n Chapter 3, you learned that in a counting experimen t, the uncerta inty 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 of the number 11 of random events that occur in a time T, then: (I) the best estimate for the true average number tha1 oocur in time T is the mean 'ii = l..vi!N of our mcasuremems,and (2) the sta11dard deviatio11 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 .,/f,. In particular, if we make only one count 11, the best estimate is just 11 and the uncertainty is the square root --fi,; this result is just the square-root rule of Chapter 3 with the additional in formation that the "uncer tainty" is actually the standard deviation and gives the margins wi thin which we can be approximately 68% confidem 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, l3, 13, 7, 8, 6, 8, I l, 12, 8, 7. Find the mean and standard deviation of these (b) The latter should be approximately equal to the square root of the former. How well is this expectation borne out?
** In Chapter 3, you learned that in a counting experimen t, the uncerta inty 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
of the number 11 of random
approximately equal to the square root of this same best estimate; that is, the uncer tainty in each measurement is .,/f,. In particular, if we make only one count 11, the best estimate is just 11 and the uncertainty is the square root --fi,; this result is just
the square-root rule of Chapter 3 with the additional in formation that the "uncer tainty" is actually the standard deviation and gives the margins wi thin which we can be approximately 68% confidem 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,
l3, 13, 7, 8, 6, 8, I l, 12, 8, 7.
- Find the mean and standard deviation of these (b) The latter should be approximately equal to the square root of the former. How well is this expectation borne out?
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