Question

According to a recent​ study, some experts believe that 26​% of all freshwater fish in a particular country have such high levels of mercury that they are dangerous to eat. Suppose a fish market has 150 fish we consider randomly sampled from the population of edible freshwater fish. Use the Central Limit Theorem​ (and the Empirical​ Rule) to find the approximate probability that the market will have a proportion of fish with dangerously high levels of mercury that is more than three standard errors above 0.26. You can use the Central Limit Theorem because the fish were randomly​ sampled; the population is more than 10 times 150​; and n times p is 39​, and n times​ (1 minus​ p) is 111​, and both are more than 10. The approximate probability that the market will have a proportion of fish with dangerously high levels of mercury that is more than three standard errors above 0.26 is

Answer 1

a) The conditions for normality are checked here as:
np = 150*0.26 = 39 >= 10 and also n(1-p) = 150*(1 - 0.26) = 111 >= 10

b) Now the probability that the market will have a proportion of fish with dangerously high levels of mercury that is more than three standard errors above 0.26 is computed here as:

Converting it to a standard normal variable, we have here:

Getting it from the standard normal tables, we get here:

Therefore 0.0013 is the required probability here.