Suppose a dataset contains the selling price for one particular product, and this product could be sold from many resellers. This product could be sold in two ways, sold without rebate to the reseller and sold with rebate to the reseller. The numbers of transaction for product sold with and without rebate are different; one has about 140 samples and the other has about 130 samples.

My goal is to perform hypothesis testing on whether the average selling price for the product sold without and with rebate is the same。

My initial thought was to use two sample t-test, but since one product could be only sold with or without rebate, they are mutually exclusive events, so they do not qualify for the independence assumption for the two sample t-test. I then wanted to use the paired t-test, but it's not a traditionally "before-after" paired data, plus the fact that the numbers of samples for two groups are not equal.

The distribution of the selling price with rebate is highly skewed, e.g., there are many points around the average selling price and there are wide spread of events toward higher selling price. (Since the reseller is given a rebate on every product they sell, the reseller would of course to sell the product as high as possible to gain more rebate on each transaction)

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    $\begingroup$ I can't understand the distinction / potential problem you are pointing out. Of course a given product can only be sold w/ or w/o the rebate, that's the way it always is. So what's the problem here? $\endgroup$ – gung - Reinstate Monica Jul 29 '17 at 0:04
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    $\begingroup$ The issue is not whether the factor with-rebate/without-rebate is dependent; you condition on that. The issue is whether the conditional distribution of the response (selling price) is dependent on the other selling prices. $\endgroup$ – Glen_b Jul 29 '17 at 5:34
  • $\begingroup$ @gung My bad, gung. I am trying to perform hypothesis testing with the null hypothesis that there is no difference between the average selling price for the product sold with-rebate and without-rebate. But I couldn't find the appropriate method. Is the problem more clear to you? $\endgroup$ – TX2013 Jul 29 '17 at 18:27
  • $\begingroup$ I'm still not quite clear on the issue here. Is it that the product types differ between rebate & not (eg, prices for TVs w/ rebates vs radios w/o rebates)? $\endgroup$ – gung - Reinstate Monica Jul 31 '17 at 12:47
  • $\begingroup$ @gung The product is the same type, and I am interested if there is a difference between prices for TVs w/ rebates vs TVs w/o rebates using hypothesis testing. $\endgroup$ – TX2013 Jul 31 '17 at 16:25

I don't really see any problem with this. In general, you could use a t-test here. The t-test does assume that the prices are normally distributed and the the standard deviations are the same for the two conditions, but it is fairly robust if the assumptions are not too badly violated and you have a decent amount of data. With 130 and 140 data, you might be OK with the skew. If you want to be more cautious, you could run a Mann-Whitney U-test, but be aware that the hypothesis tested is slightly different: you are testing if data with rebates tend to be larger (smaller) that data without, not whether the means differ. If you really want a test of means (which does make sense from a business point of view), you could also try bootstrapping, if you are familiar with that.

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  • $\begingroup$ Thanks for your answer, gung. My primary concern is that, would my two groups of data considered to be independent? My understanding is that they are mutually exclusive events and thus not independent. Please correct me if I am wrong, thanks! $\endgroup$ – TX2013 Jul 31 '17 at 20:34
  • $\begingroup$ @TX2013, I can't really figure out what you're talking about. That was the point of my questions. In a clinical trial, eg, a patient can be in the intervention arm or the control arm. Those are mutually exclusive. There is absolutely no issue there. If that's what you mean (which is what I thought from your answers), there is no problem. $\endgroup$ – gung - Reinstate Monica Jul 31 '17 at 20:41
  • $\begingroup$ @ gung I might be a little bit confused about mutually exclusive events and independent of random samples. I like your clinical trial example, and here is my understanding: the patient could be only categorized into either intervention or control arm, these two events are mutually exclusive. However, the outcome of the events, e.g. patient has certain disease or not is independent of other patient's test result. So, these two groups are suitable for the t-test assuming their distribution is normal. Or could use Mann-Whitney U-test assuming not normal distribution. Does all this make sense? $\endgroup$ – TX2013 Jul 31 '17 at 23:23
  • $\begingroup$ @TX2013, yeah, that's the gist of it. $\endgroup$ – gung - Reinstate Monica Jul 31 '17 at 23:44

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