You have received a radioactive mass that is claimed to have a mean decay rate of at least 1 particle per second. If the mean decay rate is less than 1 per second, you may return the product for a refund. Let Xbe the number of decay events counted in 10 seconds.

Question

a. If the mean decay rate is exactly 1 per second (so that the claim is true, but just barely), what is P(X ? 1)?
b. Based on the answer to part (a), if the mean decay rate is 1 particle per second, would one event in 10 seconds be an unusually small number?
c. If you counted one decay event in 10 seconds, would this be convincing evidence that the product should be returned? Explain.
d. If the mean decay rate is exactly 1 per second, what is P(X ? 8)?
e. Based on the answer to part (d), if the mean decay rate is 1 particle per second, would eight events in 10 seconds be an unusually small number?
f. If you counted eight decay events in 10 seconds, would this be convincing evidence that the product should be returned? Explain.

Answer 1

If the mean decay rate is exactly 1 per second, then the mean number of events in 10 seconds is 10, so X~Poisson(10).



(a)





(b) Yes, If the mean rate is 1 event per second, then there will be 1 or fewer events in 10 seconds only about 5 times

?n 10,000.



(c) Yes, because 1 event in 10 seconds is an unusually small number if the mean rate is 1 event per second.



(d)





(e) No. If the mean rate is1 event per second , then there will be 8 or fewer events in 10 seconds about one-third of the time.



(f) No, because 8 events in 10 seconds is not an unusually small number if the mean rate is 1 event per second.