1) While imprisoned, a mathematician suspects that a coin in his pocket is not fair. In particular, the probability that a head will be tossed is not known. Call this probability ?. In order to...

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1) While imprisoned, a mathematician suspects that a coin in his pocket is not fair. In particular, the probability that a head will be tossed is not known. Call this probability ?. In order to estimate ?, a test is run by tossing the coin 100 times and counting the number of times heads comes up. It turns out the mathematician obtains 57 heads. (a) Use the data to give an estimate for ?. (b) Find 95% confidence interval for ?. (c) How many tosses of the coin are needed in order to be 99% confident that a future sample proportion, ?̂, will be within 0.05 of the true ?? 2) One difficulty in measuring the nesting success of birds is that the researchers must count the number of eggs in the nest, which is disturbing to the parents. Even though the researcher does not harm the birds, the flight of the bird might alert predators to the presence of a nest. To see if researcher activity might degrade nesting success, the nest survival of 102 nests that had their eggs counted, was recorded. Sixty-four of the nests failed (i.e. the parent abandoned the nest.) (a) Assuming that it is reasonable to regard the 102 nests in the sample as representative of the population of nests for which the eggs have been counted, construct and interpret a 95% confidence interval for the proportion of nests that have eggs counted that are then abandoned. (b) The usual nest failure rate of these birds is 29%. Is the confidence interval from part (a) consistent with the theory that the researcher's activity affects nesting success? Justify your answer with an appropriate statistical argument. 3) For your initial post, explain why a sample size of 1000 would provide a very small margin of error when finding a confidence interval for the proportion.