in the course of a couple of hours a nuclear engineer makes

in the course of a couple of hours, a nuclear engineer makes 12 measurements of the strength of a long-lived radioactive source with the following results in milicuries: 12, 34,22,14,22,17,24,22,18,14,18,12. a) what are his mean and standard deviation. b)according to chauvenet\'s criterion, would he be justified in rejecting the value 34 as a mistake? explain your reasoning only. c)if he does reject this measurement, what does he get for his new mean and standard deviation?

Solution

a)

Getting the mean, X,          
          
X = Sum(x) / n          
Summing the items, Sum(x) =    229      
As n =    12      
Thus,          
X =    19.08333333   [ANSWER, MEAN]  
          
Setting up tables,          
x   x - X   (x - X)^2  
12   -7.083333333   50.17361111  
34   14.91666667   222.5069444  
22   2.916666667   8.506944444  
14   -5.083333333   25.84027778  
22   2.916666667   8.506944444  
17   -2.083333333   4.340277778  
24   4.916666667   24.17361111  
22   2.916666667   8.506944444  
18   -1.083333333   1.173611111  
14   -5.083333333   25.84027778  
18   -1.083333333   1.173611111  
12   -7.083333333   50.17361111  
          
Thus, Sum(x - X)^2 =    430.9166667      
          
Thus, as           
          
s^2 = Sum(x - X)^2 / (n - 1)          
          
As n =    12      
          
s^2 =    39.17424242      
          
Thus,          
          
s =    6.25893301   [ANSWER, STANDARD DEVIATION]

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b)

Here, if x = 34,

z = (x-X)/s = (34-19.08333333)/6.25893301 = 2.383260317

As for n = 15 (Closest entry in the table), the critical value is 2.13, then as z > 2.13, 34 seems to be a mistake. [ANSWER]

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c)

Getting the mean, X,          
          
X = Sum(x) / n          
Summing the items, Sum(x) =    195      
As n =    11      
Thus,          
X =    17.72727273   [ANSWER, MEAN]  
          
Setting up tables,          
x   x - X   (x - X)^2  
12   -5.727272727   32.80165289  
22   4.272727273   18.25619835  
14   -3.727272727   13.89256198  
22   4.272727273   18.25619835  
17   -0.727272727   0.52892562  
24   6.272727273   39.34710744  
22   4.272727273   18.25619835  
18   0.272727273   0.074380165  
14   -3.727272727   13.89256198  
18   0.272727273   0.074380165  
12   -5.727272727   32.80165289  
          
Thus, Sum(x - X)^2 =    188.1818182      
          
Thus, as           
          
s^2 = Sum(x - X)^2 / (n - 1)          
          
As n =    11      
          
s^2 =    18.81818182      
          
Thus,          
          
s =    4.337992833   [ANSWER, STANDARD DEVIATION]