Suppose we observe 100 alcoholics with cirrhosis of the liver, of whom 33 have hepatomas—that is, liver-cell carcinoma. Suppose we know, based on a large sample, that the risk of hepatoma among alcoholics without cirrhosis of the liver is 20%. 5.1 What is the probability that we observe exactly 33 alcoholics with cirrhosis of the liver who have hepatomas if the true rate of hepatoma among alcoholics (with or without cirrhosis of the liver) is 20%? 5.2 What is the probability of observing at least 33 hepatomas among the 100 alcoholics with cirrhosis of the liver under the assumptions in Problem 5.1? 5.3 Let p be the probability that an alcoholic with cirrhosis of the liver has hepatomas. Give an estimate of p.
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
Suppose we observe 100 alcoholics with cirrhosis of the liver, of whom 33 have
hepatomas—that is, liver-cell carcinoma. Suppose we know, based on a large
sample, that the risk of hepatoma among alcoholics without cirrhosis of the liver
is 20%.
5.1 What is the
of the liver who have hepatomas if the true rate of hepatoma among alcoholics
(with or without cirrhosis of the liver) is 20%?
5.2 What is the probability of observing at least 33 hepatomas among the 100
alcoholics with cirrhosis of the liver under the assumptions in Problem 5.1?
5.3 Let p be the probability that an alcoholic with cirrhosis of the liver has
hepatomas. Give an estimate of p.
5.4 Construct a 95% confidence interval of p.
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