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TESTING HYPOTHESES

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Two ways of arriving at a conclusion 2. Inductive inference samplepopulation samplepopulation 1. Deductive inference

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IF YOUR DATA ARE: 1. Continuous data 2. Ratio or interval 3. Approximately normal distribution 4. Equal variance (F-test) 5. Conclusions about population based on sample (inductive) 6. Sample size > 10 samplepopulation

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Imagine the following experiment: 2 groups of crickets Group 1 fed a diet with extra supplements Group 2 fed a diet with no supplements Weights 12.113.913.012.1 14.912.212.914.9 13.612.013.513.6 12.015.912.412.0 10.912.111.010.9 9.18.911.010.1 9.99.28.011.9 8.69.08.59.6 10.010.99.48.0 11.97.110.08.9 Mean = 12.8 Mean = 9.49

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What youre doing here is comparing two samples that, because youve not violated any of the assumptions we saw before, should represent populations that look like this: 9.4912.8 Are the means of these populations different?? Frequency Weight

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Are the means of these populations different?? To answer this question use a statistical test A statistical test is just a method of determining mathematically whether you definitively say yes or no to this question What test should I use??

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IF YOU HAVENT VIOLATED ANY OF THE ASSUMPTIONS WE MENTIONED BEFORE Number of groups compared 2 other than 2 T -test Direction of difference specified? YesNo One-tailedTwo- tailed Does each data point in one data set (population) have a corresponding one in the other data set? YesNo Paired t-testUnpaired t-test Are the means of two populations the same? Are the means of more than two populations the same? Number of factors being tested 12>2 Does each data point in one data set (population) have a corresponding one in the other data sets? Two way ANOVA ANOVA YesNo One way ANOVA Repeated Measures ANOVA Other tests

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A simple t-test 1. State hypotheses H o there is no difference between the means of the two populations of crickets (i.e. the extra nutrients had no effect on weight) H 1 there is a difference between the means of the two populations of crickets (i.e. the extra nutrients had an effect on weight)

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A simple t-test 2. Calculate a t-value (any stats program does this for you) 3. Use a probability table for the test you used to determine the probability that corresponds to the t- value that was calculated. (for the truly masochistic)

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A simple t-test 2. Calculate a t-value (any stats program does this for you) 3. Use a probability table for the test you used to determine the probability that corresponds to the t- value that was calculated. DataTest statisticProbability

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Unpaired t test Do the means of Nutrient fed and No nutrient differ significantly? P value The two-tailed P value is < 0.0001, considered extremely significant. t = 7.941 with 38 degrees of freedom. 95% confidence interval Mean difference = -3.307 (Mean of No nutrient minus mean of Nutrient fed) The 95% confidence interval of the difference: -4.150 to -2.464 Assumption test: Are the standard deviations equal? The t test assumes that the columns come from populations with equal SDs. The following calculations test that assumption. F = 1.192 The P value is 0.7062. This test suggests that the difference between the two SDs is not significant. Assumption test: Are the data sampled from Gaussian distributions? The t test assumes that the data are sampled from populations that follow Gaussian distributions. This assumption is tested using the method Kolmogorov and Smirnov: Group KS P Value Passed normality test? =============== ====== ======== ======================= Nutrient fed 0.1676 >0.10 Yes No nutrient 0.1279 >0.10 Yes

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Interpretation of p

X 2 = 12.52Critical value for 3 degrees of freedomat.05 level is7.82 X 2 Table Conclusion: Probability of these data fitting the expected distribution is p >.001

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A little X 2 wrinkle - the Yates correction Formula is (o -e) 2 e 2 = Except of df = 1 (i.e. youre using two categories of data) Then the formula becomes (|o -e| - 0.5) 2 e 2 =

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A second goodness-of-fit test G-test or Log-Likelihood Ratio Use if |o - e | < e e.g. if o is 12 and e is 7 G = 2 o ln= 4.60517 * o log 10 oeoe oeoe

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Type of dataNumber of samples Are data related? Test to use Nominal2YesMcNemar Nominal2NoFishers Exact Nominal>2YesCochrans Q Summary!

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Type of dataNumber of samplesAre data related?Test to use Nominal2YesMcNemar Nominal2NoFishers Exact Nominal>2YesCochrans Q Ordinal1NoKomolgorov- Smirnov Ordinal+2YesWilcoxon (paired t-test analogue) Ordinal+2NoMann Whitney U (unpaired t-test analogue) Ordinal+>2NoKruskal Wallis (analogue of one- way ANOVA Ordinal>2YesFriedman two-way ANOVA All of the parametric tests (remember the big flow chart!) have non-parametric equivalents (or analogues)