Chapter 10 Copyright © Allyn & Bacon 2008 This multimedia product and its contents are protected under copyright law. The following are prohibited by law:

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<ul><li><p>Chapter 10Copyright Allyn &amp; Bacon 2008This multimedia product and its contents are protected under copyright law. The following are prohibited by law: Any public performance or display, including transmission of any image over a network; Preparation of any derivative work, including the extraction, in whole or in part, of any images; Any rental, lease, or lending of the program.</p><p>Copyright Allyn &amp; Bacon 2008</p></li><li><p>Inferential statisticsPurposeErrorTerminologyHypothesis testingInferential testsCriteria for evaluating the inferential statistics reports in studies</p><p>Copyright Allyn &amp; Bacon 2008</p><p>Copyright Allyn &amp; Bacon 2008</p></li><li><p>The purpose of inferential statistics is to draw inferences about a population on the basis of an estimate from a sampleInferential statistics - specific statistical procedures that accomplish this purposeThe ultimate goal is to draw accurate conclusions about the populationCopyright Allyn &amp; Bacon 2008</p><p>Copyright Allyn &amp; Bacon 2008</p></li><li><p>Two types of errorsSampling errorsWithout measuring the entire population, the results can be inaccurate due to sampling errorThe larger the proportion of the population that is sampled, the lower the sampling error; the smaller the proportion of the population that is sampled, the higher the sampling errorA sample of 99% of a population is likely to show results that are very, very similar to those that would have been found if everyone in the population was measuredA sample of 1% is likely to show results that are different from those in the population - the question is how different are the sample resultsNeed to estimate the level of sampling error relative to the inferences being drawnCopyright Allyn &amp; Bacon 2008</p><p>Copyright Allyn &amp; Bacon 2008</p></li><li><p>Measurement errorsRegardless of the sample size, the results can be inaccurate due to measurement errorLack of validityLack of reliabilityNeed to estimate the level of measurement error relative to the inferences being drawnCopyright Allyn &amp; Bacon 2008</p><p>Copyright Allyn &amp; Bacon 2008</p></li><li><p>TerminologyNull hypothesisNo differences between groupsNo relationships between variablesLevel of significanceProbability of being wrong in rejecting the null hypothesisKnown as alpha (")Types of errorsType I - rejecting the null hypothesis when it is trueType II - not rejecting (i.e., accepting) the null hypothesis when it is not trueCopyright Allyn &amp; Bacon 2008</p><p>Copyright Allyn &amp; Bacon 2008</p></li><li><p>Hypothesis testing exemplified with an experimental control group comparisonThe five stages of the processState the null hypothesis - no difference between the mean scores for the experimental and control groupsAssume the null hypothesis is true to establish a base from which the statistician can workThe base is actually the sampling distribution of the test statistic, in this case the sampling distribution of the difference between two means, tThrough statistical theory we can establish the characteristics of this sampling distribution (i.e., mean; standard deviation, known as the standard error in this situation; and shape)Copyright Allyn &amp; Bacon 2008</p><p>Copyright Allyn &amp; Bacon 2008</p></li><li><p>The five stages of the process (continued)Calculate the observed difference between the mean scores for the two groupsCompare the observed difference between mean scores to the sampling distribution of the test statisticAccept or reject the null hypothesis based on this comparisonIf the observed difference is typical of the sampling distribution, the null hypothesis is likely true and it is acceptedIf the observed difference is atypical of the sampling distribution, the null hypothesis is likely untrue and it is rejected.Copyright Allyn &amp; Bacon 2008</p><p>Copyright Allyn &amp; Bacon 2008</p></li><li><p>Issues related to statistical and practical significanceStatistical significanceThe typical or atypical nature of the comparison of the observed difference to the sampling distribution can be estimated using statistical theoryThe estimate is the probability of being wrong in rejecting the null hypothesisIt is stated as p = x where x is the specific probability of the comparison (e.g., p = .001, p = .042, p = .56) or as p &lt; y where y is the alpha level (e.g., .10, .05, .01)Copyright Allyn &amp; Bacon 2008</p><p>Copyright Allyn &amp; Bacon 2008</p></li><li><p>There is always the possibility of making a mistake given that this is based on a probability modelType I error - deciding to reject the null hypothesis when in reality it is trueType II error - accepting the null hypothesis when it in reality it is falseTypical levels of significance in education - .10, .05, and .01Factors affecting the level of significanceThe actual differences between the groupsThe degree to which sampling and measurement errors existThe size of the sampleCopyright Allyn &amp; Bacon 2008</p><p>Copyright Allyn &amp; Bacon 2008</p></li><li><p>Practical significancePractical significance is related to the importance and usefulness of the resultsEstimates of