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Confidence interval for estimated mean of (discrete) uniform distribution

Say that there is a random variable X ~ U(a,b) where U is the discrete uniform distribution on integers on the interval [a,b]. Sample n such variables with the same (unknown) parameters a and b. Using those samples it's possible to estimate the mean either by taking the sample mean (sum the value of each sample and divide by n), but how would I be able to calculate a confidence interval for the estimate of the mean?

(Note: It's also possible to calculate the mean using the sample mid-range, but since the data gathering is done manually, any error which puts a value outside the interval [a,b] would with the sample mid-range forever bias the estimate of the mean, regardless of how many samples are taken.)

For the data I use this for now the sample sizes are small, typically n≤10.

Example data:
220
238
241
200
204
271
289
273
243

Any solutions or guidance to how to approach this (and resources for relevant material) are highly appreciated.
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