In my classes where we study or review statistical inference, I often give the following example of a hypothesis test concerning a single mean: you suspect your sardine cans contain fewer sardines than indicated on the packaging. The packaging says there are 10 sardines per can but you are often finding 8 or 9. So, you open up 100 cans and find the average is 9.5 with a standard deviation of 1. Is this evidence that the tins are lighter than advertised?

The other day I read a news story (here) that describes an almost identical situation, but it is was for tuna fish, not sardines (In my examples I like to pick a fish that you can count by either heads or tails.)

To learn how to do a hypothesis concerning a single mean, read the Appendix to Chapter 1 of Mastering Metrics, which you can find here.

The answer to my question above is, it is very unlikely that there are 10 sardines per can, given our findings. It could be true, and we just happened to find a mean of 9.5 by chance alone. But this would be a major fluke and it is highly, highly unlikely. To see this, use the formulas presented on page 39 of Mastering Metrics. You will find the test statistic is much larger (in absolute value) than 2. On page 41, the authors explain why the finding of a large test statistic is "...unlikely to be consistent with the null hypothesis..." In other words, it is unlikely the mean is 10 in my example.

Now, this analysis assumes the sample of cans opened was a random sample (you didn't pick an unrepresentative sample of cans to open.) I'm no lawyer, but I would guess a finding like the one in my (fictional) example is probably grounds for a class action lawsuit.

In the analysis reported in the tuna fish story linked to above, in one of their samples, they didn't find a

p.s. If you want to calculate the test statistic, and the associated p-value, post your answer as a comment and I will grade it for you for free!

The other day I read a news story (here) that describes an almost identical situation, but it is was for tuna fish, not sardines (In my examples I like to pick a fish that you can count by either heads or tails.)

To learn how to do a hypothesis concerning a single mean, read the Appendix to Chapter 1 of Mastering Metrics, which you can find here.

The answer to my question above is, it is very unlikely that there are 10 sardines per can, given our findings. It could be true, and we just happened to find a mean of 9.5 by chance alone. But this would be a major fluke and it is highly, highly unlikely. To see this, use the formulas presented on page 39 of Mastering Metrics. You will find the test statistic is much larger (in absolute value) than 2. On page 41, the authors explain why the finding of a large test statistic is "...unlikely to be consistent with the null hypothesis..." In other words, it is unlikely the mean is 10 in my example.

Now, this analysis assumes the sample of cans opened was a random sample (you didn't pick an unrepresentative sample of cans to open.) I'm no lawyer, but I would guess a finding like the one in my (fictional) example is probably grounds for a class action lawsuit.

In the analysis reported in the tuna fish story linked to above, in one of their samples, they didn't find a

*single*can that was over the advertised amount. That will certainly make the job of explaining the analysis to a jury much, much easier!p.s. If you want to calculate the test statistic, and the associated p-value, post your answer as a comment and I will grade it for you for free!

I wonder if the decision to undersell a product will yield enough of a profit to justify false advertisement. This also exemplifies the value of having government in a capitalistic society, or just government in general. The only question I really have is whether or not it is valuable enough for the consumer to sue. Say a pack of sardines cost $1. That means that if the product said it was 10 sardines and on average you got 9.5, you would be getting 5 cents "ripped off". And this is without taking into consideration that the value of 0.5 sardines is worth significantly less than 5 cents for the producer. I wonder if companies blur these lines to make a larger profit with the assumption that no one would really care enough to adjudicate them. It just seems baffling that these small margins ever amount to a large enough sum of money to be worthwhile.

ReplyDeleteThis is a fascinating discovery but unfortunately in today's world, a reality in many places. This being not just at Safeway but many supermarket brands and various products too. I'm curious to see if people did studies like this on a range of products how much legal trouble producers of these goods would be in. This would have the potential to change the production line and impact the unemployment rate significantly in a negative way. Firstly the backlash of company's being fined due to the legal troubles of ripping off their customers. Secondly moving forward to cover legal costs or to cover the extra amount of expenses with providing for example the full amount of 10 sardines as mentioned above.

ReplyDeleteShould the laws surrounding this issue be clamped down on? Would this do more bad then good in the wider scheme of the economy?

Justin Pickering.