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I have a sample with a size of n=100, and I want to show that the minimum value of the underlying distribution is not less than a certain threshold, with a confidence level of 95%.

The distribution of the data appears to follow a Normal Mixture.

Initially, I considered using hypothesis testing to establish that the minimum value wouldn't fall below the threshold. Is this the appropriate method to demonstrate this and how can I do that?

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    $\begingroup$ Welcome to CV, Joe. By comparing the smallest number in the dataset to the threshold you can be 100% confident about the result! If you wish to test whether the distribution supposed to generate the data has a minimum and that minimum is not less than a threshold, then even to formulate a test statistic you must postulate some specific set of distributions, for otherwise your question is unanswerable. Please, then, edit your post to provide more information about your dataset and your assumptions. $\endgroup$
    – whuber ♦
    Commented Nov 27, 2023 at 19:01
  • $\begingroup$ "the minimum value" suggests that there is a True Minimum value in the underlying population being sampled. It's a value that could conceivably show up in future samples. If the random variable being sampled is Age then we can announce "minimum value is zero!" without resorting to statistics, assuming the generative process didn't filter at some threshold. The question that perhaps you really wanted to ask is: I will take K samples of size N in the future, producing K minima drawn from a distribution. Can I usefully estimate (mu, sigma) for that distribution? $\endgroup$
    – J_H
    Commented Nov 27, 2023 at 19:48
  • $\begingroup$ @whuber thanks for the response, I updated my question. $\endgroup$
    – Joe the Second
    Commented Nov 27, 2023 at 20:24
  • $\begingroup$ @J_H, Thanks for the response, I updated my question. $\endgroup$
    – Joe the Second
    Commented Nov 27, 2023 at 20:24
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    $\begingroup$ Is it worth noting that a normal distribution extends from negative infinity to positive infinity? It is possible to show that the mean of the distribution is above a threshold with 95% confidence even though the smallest possible value is far, far below that threshold. $\endgroup$
    – Michael Lew
    Commented Nov 27, 2023 at 20:28

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