Inferential Stats in Educational Research: Samples to Populations & Hypothesis Testing - P, Study notes of History of Education

Download Inferential Stats in Educational Research: Samples to Populations & Hypothesis Testing - P and more Study notes History of Education in PDF only on Docsity! 1 Inferential Statistics Katie Rommel-Esham Education 504 Probability • Probability is the scientific way of stating the degree of confidence we have in predicting something • Tossing coins and rolling dice are examples of probability experiments • The concepts and procedures of inferential statistics provide us with the language we need to address the probabilistic nature of the research we conduct in the field of education 2 From Samples to Populations • Probability comes into play in educational research when we try to estimate a population mean from a sample mean • Samples are used to generate the data, and inferential statistics are used to generalize that information to the population, a process in which error is inherent • Different samples are likely to generate different means. How do we determine which is “correct?” The Role of the Normal Distribution • If you were to take samples repeatedly from the same population, it is likely that, when all the means are put together, their distribution will resemble the normal curve • The resulting normal distribution will have its own mean and standard deviation • This distribution is called the sampling distribution and the corresponding standard deviation is known as the standard error 5 How does all this fit together? • Researchers use inferential statistics to determine the probability that the null hypothesis is untrue • Recall that if the null hypothesis is untrue, that is if it is “not true that there is no difference,” the most plausible conclusion is that there is indeed a difference • We never prove that anything is true, we only fail to disprove Levels of Significance • Used to indicate the chance that we are wrong in rejecting the null hypothesis • Also called the level of probability or p level • p=.01, for example, means that the probability of finding the stated difference as a result of chance is only 1 in 100 6 Errors in Hypothesis Testing • A type I error is made when a researcher rejects the null hypothesis when it is true • The probability of making this type of error is equal to the level of significance • A type II error is made when a researcher accepts the null hypothesis when it is false • As the level of significance increases, the likelihood of making a Type II error decreases In summary… Correct decision: there are differencesType II errorNull is false Type I error Correct decision: there is no differenceNull is true Reject NullAccept Null 7 Interpreting Level of Significance • Researchers generally look for levels of significance equal to or less than .05 • If the desired level of significance is achieved, the null hypothesis is rejected and the result is that there is a “statistically significant” difference in the means Some notes on p-values • Acceptable levels of significance are situation specific • p = .05 is fine for most educational research • p = .05 is not an acceptable level if we are considering the error in a test concerning usage of a drug that might cause death 10 Variations on the t • An independent samples t-test is used when the groups have no relationship to one another, as would an experimental group and control group • You may also encounter literature that references a dependent sample, paired, correlated, or matched t- test • These are used if the subjects in the two groups are matched in some way, as they would be matched with themselves in a pretest-posttest situation Analysis of Variance (ANOVA) • Similar to a t-test, but used when there are more than two groups being compared • ANOVA is an extension of the t-test • Addresses the question "Is there a significant difference between any two population means?” 11 How ANOVA Works • Analysis of variance allows a researcher to examine differences in all population means simultaneously rather than conducting a series of t-tests • It uses variances (rather than means) of groups to calculate a value that reflects the degree of differences in the means Interpreting ANOVA • Produces an F statistic (or F ratio) which is analogous to the t-statistic • A "1x4 ANOVA" is a one-way (i.e. one independent variable) ANOVA that is comparing four group means 12 1x4 ANOVA Math achievement testClassroom #4 Math achievement testClassroom #3 Math achievement testClassroom #2 Math achievement testClassroom #1 Factorial Analysis of Variance • A factorial analysis of variance is used when there are two or more independent variables being analyzed simultaneously • A "2x3 ANOVA" indicates that there are three groups being compared on 2 variables 15 Post hoc tests • Statistical tests that tell the researcher which means are different • Common post hoc comparisons include Fisher’s LSD (least-significant difference), Duncan’s new multiple range test, the Newman-Keuls, Tukey’s test, and the Scheffe’s test • The choice of test depends on characteristics of the data sets ANCOVA • Analysis of covariance (ANCOVA) is used when the researcher needs to adjust initial group differences statistically on one or more variables that are related to the independent variable but uncontrolled, and to increase the likelihood of finding a significant difference between two group means 16 For example, • Two groups are pretested, group A’s mean is higher. The same two groups are posttested, group A’s mean is still higher. • Is the higher posttest mean due to the fact that group A’s pretest mean was higher (i.e. are they “smarter?”) • ANCOVA adjusts for these initial pretest score differences Multivariate Analyses • Used to investigate problems in which the researcher is interested in studying more than one dependent variable 17 An Example • “Attitudes towards science” is a complex construct that might involve things like enjoying science, valuing science, attitudes towards different branches of science (Earth Science, Biology, Chemistry), lab work, science field trips, etc. • Multivariate methods allow researchers to look at each of these components separately Multivariate Tests MANCOVAANCOVA MANOVAANOVA Hotelling’s T2t-test Multivariate TestUnivariate Test 20 Chi-Square • Nonparametric procedure used when data are in nominal form • It is a way of answering questions about relationship based on frequencies of observations in categories An Example • What is the relationship between year in college (freshman, sophomore, junior, senior) and use of campus counseling services? • Responses to this question will involve a count of how many in each group use the counseling service • The independent variable is year in college which has four categories 21 Other Procedures You May Encounter • Repeated Measures Analysis of Variance (measure repeated across subjects) • Factor Analysis (how well items are related to one another and form clusters or factors) • Path Analysis (examines correlations with a hint of directionality) • Multiple Regression (using multiple factors to increase predictive power) • Rasch Analysis (allows for direct comparison between the difficulty of an item and the probability of a student at any ability level getting it correct)