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I'm really new to statistics and I'm working on a project where I'm basically trying to rig a coin toss. I have three groups and $n = 50$ tosses for each group: Control (24/50 success) vs treatment 1 (28/50 success) vs treatment 2 (30/50 success). Success = 1 and is for Heads, failure = 0 and is for tails.

My problem is that I don't know how to analyze this data. I just can't figure out what tests to do and what is appropriate to test my alternative hypothesis that the groups are NOT equal to each other. I'm also trying to figure out what power would be appropriate in this test.

Someone, please lead me in the right direction!

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The question is not entirely clear, but I interpret it as asking how to test the hypothesis that the binomial $p$ (probability of success) is the same in the three groups. This can be formulated as a logistic regression, but in this case there is simpler (approximate) ways. I will only now show how you can do it in R:

  yourtab <- as.table(cbind(succ=c(24, 28, 30), fail=c(26, 22, 20)))
  yourtab
  succ fail
A   24   26
B   28   22
C   30   20
  prop.test(yourtab)

    3-sample test for equality of proportions without continuity
    correction

data:  yourtab
X-squared = 1.5065, df = 2, p-value = 0.4708
alternative hypothesis: two.sided
sample estimates:
prop 1 prop 2 prop 3 
  0.48   0.56   0.60 

This uses an approximate chi-squared test. Using logistic regression we can do:

groups <- as.factor(1:3)
 mod0 <- glm(yourtab ~ 1,  family=binomial)
 mod1 <- glm(yourtab ~ groups,  family=binomial)
 anova(mod0,  mod1)
Analysis of Deviance Table

Model 1: yourtab ~ 1
Model 2: yourtab ~ groups
  Resid. Df Resid. Dev Df Deviance
1         2     1.5067            
2         0     0.0000  2   1.5067

Compare the deviance above with the chisquared (X-squared) from prop.test. This is effectively the same test.

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