Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Psychometrics wikipedia , lookup
Taylor's law wikipedia , lookup
Bootstrapping (statistics) wikipedia , lookup
Sufficient statistic wikipedia , lookup
History of statistics wikipedia , lookup
Foundations of statistics wikipedia , lookup
Omnibus test wikipedia , lookup
Statistical inference wikipedia , lookup
Student's t-test wikipedia , lookup
Education Research 250:205 Writing Chapter 3 Objectives Subjects Instrumentation Procedures Experimental Design Statistical Analysis Displaying data Analyzing data Descriptive statistics Derived scores Inferential statistics Introduction Confidence intervals Comparison of means Correlation and regression Introduction Statistical inference: A statistical process using probability and information about a sample to draw conclusions about a population and how likely it is that the conclusion could have been obtained by chance Distribution of Sample Means Assume you took an infinite number of samples from a population What would you expect to happen? Assume a population consists of 4 scores (2, 4, 6, 8) Collect an infinite number of samples (n=2) Total possible outcomes: 16 p(2) = 1/16 = 6.25% p(3) = 2/16 = 12.5% p(4) = 3/16 = 18.75% p(5) = 4/16 = 25% p(6) = 3/16 = 18.75% p(7) = 2/16 = 12.5% p(8) = 1/16 = 6.25% Central Limit Theorem The CLT describes ANY sampling distribution in regards to: Shape 2. Central Tendency 3. Variability 1. Central Limit Theorem: Shape All sampling distributions tend to be normal Sampling distributions are normal when: The population is normal or, Sample size (n) is large (>30) Central Limit Theorem: Central Tendency The average value of all possible sample means is EXACTLY EQUAL to the true population mean µ = 2+4+6+8 / 4 µ=5 µM = 2+3+3+4+4+4+5+5+5+5+6+6+6+7+7+8 / 16 µM = 80 / 16 = 5 Central Limit Theorem: Variability The standard deviation of all sample means is = SEM/√n Also known as the STANDARD ERROR of the MEAN (SEM) Central Limit Theorem: Variability SEM Measures how well statistic estimates the parameter The amount of sampling error that is reasonable to expect by chance Central Limit Theorem: Variability SEM decreases when: decreases Sample size increases Population SEM = /√n Other properties: When As n=1, SEM = population SD SEM decreases the sampling distribution “tightens” So What? A sampling distribution is NORMAL and represents ALL POSSIBLE sampling outcomes Therefore PROBABILITY QUESTIONS can be answered about the sample relative to the population Introduction 1. 2. Two main categories of inferential statistics Parametric Nonparametric Introduction Parametric or nonparametric? What is the scale of measurement? or ordinal Nonparametric Interval or ratio Answer next question Nominal Is the distribution normal? Parametric No Nonparametric Yes Objectives Subjects Instrumentation Procedures Experimental Design Statistical Analysis Displaying data Analyzing data Descriptive statistics Derived scores Inferential statistics Introduction Confidence intervals Comparison of means Correlation and regression Confidence Intervals Application: Estimation of an unknown variable that is unable or undesirable to be measured directly Confidence intervals estimate with a certain amount of confidence Confidence Intervals 1. Components of a confidence interval: The level of confidence -Chosen by researcher -Typically 95% -What does it mean? 2. 3. The estimator (point estimate) The margin of error X% CI = Estimator +/- Margin of error Confidence Intervals: Example A researcher is interested in the amount of $ budgeted for special education by elementary schools in Iowa Select a random sample from the population and collect appropriate data Results: average $ spent was $56,789 (95% CI: $51,111 – 62,467) The average$ spent was $56,789 +/- 5,678 (95% CI) The Objectives Subjects Instrumentation Procedures Experimental Design Statistical Analysis Displaying data Analyzing data Descriptive statistics Derived scores Inferential statistics Introduction Confidence intervals Comparison of means Correlation and regression Comparing Means Hypothesis Tests Compare two means Compare Compare Compare Compare three or more means Compare Compare means between groups means within groups Compare means as a function of two or more factors (independent variables) Factorial a mean two a known value means between groups means within groups designs Compare means of multiple dependent variables Multivariate designs Hypothesis Tests – A Step by Step Process Step 1: State the null hypothesis Step 2: Select level of significance Step 3: Sample data Step 4: Choose statistic Step 5: Calculate the statistic Step 6: Interpret the statistic Step 1: Null Hypothesis