You are designing an experiment to see if a newly-designed early childhood education program (NEW) for 4 year-olds has an effect on a child's test score for kindergarten school readiness. You will recruit families to be part of the study and randomly assign them to a Control group that receives the status quo program (SQ) and a Treatment group that has the NEW program. You are trying to determine the sample size you need to detect treatment effects on test scores. You fix the probability of a Type 1 error, , to be 0.05, and the probability of a Type 2 error, , to be 0.20 A) Suppose previous studies on early childhood programs have found treatment effects of a 0.3 standard deviation (SD) increase in average school readiness test scores. If you assume your NEW program could have similar effect sizes, what sample size do you need in your Control group and Treatment group? Explain your reasoning.

B) Now suppose you learn from your research partners that getting a treatment effect of 0.3 SD in average test scores is too ambitious. It would likely be a 0.1 SD increase. What is the sample size you would need in each group to detect this effect? Explain your reasoning.

C) Your research partners now tell you that you cannot choose any sample size because they have limited resources. Suppose you can have a sample size of 90 children in the Control and 90 in the Treatment group. What is the minimal treatment effect (in standard deviations of average test scores) that you can detect with this sample? Explain your reasoning.

D) ou have a brilliant idea to add an additional treatment that is the NEW program plus a parenting program to help improve parenting skills (NEW+). Your research partners say you can choose whatever sample size you need for this design. What is the sample size you would need in the SQ, NEW and NEW+ groups if you think test scores would be 0.2 SD higher in the NEW compared to the SQ group, 0.3 SD higher in the NEW+ compared to the SQ group and 0.1 SD higher in the NEW compared to the NEW+ group? Explain your reasoning.