Chapter 4.11, Problem 20E

Radiocarbon dating: Carbon-14 is a radioactive isotope of carbon that decays by emitting a beta particle. In the earth’s atmosphere, approximately one carbon atom in 10¹² is carbon-14. Living organisms exchange carbon with the atmosphere, so this same ratio holds for living tissue. After an organism dies, it stops exchanging carbon with its environment, and its carbon-14 ratio decreases exponentially with time. The rate at which beta particles are emitted from a given mass of carbon is proportional to the carbon-14 ratio, so this rate decreases exponentially with time as well. By measuring the rate of beta emissions in a sample of tissue, the time since the death of the organism can be estimated. Specifically, it is known that t years after death, the number of beta particle emissions occurring in any given time interval from 1 g of carbon follows a Poisson distribution with rate $\lambda =15.3e^{-0.0001210t}$ events per minute. The number of years t since the death of an organism can therefore be expressed in terms of λ:

$t=\frac{\ln 15.3-\ln \lambda }{0.0001210}$

An archaeologist finds a small piece of charcoal from an ancient campsite. The charcoal contains 1 g of carbon.

a.    Unknown to the archaeologist, the charcoal is 11,000 years old. What is the true value of the emission rate λ?

b.    The archaeologist plans to count the number X of emissions in a 25 minute interval. Find the mean and standard deviation of X.

c.    The archaeologist then plans to estimate λ with $\hat{\lambda }=X/25$ . What is the mean and standard deviation of $\hat{\lambda }$?

d.    What value for $\hat{\lambda }$ would result in an age estimate of 10.000 years?

e.    What value for $\hat{\lambda }$ would result in an age estimate of 12,000 years?

f.    What is the probability that the age estimate is correct to within ±1000 years?