Recall Null hypothesis is a statement of no effect The test statistic either accepts or rejects the H0 Create H0 for following tests: Are females in Iowa taller than 6 feet? Do 6th grade boys score differently than 6th grade females on math tests? Does an 8-week reading program affect reading comprehension in 3rd graders? Step 1: Null Hypothesis The statistic will “test” the H0 based on data No statistic is perfect The probability of error always exists There are two types of error: I error Reject a true H0 Type II error Accept a false H0 Type Step 1: Null Hypothesis How does one control for Type I and II error? Researcher Conclusion Accept H0 Reality No real difference About exists Test Real difference exists Reject H0 Correct Type I Conclusion error Type II error Correct Conclusion Hypothesis Tests – A Step by Step Process Step 1: State the null hypothesis Step 2: Select level of significance Step 3: Sample data Step 4: Choose statistic Step 5: Calculate the statistic Step 6: Interpret the statistic Step 2: Significance Level Level of significance: Criterion that determines acceptance/rejection of H0 Level of significance denoted as alpha (a) a = the probability of a type I error a can range between >0.0 – <1.0 Typical values: 10% chance of type I error 0.05 5% chance of type I error 0.01 1% chance of type I error 0.10 Step 2: Significance Level How to determine a? Exploratory research: Type I error is acceptable therefore set higher a 0.05 – 0.10 When is type I error unacceptable? Hypothesis Tests – A Step by Step Process Step 1: State the null hypothesis Step 2: Select level of significance Step 3: Sample data Step 4: Choose statistic Step 5: Calculate the statistic Step 6: Interpret the statistic Step 3: Sample Data Parametric statistics assume that data were randomly sampled from population of interest Generalization is limited to population that was sampled Hypothesis Tests – A Step by Step Process Step 1: State the null hypothesis Step 2: Select level of significance Step 3: Sample data Step 4: Choose statistic Step 5: Calculate the statistic Step 6: Interpret the statistic Step 4: Choose the Statistic Parametric or nonparametric? Scale How many means are being compared? Two, of measurement and distribution three or more? How are the means being compared? Between How many independent variables (factors) are being tested? Factorial or within group? design? How many dependent variables are there? Multivariate design? Hypothesis Tests – A Step by Step Process Step 1: State the null hypothesis Step 2: Select level of significance Step 3: Sample data Step 4: Choose statistic Step 5: Calculate the statistic Step 6: Interpret the statistic Step 5: Calculate the Statistic Recall: exp. design statistic The statistic tests the H0 H0 A test statistic can be considered as a ratio between: Between variance (difference b/w means) Within variance (variability w/n means) Statistic = BV/WV Large test statistics imply that: The The difference between the means is relatively large variance within the means is relatively small Example: Researchers compare IQ scores between 6th grade boys and girls. Results: Girls (150 +/- 50), boys (75 +/- 50) Between Variance Distribution overlap? Within Variance 0 50 150 200 Statistic = BV/WV Statistic = Big / Big = small value Statistic = Small / Small = small value Statistic = Small / Big = small value Statistic = Big / Small = Big value Step 5: Calculate the Statistic How does sample size affect the statistic? As sample size increases, the within variance decreases increases size of test statistic Hypothesis Tests – A Step by Step Process Step 1: State the null hypothesis Step 2: Select level of significance Step 3: Sample data Step 4: Choose statistic Step 5: Calculate the statistic Step 6: Interpret the statistic Step 6: Interpret the Statistic Calculation of the test statistic also yields a p-value The p-value is the probability of a type I error The p-value ranges from >0.0 – <1.0 Recall alpha (a) a represents the maximum acceptable probability of type I error therefore . . . Step 6: Interpret the Statistic If the p-value > a accept the H0 Probability of type I error is higher than accepted level Researcher is not “comfortable” stating that any differences are real and not due to chance If the p-value < a reject the H0 Probability of type I error is lower than accepted level Researcher is “comfortable” stating that any differences are real and not due to chance Statistical vs. Practical Significance 1. 2. Distinction: Statistical significance: There is an acceptably low chance of a type I error Practical significance: The actual difference between the means are not trivial in their practical applications