An environmental group is suin
An environmental group is suing a manufacturer because chemicalsdumped into a nearby river may be harming fish. A sample of fishfrom upstream (no chemicals) is compared with a sample fromdownstream (chemicals), and a 95% confidence interval for thedifference in proportions of healthy fish is .01 to .11 (with ahigher proportion of healthy fish upstream). Interpret thisinterval. The statistician for the manufacturer produces a 99%confidence interval ranging from –0.01 to +0.13. Hetells the judge that because the interval includes 0, and becauseit has higher accompanying confidence than the other interval, wecan’t conclude that there is a problem. Comment on this from boththe perspective of the lawyer for the environmental group and asthe judge hearing the case.
Answer:
To answer this question, we should first understand what levelof significance(
)is. It is the probability of rejecting the null hypothesis when itis actually true.
The level of significance and confidence interval are linked asfollows:
Confidence Interval = 1 –
Now we want alpha to be as small as possible. The smaller thelevel of significance would be the smaller would be the chance ofrejecting the null hypothesis when it is true. Consequently thehigher would be the confidence interval. By higher we mean that thewidth of the confidence interval would be larger. So we would get abigger range of values the population parameter my be taking.
Now a 95% confidence interval for the difference in proportionsof healthy fish is .01 to .11 (with a higher proportion of healthyfish upstream) means that we can say with 95% confidence that thetrue value of the difference between the mean – lies between 0.01and 0.11. This in turn means that the probability of rejecting thenull hypothesis when it is true is 5%.
Now if we increase the confidence interval to 99% from 95% itmeans that we are 99% confident that the true value of thedifference between the mean – lies between –0.01 to +0.13. As youmay noticed to increase higher confidence and lower alpha, thewidth of the confidence interval increases.
Now from both the perspective of the lawyer for theenvironmental group it makes sense to use 95% confidence interval.We are nearly certain that the chemicals dumped into a nearby riverare harming fish. In fact we are 95% confident of the same. Nowfrom the judge perspective, increasing the confidence interval haswidespread the confidence interval and there is reasonableuncertainty that there is a small chance that themanufacturer is actually not harming the fish by dumping chemicalsin the river.
Let me know incomments if anything is not clear. Will reply ASAP. Please doupvote if satisfied.
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