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Could anyone enlighten me on what will be the appropriate statistics to test whether a series of events is significantly different from the expected or not.

For example, there is a pool of infinitive balls, and one can use a basket to retrieve balls from the pool. If the basket is functioning well, each time it should get an average of 10 balls (sd = 3). But if it has some problems, the number might be less or more.

In this case, if I did 5 retrievals, and I observe 3, 5, 10, 10, 2 balls, how can I tell if the basket functions well or not?

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  • $\begingroup$ If "functions well" means it needs to get 10 balls, then obviously it isn't functioning well. If, on the other hand, "functions well" means it always gets some balls, then it is functioning well. This shows that you need to defined what "functions well" might mean for your application before statistics can help you determine (from data) whether the process functions well. $\endgroup$
    – whuber
    Feb 9, 2021 at 14:07
  • $\begingroup$ Thanks for the comment. "functions well" what I intend to say is that on average 10 balls (SD = 3) should be in the basket for each retrieval. $\endgroup$
    – Psytky
    Feb 9, 2021 at 14:35

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Several options to consider among others:

Z-test can be used to test whether the mean is 10 assuming the sd is 3. The standard error of the mean is $3/\sqrt{5}$. The sample mean (6) is 3 standard errors below 10. A mean this far away from 10 only happens about once or twice in a thousand times. There is moderately strong evidence that the mean is not 10 assuming the distribution is normal and the sd. is 3. I would only conclude that either one of the assumptions is false or the mean is not 10.

t-test can be used to test whether the mean is 10 assuming the data are normal. The p-value is about 0.08 so not much evidence against the null hypothesis that the mean is 10.

Chi-square test can be used to test whether the variance is 3^2. The statistic won't have a chi-square distribution unless the data are normal. But, a large sample test can be made using the mean and variance of $s^2$ found here.
Observed variance of 14.5 is not very far from hypothesized variance of 3^2 in any case.

If you think the counts have a specific distribution such as the Poisson distribution with mean and variance 10, then you can directly test whether the mean is 10. A sample mean of 6 only happens about once or twice in a thousand times for samples with n=5 from the Poisson distribution with mean 10.

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