EXAMPLESX = blood serum cholesterol levelC) Run The Experiment And Collect The DataTopic (11) – CONCEPTS OF HYPOTHESIS TESTING 11 - 1 Topic (11) – CONCEPTS OF HYPOTHESIS TESTING Recall the definition of Scientific Method: 1. knowledge is obtained in a systematic and objective manner in order to extend our understanding. 2. Based on this knowledge we form a HYPOTHESIS – a tentative or postulated explanation of the phenomenon. Hence it is a statement about a population characteristic (eg, a mean µ or proportion π or the difference between population means 21µµ−) 3. To evaluate the hypothesis we DESIGN and execute an objectively planned experiment. 4. The resulting data are TESTED to determine if they support or do not support the hypothesis.Topic (11) – CONCEPTS OF HYPOTHESIS TESTING 11 - 2 A) Construct Hypotheses Almost all statistical testing procedures are based on testing two competing claims: the null and alternative hypotheses Defn: Ho is the NULL HYPOTHESIS. This is the status quo, i.e. it is the truth until disproven by testing. HA is the ALTERNATIVE HYPOTHESIS. This is the competing claim made by the researcher*. *There are exceptions, usually for testing equivalency or goodness of fit to a probability distribution. The alternative hypothesis, HA, lists the outcomes claimed to be true by the scientist. The null hypothesis, H0, then lists the remaining or unclaimed cases. The testing procedure results in either 1) rejecting the null hypothesis in favor of the alternative hypothesis because the data support the alternative, or 2) failing to reject the null hypothesis because there is insufficient evidence to show it is wrongTopic (11) – CONCEPTS OF HYPOTHESIS TESTING 11 - 3 Step 1) State Your Claim In Words EXAMPLES: A. In a study of the effect of the new drug for reducing serum cholesterol levels in men at risk of heart disease, the company’s claim is that the drug reduces serum cholesterol levels in the target population. So we might write: Ho: the drug does not reduce serum cholesterol levels HA: the drug reduces serum cholesterol levels B. An entomologist believes that there is sexual dimorphism in body size of the periodical cicada. One measure of size of the length of the hind tibia. So her hypothesis might be stated: Ho: there is no sexual dimorphism in the periodical cicada HA: there is sexual dimorphism in the periodical cicada Note that these are informal statements that need to be clarified and made more rigorousTopic (11) – CONCEPTS OF HYPOTHESIS TESTING 11 - 4 Step 2) State The Hypotheses In Terms Of The Relevant Population Characteristics (parameters) EXAMPLES A. In the study of the effect of the new drug for reducing serum cholesterol levels we had Ho: the drug does not reduce serum cholesterol levels HA: the drug reduces serum cholesterol levels What variable(s) are being measured? X = blood serum cholesterol level What population characteristic is being modified by the drug? If the drug leads to a reduction in blood levels, the population mean should go down. (We assume variability of values doesn’t change.) What is the value of the characteristic without the drug? Without the drug, the target population has a mean blood serum level of 250. So we can write the hypotheses more specifically as Ho: µ = 250 HA: µ < 250Topic (11) – CONCEPTS OF HYPOTHESIS TESTING 11 - 5 B. sexual dimorphism in the morphology of the periodical cicada Since they are studying the hind tibia length we should first clarify the hypotheses in words: HA: there is a difference in hind tibia lengths between males and females Ho: there is no difference in hind tibia lengths between the sexes What population characteristic(s) is(are) different between the 2 sexes? The means for hind tibia length in the 2 genders. Hence we can write the hypotheses as: Ho: µfemales = µmales HA: µfemales ≠ µmalesTopic (11) – CONCEPTS OF HYPOTHESIS TESTING 11 - 6 C. The scientist studying the proportion of fiddler crabs with dominant left pincers hypothesized that isolation on the island led to a proportion of left-pincered crabs larger than the typical 10%. Her hypotheses in words are: Ho: 10% of the population of fiddler crabs on the island are left-pincered HA: more than 10% of the population of fiddler crabs on the island are left-pincered and in symbols are Ho: π = 0.10 HA: π > 0.10 Important Point: Hypotheses must be carefully structured since, as was implied earlier, statistical tests can only disprove the null hypothesis; they CANNOT prove that Ho is true.Topic (11) – CONCEPTS OF HYPOTHESIS TESTING 11 - 7 Forms Of Hypotheses For Single Populations The null hypothesis is always stated as H0: population parameter = hypothesized value where the hypothesized value is given by the problem (usually it’s the value being challenged). The alternative has one of the following forms: 1) 2-sided test alternative HA: parameter ≠ hypothesized value 2) 1-sided upper tail alternative HA: parameter > hypothesized value 3) 1-sided lower tail alternative HA: parameter < hypothesized value Note that the alternative never includes the hypothesized value. Having defined your hypotheses, the next step is:Topic (11) – CONCEPTS OF HYPOTHESIS TESTING 11 - 8 B) Design The Experiment: a) identify the appropriate statistical test to be used b) identify type of data to be collected (variables) c) construct the sampling or experimental design so that it actually provides the data needed for the test Comment: These three cannot be separated as distinct activities Point: most of the tests we’ll learn require random sampling, independent sampling among different populations, and sample sizes sufficiently large so we can argue that our statistics are approximately normally distributed. Point: if the data are categorical then the test is different from that for continuous data Example of Experimental Design: in a study of the effect of temperature on seedling growth, the scientist put 100 plants at one temperature in a greenhouse with its windows painted over and another
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