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I have a question here and not sure how to solve it, perhaps I'm overthinking it too much !

A researcher believes that the number of customers who enter a shop is uniformly distributed over 5 days and a sample week yields (15,12,14,15,11). Perform a hypothesis test to investigate this claim.

X bar is 14 which is obvious but not sure what value to start or should I say I have no idea what my null hypothesis is here i.e. That u = ?

XeroPhobous
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    Note number of customers that enter a store each day is a discrete random variable. I would use Goodness of Fit to test $$H_0: \text{The distribution is } \mathcal{U}{11,12,13,14}$$ against $$H_a: \text{The distribution isn't } \mathcal{U}{11,12,13,14}$$ – Matthew H. Jul 05 '22 at 14:27

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It sounds like you're being asked to accept or reject whether the data fits a distribution as a whole, not just some condition on the parameters of the Uniform distribution.

In this case maybe you should look at the Kolmogorov-Smirnov tests where $H_0$: data fits a Uniform CDF and $H_a$: data does not fit a Uniform CDF. The test statistic is related to the maximum difference between the reference CDF and the empirical CDF calculated from the data.

More info here: Is there a simple test for uniform distributions?

For what it's worth, the sample mean is not a particularly good estimator of the parameters in uniform distribution: Maximum likelihood estimation of $a,b$ for a uniform distribution on $[a,b]$

nimish
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