# "Chi-squared approximation may be incorrect" on hypothesis test

I'm doing a hypothesis test to determine if the data follows binomial distribution with p=0.31, and I'm getting a warning when running this:

chisq.test(x = c(36,48,38,23,10,3),p = dbinom(0:5,5,0.31))

Chi-squared test for given probabilities

data:  c(36, 48, 38, 23, 10, 3)
X-squared = 28.265, df = 5, p-value = 3.23e-05


And the warning message I'm getting:

Warning message:
In chisq.test(x = c(36, 48, 38, 23, 10, 3), p = dbinom(0:5, 5, 0.31)) :
Chi-squared approximation may be incorrect


I believe what you were trying to do was to check if given the size of 100 observation from a binomial distribution with the probability of 0.31, you would get the same result. In this case R's chisq.test() is not a good choice, first because you have stated the probability wrong. You're giving the probabilities of each observation one-by-one, element-wise and that's not what you have meant. Here's an example:

x <- c(36,48,38,23,10,3)

binom(0:5, 5, 0.31)
[1] 0.156403135 0.351340375 0.315697149 0.141834951 0.031861475 0.002862915


36 is not equal to 0.15 anyway, the same goes with the the other elements. You could have divided x by 100 but still you're not applying the right statistical test here.

One option is to use Kolmogorov–Smirnov test which is used to test whether or not a sample comes from a certain distribution. To perform a one-sample or two-sample Kolmogorov-Smirnov test in R we can use the ks.test() function. Here we assume that:

$$H_0:$$ the two dataset values are from the same distribution - Binomial in this case

ks.test(c(36,48,38,23,10,3), pbinom(5, 100, 0.31))
Two-sample Kolmogorov-Smirnov test

data:  c(36, 48, 38, 23, 10, 3) and pbinom(5, 100, 0.31)
D = 1, p-value = 0.2857
alternative hypothesis: two-sided


Voila! Turns our your observations may come from a Binomial distribution with p=0.31. Still be aware that with these few observations, the test result couldn't be much reliable after all.

• Thank you very very much for taking the time to answer this! Sincerely appreciate it Commented Oct 10, 2022 at 22:15
• @AlePouroullis You're very welcome. I hope this answered what you were exactly looking for. If it did, please mark the question as answered. Commented Oct 10, 2022 at 22:23