Statistical difference among groups I have a sample of people answering a purchase intent question for 3 products. I tabulate the data and get this for definitely would buy:
prod1 20% definitely would buy
prod2 15% definitely would buy
prod3 25% definitely would buy

So the average across the 3 products is 16,7%. What I would like to know is, given the 16,7% average how can I test if for example prod1 is statistically higher than the average across the 3 products. My sample is 250 respondents. Calculations can be done manually but I prefer if done in R (95% confidence).
 A: Test of hypothesis. An exact binomial test, not using a normal approximation, of
$H_0: p = .167$ against $H_a: p \ne .167$ for $x = 50$ who would buy
out of $n = 250$ does not reject $H_0$ at the 5% level of significance.
binom.test(50, 250, p=.167)

        Exact binomial test

data:  50 and 250
number of successes = 50, number of trials = 250, 
  p-value = 0.1743
alternative hypothesis: 
  true probability of success is not equal to 0.167
95 percent confidence interval:
 0.1522402 0.2550261
sample estimates:
probability of success 
               0.2 

Confidence intervals. A 95% confidence interval for the population proportion $p$ is $\hat p \pm 1.96\sqrt{\frac{\hat p(1-\hat p)}{n}}.$ For $\hat p = 0.2$ and $n = 250,$ the CI is $(0.1504, 0.2496),$ which includes 0.167. So I don't think you can reasonably exclude $p = 0.167.$ This method uses a normal approximation.
The somewhat more accurate Agresti-Coull 95% CI for $p$ based on $\hat p = 52/254$ is $(0.1551, 0.2543),$ which also includes 0.167.
Note: You are trying to distinguish among proportions in the neighborhood of .15 to .25 that differ by about 0.03, which would require roughly $n = 800$ subjects.
