(Continues Exercise 20 in Section 4.11.) The age of an

QUESTION:

(Continues Exercise 20 in Section 4.11.) 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 1g of carbon, then X has a Poisson distribution with mean \(\lambda=45.62\). An archaeologist has found a small bone fragment that contains exactly 1g 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 \(\hat{t}\) with the formula

\(\hat{t}=\frac{\ln 15.3-\ln (X / 10)}{0.0001210}\)

Unknown to the archaeologist, the bone is exactly 10,000 years old, so X has a Poisson distribution with \(\lambda=45.62\).

a. Generate a simulated sample of 1000 values of X, and their corresponding values of \(\hat{t}\).

b. Estimate the mean of \(\hat{t}\).

c. Estimate the standard deviation of \(\hat{t}\).

d. Estimate the probability that \(\hat{t}\) will be within 1000 years of the actual age of 10,000 years.

e. Estimate the probability that \(\hat{t}\) will be more than 2000 years from the actual age of 10,000 years.

f. Construct a normal probability plot for \(\hat{t}\). Is \(\hat{t}\) approximately normally distributed?

Equation Transcription:

Text Transcription:

Lambda = 45.62

t hat

t hat = ln 15.3-ln(X/10)/0.0001210