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 λ = 15.3e^{ˆ’0.0001210t}events per minute. The number of years t since the death of an organism can therefore be expressed in terms of λ:

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 λ̂? 

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 λ̂ = X/25. What is the mean and standard deviation  of λ̂?  

d. What value for λ̂ would result in an age estimate of 10,000 years? 

e. What value for λ̂ 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?

Transcribed Image Text:

In 15.3 – In A 0.0001210