Give FULL step by step solutions with formulas and explanation. Topic: Statistical Analysis of

Random Uncertainties [Book: John R. Taylor - Introduction to Error Analysis..]

4.6. * * In Chapter 3, you learned that in a counting experiment, the uncertainty associated with a counted number is given by the "square-root rule" as the square root of that number. This rule can now be made more precise with the following statements (proved in Chapter 11): If we make several counts V1, Vy, . . ." VNof the number v of random events that occur in a time 7, then: (1) the best estimate for the true average number that occur in time 7 is the mean v = Ev;/N of our measurements, and (2) the standard deviation of the observed numbers should be approximately equal to the square root of this same best estimate; that is, the uncer- tainty in each measurement is vv. In particular, if we make only one count v, the best estimate is just v and the uncertainty is the square root vv; this result is just the square-root rule of Chapter 3 with the additional information that the "uncer- tainty" is actually the standard deviation and gives the margins within which we can be approximately 68% confident the true answer lies. This problem and Problem 4.7 explore these ideas. A nuclear physicist uses a Geiger counter to monitor the number of cosmic-ray particles arriving in his laboratory in any two-second interval. He counts this num- ber 20 times with the following results: 10, 13, 8, 15, 8, 13, 14, 13, 19, 8, 13, 13, 7, 8, 6, 8, 11, 12, 8, 7. (a) Find the mean and standard deviation of these numbers. (b) The latter should be approximately equal to the square root of the former. How well is this expectation borne out