1. If you take only a single water sample and process it with you pH kit, what\'
ID: 3251081 • Letter: 1
Question
1. If you take only a single water sample and process it with you pH kit, what's the probability that you'll incorrectly conclude that this lake is threatened?
2. You're not happy with the misclassification probability in (1), and you decide to remedy this by taking n > 1 independent water samples from the kale and basing your assessment on their mean pH value y bar. How large does n need to be to make the probability of {incorrectly concluding that his lake is threatened} 0.5% or less?
Unpolluted deposition (or rain in balance with atmospheric carbon dioxide, has a pH of 5.6. Almost everywhere in the world the pH lower than this. The main pollutants responsible deposition (or acid rain) are sulfur dioxide (so2) and for nitrogen oxides (NO.). Acid deposition influences mainly the Most freshwater lakes, streams, and ponds have a natural pH in the range of 6 to 8. Acid deposition has many harmful ecological effects when the pH of aquatic systems falls below 6 and especially below 5. Here are some effects of increased acidity o aquatic systems: As the pH approaches 5, non-desirable species of plankton and mosses may begin to invade, and populations of fish such as small-mouth bass disappear. Below a pH of 5, fish populations begin to disappear, the bottom is covered with undecayed material, and mosses may dominate near-shore Below a pH of 4.5, the water is essentially devoid of fish You're a out in the field studying a lake ently remote that you had to backpack in to get to it and this lake looks like it may already have been damaged by acid rain. The only pH measurement kit you could bring with you in your backpack is rather crude: it's to unbiased pH measurements that fluctuate around the true value with an SD of 0.15 and an give approximately normal for its measurement errors. You'll be surveying enough lakes distribu on this trip that you can't bring water samples back with you; need to estimate their pH values in the field You're wondering if the pH of the lake you're now standing front is bclow 5: call any such lake threatened. Suppose for the sake of this problem that the true pH of this lake is 5.1, so that in fact it's not actually threatenedExplanation / Answer
1) Probability of the left tail below value 5. z = (5 - 5.1)/(0.15/sqrt(1)) = -2/3, p(z<-2/3) = 0.2524925
2) p(z < -2/3*sqrt(n)) = 0.05, z = -1.644854 = -2/3*sqrt(n) -> n = 6.0875 = 7
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