practical significanceFor correlations the coefficient of determination (i.e., r2) is usedFor comparisons an effect size is usedEffect size is the difference between two group means in terms of the control group standard deviationEvaluating effect sizes small (.30), moderate (.50), and large (.75)Copyright Allyn &amp; Bacon 2008</p><p>Copyright Allyn &amp; Bacon 2008</p></li><li><p>Each consumer of the research must judge the balance between the statistical significance and the practical significance of the statistical results given the context in which the results might be usedCopyright Allyn &amp; Bacon 2008</p><p>Copyright Allyn &amp; Bacon 2008</p></li><li><p>Two types of inferential testsParametric - inferential procedures using interval or ratio level dataNon-parametric - inferential procedures using nominal or ordinal dataCopyright Allyn &amp; Bacon 2008</p><p>Copyright Allyn &amp; Bacon 2008</p></li><li><p>T-testA comparison of the means for two groupsDo the mean scores on the final exam differ for the experimental and control groups?Independent samples t-test - compares the means of two separate groups on one variablePosttest means for Group 1 and Group 2Dependent sample t-test - compares the means of two variables for one groupPre-test and posttest means for Group 1Copyright Allyn &amp; Bacon 2008</p><p>Copyright Allyn &amp; Bacon 2008</p></li><li><p>T-test (continued)A determination of whether a relationship existsDoes a correlation of +.63 between students math attitudes and math achievement indicate a relationship exists between these two variables?Correlation t-test - compares the magnitude of the difference between a correlation coefficient and 0.00Copyright Allyn &amp; Bacon 2008</p><p>Copyright Allyn &amp; Bacon 2008</p></li><li><p>Analysis of variance (ANOVA)A comparison of the means for two or more groupsOmnibus ANOVA - a procedure that indicates whether one of more pairs of means are differentDo the mean scores differ for the groups using co-operative group, lecture, or web-based instruction?Copyright Allyn &amp; Bacon 2008</p><p>Copyright Allyn &amp; Bacon 2008</p></li><li><p>ANOVA (continued)Multiple comparisons (i.e., post-hoc)Procedures that indicate which specific pairs of means are different as a follow-up to a significant omnibus ANOVA resultDo the mean scores differ between the co-operative group and lecture, co-operative group and web-based, and lecture and web-based instruction?Two common testsTukeyScheffeCopyright Allyn &amp; Bacon 2008</p><p>Copyright Allyn &amp; Bacon 2008</p></li><li><p>Factorial ANOVAA procedure that analyzes the difference between groups across two or more independent variablesDo the mean scores differ for co-operative group, lecture, and web-based instruction for males and females?EffectsMain effects - differences between the levels of each independent variableInteraction effects - differences between combinations of the levels of each independent variableCopyright Allyn &amp; Bacon 2008</p><p>Copyright Allyn &amp; Bacon 2008</p></li><li><p>Analysis of covariance (ANCOVA)A procedure that compares means after statistically adjusting them for pretest differences between groupsVery stringent assumptions that must be met to use this procedureAdjusts for small to moderate - not large - pretest differencesCopyright Allyn &amp; Bacon 2008</p><p>Copyright Allyn &amp; Bacon 2008</p></li><li><p>Multivariate statisticsComparisons or relationships involving two or more dependent variablesComparison of meansAre there differences in the attitudes and performances of students being taught with lecture or web-based instruction?Specific testsMultivariate ANOVA (MANVOA)Multivariate ANCOVA (MANCOVA)Hotellings TCopyright Allyn &amp; Bacon 2008</p><p>Copyright Allyn &amp; Bacon 2008</p></li><li><p>Multivariate statistics (continued)RelationshipsAre students affective traits (e.g., attitudes, self-esteem, preferences, etc.) predictive of their knowledge (i.e., test scores) and skills (i.e., performances)?Canonical correlationCopyright Allyn &amp; Bacon 2008</p><p>Copyright Allyn &amp; Bacon 2008</p></li><li><p>Chi-square - differences in frequencies across different categoriesDo mothers and fathers differ in their support of a year-round school calendar?Do the percentages of undergraduate, graduate, and doctoral students differ in terms of their support for the new class attendance policy?Copyright Allyn &amp; Bacon 2008</p><p>Copyright Allyn &amp; Bacon 2008</p></li><li><p>Comparison of meansMann Whitney U-testWilcoxon testKruskal-Wallis ANOVARelationshipsSpearman rCopyright Allyn &amp; Bacon 2008</p><p>Copyright Allyn &amp; Bacon 2008</p></li><li><p>Basic descriptive statistics are needed to evaluate the inferential resultsInferential analyses report statistical significance, not practical significanceInferential analyses do not indicate internal or external validityThe results depend on sample sizesCopyright Allyn &amp; Bacon 2008</p><p>Copyright Allyn &amp; Bacon 2008</p></li><li><p>The appropriate statistical procedures are usedThe level of significance is interpreted correctlyCaution is used to interpret non-parametric results from studies with few subjects in one or more groups or categoriesCopyright Allyn &amp; Bacon 2008</p><p>Copyright Allyn &amp; Bacon 2008</p><p>*</p></li></ul>

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