for Hypotheses H0 is always stated as an equality claim involving parameters Ha is an inequality claim that contradicts H0 It may be one sided using either or or two sided using A Test of Hypotheses A test of hypotheses is a method for using sample data to decide whether the null hypothesis should be rejected Test Procedure A test procedure is specified by 1 A test statistic a function of the sample data on which the decision is to be based 2 Sometimes not always A rejection region the set of all test statistic values for which H0 will be rejected Errors in Hypothesis Testing A type I error consists of rejecting the null hypothesis H0 when it was true A type II error consists of not rejecting H0 when H0 is false a and b are the probabilities of type I and type II error respectively Level a Test Sometimes the experimenter will fix the value of a also known as the significance level A test corresponding to the significance level is called a level a test A test with significance level a is one for which the type I error probability is controlled at the specified level Rejection Region a and b Suppose an experiment and a sample size are fixed and a test statistic is chosen Decreasing the size of the rejection region to obtain a smaller value of a results in a larger value of b for any particular parameter value consistent with Ha 8 2 Tests About a Population Mean Case I A Normal Population With Known s Null hypothesis Test statistic value H 0 m m0 x m0 z s n Case I A Normal Population With Known s Alternative Hypothesis H a m m0 H a m m0 H a m m0 Rejection Region for Level a Test z za z za z za 2 or z za 2 Recommended Steps in Hypothesis Testing Analysis 1 Identify the parameter of interest and describe it in the context of the problem situation 2 Determine the null value and state the null hypothesis 3 State the alternative hypothesis Hypothesis Testing Analysis 4 Give the formula for the computed value of the test statistic 5 State the rejection region for the selected significance level 6 Compute any necessary sample quantities substitute into the formula for the test statistic value and compute that value Hypothesis Testing Analysis 7 Decide whether H0 should be rejected and state this conclusion in the problem context The formulation of hypotheses steps 2 and 3 should be done before examining the data for a Level a Type II Probability b m Test Type II Alt Hypothesis H a m m0 H a m m0 H a m m0 Probability b m m0 m F za s n m0 m 1 F za s n m0 m F za 2 F s n m0 m za 2 s n Sample Size The sample size n for which a level a test also has b m b at the alternative value m is s za z b one tailed test m0 m n 2 s za 2 z b two tailed test m0 m 2 Case II Large Sample Tests When the sample size is large the z tests for case I are modified to yield valid test procedures without requiring either a normal population distribution or a known s Large Sample Tests n 40 For large n s is close to s Test Statistic Z X m0 S n The use of rejection regions for case I results in a test procedure for which the significance level is approximately a Case III A Normal Population Distribution If X1 Xn is a random sample from a normal distribution the standardized variable X m T S n has a t distribution with n 1 degrees of freedom The One Sample t Test Null hypothesis Test statistic value H 0 m m0 t x m0 s n The One Sample t Test Alternative Hypothesis Rejection Region for Level a Test H a m m0 t ta n 1 H a m m0 t ta n 1 H a m m0 t ta 2 n 1 or t ta 2 n 1 A Typical b Curve for the t Test b curve for n 1 df b when m m 0 Value of d corresponding to specified alternative to m 8 3 Tests Concerning a Population Proportion A Population Proportion Let p denote the proportion of individuals or objects in a population who possess a specified property Large Sample Tests Large sample tests concerning p are a special case of the more general large sample procedures for a parameter q Large Samples Concerning p Null hypothesis H 0 p p0 Test statistic value z p p0 p0 1 p0 n Large Samples Concerning p Alternative Hypothesis Rejection Region H a p p0 z za H a p p0 z za H a p p0 Valid provided z za 2 or z za 2 np0 10 and n 1 p0 10 General Expressions for b p Alt Hypothesis H a p p0 H a p p0 b p p0 p za p0 1 p0 n F p 1 p n p0 p za p0 1 p0 n 1 F p 1 p n General Expressions for b p Alt Hypothesis H a p p0 b p p0 p za p0 1 p0 n F p 1 p n p0 p za p0 1 p0 n F p 1 p n Sample Size The sample size n for which a level a p test also has b p 2 z p 1 p z p 1 p a 0 0 b p p 0 n 2 za 2 p0 1 p0 z b p 1 p p p 0 one tailed test two tailed test Small Sample Tests Test procedures when the sample size n is small are based directly on the binomial distribution rather than the normal approximation P type I 1 B c 1 n p0 B p B c 1 n p 8 4 P Values P Value The P value is the smallest level of significance at which H0 would be rejected when a specified test procedure is used on a given data set 1 P value a reject H 0 at a level of a 2 P value a do not reject H 0 at a level of a P Value The P value is the probability calculated assuming H0 is true of obtaining a test statistic value at least as contradictory to H0 as the value that actually …