How do I prove one company is actually doing better than another?

Question

Suppose I have the return information for two reverse logistic companies--See example data. For certain groups companies A and B each receive 50% of the work from a client, but for another group company A receives 60% of the work and company B receives 40% of the work. They are assigned work from clients based on the last digit of an account number and are told it is random.

1. Is there any way to find out that the account numbers are really randomly assigned?
2. How would I prove/disprove to someone if one company is actually doing better than another company or if it is probably random chance that one company's rate of return is ever so slightly better than the other company's? My initial thoughts are a t-test or chi-squared test, but in some groups the data isn't evenly divided between the two companies and the data isn't normally distributed. EDIT: In samples of the data the amount of work isn't evenly divided between the two companies and the rate of return isn't normally distributed.

Answer 1

1. On testing performance of two group

Since, task is assigned randomly to each of the company, and normality is suspicious, therefore, I would recommend you to use non-parametric test to compare the performance of the group. One option is Mann Whitney Test. Unlike the t-test it does not require the assumption of normal distributions. It is nearly as efficient as the t-test on normal distributions.

2. Is there any way to find out that the account numbers are really randomly assigned?

For this, start with plotting (You can use this for part 1 too). Make a line chart, or frequency distribution. If you find any strong pattern in the plot (like 70% of the numbers less than 5 are assigned to company A etc), then you can infer that tasks are not assign randomly to both the company (provided sample size is large enough). Otherwise, you can use Run Test to test the hypothesis. Again Run test is non-parametric test, therefore, it does not make any assumption about the distribution of the data.

Answer 2

Welch's T test can control for uneven groups, and if you have large enough samples then central limit theorem applies so no need to worry if the original data i normal or not.

EDIT: sigh, you try to help people and get thumbed down for it. Central limit theorem isn't just some 'theory', it's one of the most fundamental theories in stats. It says that for any distribution with variance less than infinity, the distribution of sample means will be normally distributed. That's why you can use a T test on it, because a T test tests sample means against other sample means. The original distribution drops away. But CLT requires large samples to converge--larger for more non normal distributions, but usually only around 30+. The T test is robust with large sample sizes. But feel free to use other non parametric tests, but they tend to pay a price in power by having fewer assumptions. The other suggestion of the Mann Whitney U is fine if you want to go with that; I was just suggesting a simple familiar test.

Answer 3

Is there any way to find out that the account numbers are really randomly assigned?

I believe you can use tests available in this link, here are four tests that you can do on frequency counts.

Now for this question-

How would I prove/disprove to someone if one company is actually doing better than another company or if it is probably random chance that one company's rate of return is ever so slightly better than the other company's? In samples of the data the amount of work isn't evenly divided between the two companies

Use t test on mean rate of return between two companies which has lesser rate of return is doing good, If the amount not evenly distributed it doesn't matter, and the problem with normal distribution of rate of return use transformations here is an outstanding blog on how to transform data to normal distribution.

Please comment if you have any doubts or i didn't met the question properly. Thanks!