Beyond the Formula 2005 Investigating Statistical Assumptions Using Fathom 2 Sponsored by Key Curriculum Press Tom Short Indiana University of Pennsylvania tshort@iup.edu

Outline + Introduction + Inference for one proportion + Visualizing confidence intervals + Simulation studies of coverage + Inference for means + Questions and discussion

Inference for One Proportion What is the true proportion, p, of BTF participants who are currently high school classroom teachers? Collect a sample of size n Count the proportion of “successes,” x Compute the sample proportion,

Confidence Interval for p An approximate 95% confidence interval for the true proportion, p, is given by: Note that for 95% confidence, z* = 1.96

Rules for Sample Proportions + Number of trials, n, is fixed + Trials are independent + There is a constant probability of success on each trial, p + The number of observed successes, np >= 10 (or 5 or 15 or …) + The number of observed failures, n(1-p) >= 10 (or 5 or 15 or …)

Quantifying Skewness Skewness Large negative values indicate a tail to the left. Large positive values indicate a tail to the right. Skewness is the third moment about the mean.

What are the consequences? + Misleading or incorrect coverage percentages + Inflated Type I error probabilities