Question

An environmental group is suing a manufacturer because chemicals dumped into a nearby river may be harming fish. A sample of fish from upstream (no chemicals) is compared with a sample from downstream (chemicals), and a 95% confidence interval for the difference in proportions of healthy fish is .01 to .11 (with a higher proportion of healthy fish upstream). Interpret this interval. The statistician for the manufacturer produces a 99% confidence interval ranging from   –0.01 to +0.13. He tells the judge that because the interval includes 0, and because it has higher accompanying confidence than the other interval, we can’t conclude that there is a problem. Comment on this from both the perspective of the lawyer for the environmental group and as the judge hearing the case.

Answer 1

To answer this question, we should first understand what level of significance() is. It is the probability of rejecting the null hypothesis when it is actually true.

The level of significance and confidence interval are linked as follows:

Confidence Interval = 1 -

Now we want alpha to be as small as possible. The smaller the level of significance would be the smaller would be the chance of rejecting the null hypothesis when it is true. Consequently the higher would be the confidence interval. By higher we mean that the width of the confidence interval would be larger. So we would get a bigger range of values the population parameter my be taking.

Now a 95% confidence interval for the difference in proportions of healthy fish is .01 to .11 (with a higher proportion of healthy fish upstream) means that we can say with 95% confidence that the true value of the difference between the mean - lies between 0.01 and 0.11. This in turn means that the probability of rejecting the null hypothesis when it is true is 5%.

Now if we increase the confidence interval to 99% from 95% it means that we are 99% confident that the true value of the difference between the mean - lies between –0.01 to +0.13. As you may noticed to increase higher confidence and lower alpha, the width of the confidence interval increases.

Now from both the perspective of the lawyer for the environmental group it makes sense to use 95% confidence interval. We are nearly certain that the chemicals dumped into a nearby river are harming fish. In fact we are 95% confident of the same. Now from the judge perspective, increasing the confidence interval has widespread the confidence interval and there is reasonable uncertainty that there is a small chance that the manufacturer is actually not harming the fish by dumping chemicals in the river.

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