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?

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** 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 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?
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