Comparing percentages with unequal sample sizes

I've spent some time collecting participation data on a schools competition.

To sum it up:

Roughly speaking 1000 schools take part in the initiative every year. Only 10 schools become finalists.

The naked eye shows the % of Independent Schools increases significantly in the Finals.

The pattern is evident in the last 5 editions of the competition.

My intention is to prove independent schools are overrepresented in the finals. ... null hypothesis: both percentages are the same

Question(s)

What sort of statistical analysis shall I carry out to discard the null hypothesis?

Basically I want to compare percentages (% participant indep schools vs % finalist independent schools)...

is the analysis somehow compromised by the highly dissimilar sample sizes (n=1000 vs n=10)?

As I said the pattern is evident in all editions.. year after year... would this be an advantage to overcome the sample size problem?

If I understand your question correctly, a simple approach is to ignore the fact that there are multiple editions of the competition. In this case you want to compare the proportion of independent schools in the finals to the proportion of independent schools in the population of participants.

Assume: Independent schools are 0.35 of the population of participants

Assume: Out of 50 finalist (in the last five episodes), 28 are independent schools.

Then you just have a binomial test, or a chi-square goodness-of-fit test.

The following code will run in R or at: rdrr.io/snippets/

Population = 0.35
n = 50
x = 28

binom.test(x, n, Population)

# Exact binomial test
#
# data:  x and n
# number of successes = 28, number of trials = 50, p-value = 0.002713
# alternative hypothesis: true probability of success is not equal to 0.35
# 95 percent confidence interval:
# 0.4125441 0.7000928
# sample estimates:
# probability of success
#                  0.56

Counts = c(x, n-x)
Props  = c(Population, 1-Population)

# chisq.test(Counts, p = Props)
#
# Chi-squared test for given probabilities
#
# data:  Counts
# X-squared = 9.6923, df = 1, p-value = 0.00185

• Thanks for the tip, Sal. Ignoring the fact that there are multiple editions sounds like a plan but bear in mind the population of participants and the proportion of participant independent schools does slightly change from year to year. Do you see this as an obstacle to follow your plan? – Iván Diego May 8 '19 at 13:31
• No, you could use the sum of values for the e.g. five editions I think without issue.... But If desired I'm sure you could take into account the differences in the e.g. five editions with a more complex model, but if that's necessary, maybe the simplest way is to analyze the editions individually. – Sal Mangiafico May 8 '19 at 14:40