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Chapter 7

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Aaron Childs

Stats 2B03: Statistical Methods for Science Chapter 7: Hypothesis Testing 7.1 Introduction - Purpose of hypothesis testing is to aid the clinician, researcher, or administrator in reaching a conclusion concerning a population by examining a sample from that population - Hypothesis may be defined simply as a statement about one or more populations - Research hypothesis is the conjecture or supposition that motivates the research leading to statistical hypotheses - Statistical hypotheses are hypotheses that are stated in such a way that they may be evaluated by appropriate statistical techniques - Hypothesis testing steps:  Data: the nature of the data that form the basis of the testing procedures must be understood, since this determines the particular test to be employed.  Assumptions: a general procedure is modified depending on the assumptions. The same assumptions that are of importance in estimation are important in hypothesis testing. Including assumptions about the normality of the population distribution, equality of variances, and independence of samples.  Hypotheses:  Null hypothesis: is the hypothesis to be tested, it is a statement of agreement with conditions presumed to be true in the population of interest  Alternative hypothesis: statement of what we will believe is true if our sample data causes us to reject the null hypothesis  Test statistic: some statistic that may be computed from the data of the sample  Distribution of test statistic: key to statistical inference is the sampling distribution  Decision rule: all possible values that the test statistic can assume are points on the horizontal axis of the graph of the distribution of the test statistic and are divided into two groups; one group constitutes what is known as the rejection region and the other group makes up the nonrejection region. The values of the test statistic forming the rejection region are those values that are less likely to occur if the null hypothesis is true, while the values making up the acceptance region are more likely to occur if the null hypothesis is true. The decision rule tells us to reject the null hypothesis if the value of the test statistic that we compute from our sample is one of the values in the rejection region and to not reject the null hypothesis if the computed value of the test statistic is one of the values in the nonrejection region.  Calculation of test statistic: from data contained in the sample we compute a value of the test statistic and compare it with the rejection and nonrejection regions that have been specified  Statistical decision: consists of rejecting or not rejecting the null hypotheses.  Conclusion: if H0is rejected, we conclude that H iA true. If H i0 not rejected, we conclude that H m0y be true.  P values: number that tells us how unusual our sample results are, given that the null hypothesis is true. - Rules for stating statistical hypotheses:  For deciding what statement does in the null hypothesis and what statement does in the alternative hypothesis:  What you hope or expect to be able to conclude as a result of the test usually should be placed in the alternative hypothesis  The null hypothesis should contain a statement of equality, either =, ≤, or ≥.  The null hypothesis is the hypothesis that is tested  The null and alternative hypotheses are complementary. That is, the two together exhaust all possibilities regarding the value that the hypothesized parameter can assume. - A precaution: neither hypothesis testing nor statistical inference leads to the proof of a hypothesis - General formula for test statistic: - Significance level: the level of significance α is a probability and, in fact, is the probability of rejecting a true null hypothesis - Types of errors:  Type I error: error committed when a true null hypothesis is rejected  Type II error: error committed when a false null hypothesis is not rejected, designated by β (generally larger than α) - Confusion matrix Condit
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