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For an experiment, I want to test if people at each condition have performed significantly above chance level on average, in which they answer 3 questions, each with two options. I will compare each group to chance level performance, without comparing each group to each other.

Then, should I do a one-sampled t-Test by doing these steps, for each single group?

  1. Turn each individual's performance to a proportion (0, 0,3333333, 0,66666666, 1)
  2. Turn it into a data frame.
  3. run it in R with the code t.test(my_dataframe, mu = 0.5, alternative = "two.sided")

I can imagine how trivial the question looks, but it's the first time I'm doing it, so I wanted to make sure. Thank you!

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  • $\begingroup$ Do you mean $H_0: \mu_1=\mu_2=\mu_3 = 0.5$ vs $H_a: H_0 \text{ is false}$? $\endgroup$ – Dave Jun 8 at 17:36
  • $\begingroup$ The second one! I don't compare the groups. I just want to compare each group to chance level perfromance, without comparing the groups to each other. $\endgroup$ – breathe Jun 8 at 17:54
  • $\begingroup$ That is one test. You do not choose between testing $H_0$ and $H_a$. Those are the null and alternative hypotheses. Do you not want to test that $\mu_1=0.5$, $\mu_2=0.5$, and $\mu_3=0.5$? $\endgroup$ – Dave Jun 8 at 17:58
  • $\begingroup$ Yes, that's what I want to do exactly! $\endgroup$ – breathe Jun 8 at 18:32
  • $\begingroup$ How many subjects in the various groups? $\endgroup$ – BruceET Jun 8 at 20:20
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Some ideas: Looking at one group only.

Suppose you have 50 subjects in a group. Then each subject will give you a binomial score $X_i \sim \mathsf{Binom}(3, p_i).$ If all subjects had the same probability of answering correctly $(p_i \equiv p),$ then you could say that the total score for a group is $T \sim \mathsf{Binom}(150, p)$ and you could do a binomial test of $H_O: p = 0.5$ against $H_a: p >0.5.$

However, there is no reason to suppose all 50 subjects are equally capable. so a t test of $H_0: \mu=1.5$ against $H_a: \mu > 1.5$ for $n = 50$ individual scores should give reasonably accurate results.

Example: In R, suppose I have $n = 50$ (fictitious) scores in vector x as summarized and plotted below.

summary(x); sd(x)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
   0.00    1.00    2.00    1.98    3.00    3.00 
[1] 0.9145089
hist(x, prob=T, br=seq(-.5,3.5), col="skyblue2")

enter image description here

Although the sample is not normally distributed, a t test on $n = 50$ such values without outliers should be reasonably accurate. The one-sided test shows a highly significant result with P-value near $0.$

t.test(x, mu=1.5, alt="g")

        One Sample t-test

data:  x
t = 3.7114, df = 49, p-value = 0.0002634
alternative hypothesis: true mean is greater than 1.5
95 percent confidence interval:
 1.76317     Inf

sample estimates:
mean of x 
     1.98 

If one has qualms about using a t test, a less-powerful one-sided sign test based on 35 of 50 observations in x above $1.5$ rejects that the population median score is $1.5$ with P-value 0.0033.

y = sum(x > 1.5); y
[1] 35
sum(dbinom(y:50, 50, .5))
[1] 0.003300224

Also, a simple quantile 95% nonparametric bootstrap CI for $p$ is $(1.72, 2.22),$ which does not contain $1.5.$

set.seed(1234)
a.re = replicate(5000, mean(sample(x,50,rep=T)))
quantile(a.re, c(.025,.975))
 2.5% 97.5% 
 1.72  2.22 

Note: The fictitious data in x were generated as follows in R. The mean of $p \sim \mathsf{Beta}(8,5)$ is $\mu = 8/13 = 0.6154 > 0.5.$

set.seed(2021)
x = rbinom(50, 3, rbeta(50,8,5))
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  • $\begingroup$ What is the theory behind the mean of $p$ following a beta distribution? $\endgroup$ – Galen Jun 8 at 22:59
  • $\begingroup$ @Galen: $p$ can take different values in $(0,1)$ for different subjects. Beta distributions have support $(0,1).$ BETA(9,5) averages $p \approx 0.61.$ It's not the only way to model fictitious data that might match this experiment, but it is a reasonable one. // My fictitious $X$ will tend to have a larger variance than a binomial dist'n with the same mean. // In Bayesian statistics beta distributions are often used as prior distributions on binomial success probability. $\endgroup$ – BruceET Jun 9 at 1:56

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