The age of an ancient piece of organic matter can be estimated from the rate at which it emits beta particles as a result of carbon-14 decay. For example, if X is the number of particles emitted in 10 minutes by a 10,000-year-old bone fragment that contains 1 g of carbon, then X has a Poisson distribution with mean λ = 45.62. An archaeologist has found a small bone fragment that contains exactly 1 g of carbon. If t is the unknown age of the bone, in years, the archaeologist will count the number X of particles emitted in 10 minutes and compute an estimated age with the formula

Unknown to the archaeologist, the bone is exactly 10,000 years old, so X has a Poisson distribution with λ = 45.62.
a. Generate a simulated sample of 1000 values of X, and their corresponding values of .
b. Estimate the mean of .
c. Estimate the standard deviation of .
d. Estimate the probability that will be within 1000 years of the actual age of 10,000 years.
e. Estimate the probability that will be more than 2000 years from the actual age of 10,000 years.
f. Construct a normal probability plot for . Is approximately normally distributed?

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In 15.3 In(X/10) 0.0001210