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The essential features of this question are: It does not make strong distributional assumptions, lending it a non-parametric flavor. It concerns only tail behavior, not the entire distribution. With some diffidence--because I have not studied my proposal theoretically to fully understand its performance--I will outline an approach that might be practicable....


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I would suggest to fit different distributions on your observations, and to perform model selection to find the distribution that fits your observations the best. Exponential and Pareto distributions seem to be the best candidates given your hypotheses (positivity, monotone decrease). Once you have fitted these candidates distributions, model selection ...


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As you expected, the chi-squared test seems fit for your problem. The null hypothesis would be $$ H_0: p_1=p_2=p_3 $$ where $p_i$ is the proportion of people in group $i$ who decided to buy an item. The alternative would be not $H_0$.


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Simply: you use the same language as you would to report a significant result, altering as necessary. I had the honor of collaborating with a much regarded biostatistical mentor who wrote an entire manuscript prior to performing final data analysis, with just a placeholder for discussion, as that's truly the only place where discourse diverges depending on ...


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