Download Hypothesis Testing with t: A Review and more Slides Statistics in PDF only on Docsity! 1 Introduction to t I. Hypothesis testing with Z, a review II. Estimating pop. Standard Deviation III. The t distribution & table IV. The t formula V. Steps VI. Examples VII. Effect Size Hypothesis testing with Z, a review 1. State H0, H1, and choose α 2. Determine what type of observation it would take to reject H0 a) What is the appropriate test statistic? b) What is the critical value? 3. Evaluate the sample data 4. Reach a conclusion Hypothesis testing with Z, a review 2. Determine what type of observation it would take to reject H0 a) What is the appropriate test statistic? b) What is the critical value? With a known µ, and σ, we can calculate probabilities with Z Z = M - µ σM σ M = σ √ n Docsity.com 2 Estimating pop. Standard Deviation But what if do not know σ, the population standard deviation? We have to estimate it from our sample: Remember: SS = ∑(x-M)2 Population Sample Variance: σ 2=SS/N s2=SS/n-1 Standard Deviation: σ =√SS/N s= √SS/n-1 Estimating pop. Standard Deviation Similarly, estimate the standard error: Also, remember your estimates of population values get better with sample size. Also, df = n-1 σ M = σ √ n s M = s √ n s2 n= √ The t distribution & table Previously we argued that the distribution of sample means was normal (central limits theorem), so we could use Z to test hypotheses about those means. The t distribution is a probability distribution (similar to Z) that takes into account that we are estimating the standard deviation, and this estimate changes with degrees of freedom (sample size). The distribution is normal when n is large, but flatter than normal for small values of n. Docsity.com 5 Other examples These will be done in class as appropriate. Effect Size What if we find a statistically significant effect, is the effect “meaningful.” Effect size is a way to quantify the magnitude of a treatment effect. Two measures: Cohen’s d r2 Effect Size Cohen’s d Measures how big the effect is by comparing it to the standard deviation. Cohen’s d = mean difference/standard dev. In our basketball example: Cohen’s d = 76-69/1.63 = 4.29 Docsity.com 6 Effect Size Interpreting Cohen’s d: Cohen’s d seems arbitrary. Effect Size r2 as a measure of effect size r2 measures the proportion of variance in the data that is accounted for by the treatment effect. total variance in the data = treatment effect + unexplained variance (error) Effect Size variance explained by treatment total variance in the data Two ways to calculate r2 1) calculate SStotal, SStreatment, SSerror r2 = SStreatment/ Sstotal (the hard way) 2) r2 = t2/(t2+df) r2 = Docsity.com 7 Effect Size Basketball example: r2 = t2/(t2+df) = 8.542/(8.542+3) = .96 That is, 96% of the variance from the population mean can be explained by the fact that my sample is basketball players. Effect Size Interpreting r2 Docsity.com
