In 1798 the English scientist Henry Cavendish measured the d

In 1798, the English scientist, Henry Cavendish, measured the density of the earth by careful work with a torsion balance. The variable recorded was the density of the earth as a multiple of the density of water. Here is the Cavendish\'s 29 measurements: 5.50, 5.61, 4.88, 5.07, 5.26, 5.55, 5.36, 5.29, 5.58, 5.65, 5.57, 5.53, 5.62, 5.29, 5.44, 5.34, 5.79, 5.10, 5.27, 5.39, 5.42, 5.47, 5.63, 5.34, 5.46, 5.30, 5.75, 5.68, 5.85 A) Determine the sample mean. B) Determine sample standard deviation C) Determine the median value of the data D) Is there any data that falls beyond the plus or minus 3a? If so, discard that data and recalculate both mean and standard deviation. E) Adopting a class size of 0.1 draw the histogram of the data. F) Considering the histogram, determine the mode of the data in terms of class G) If it is \'normally\' distributed data, determine the percentage of data that will be below 5.

Solution

a)

Getting the mean, X,          
          
X = Sum(x) / n          
Summing the items, Sum(x) =    157.99      
As n =    29      
Thus,          
X =    5.447931034   [ANSWER]

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B)

Consider:

Thus, Sum(x - X)^2 =    1.366875862  
      
Thus, as       
      
s^2 = Sum(x - X)^2 / (n - 1)      
      
As n =    29  
      
s^2 =    0.048816995  
      
Thus,      
      
s =    0.220945684   [ANSWER]

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c)

Ordering the data,

4.88
5.07
5.1
5.26
5.27
5.29
5.29
5.3
5.34
5.34
5.36
5.39
5.42
5.44
5.46
5.47
5.5
5.53
5.55
5.57
5.58
5.61
5.62
5.63
5.65
5.68
5.75
5.79
5.85

Thus, the median is the middle entry, the 15th entry,

Median = 5.46 [ANSWER]

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d)

Here,

u - 3*Sigma = 5.447931034 - 3*0.220945684 = 4.785093982

u + 3*sigma = 5.447931034 + 3*0.220945684 = 6.110768086

As we can see, there\'s no data beyond these numbers here.

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x	x - X	(x - X)^2
5.5	0.052069	0.002711
5.61	0.162069	0.026266
4.88	-0.56793	0.322546
5.07	-0.37793	0.142832
5.26	-0.18793	0.035318
5.55	0.102069	0.010418
5.36	-0.08793	0.007732
5.29	-0.15793	0.024942
5.58	0.132069	0.017442
5.65	0.202069	0.040832
5.57	0.122069	0.014901
5.53	0.082069	0.006735
5.62	0.172069	0.029608
5.29	-0.15793	0.024942
5.44	-0.00793	6.29E-05
5.34	-0.10793	0.011649
5.79	0.342069	0.117011
5.1	-0.34793	0.121056
5.27	-0.17793	0.031659
5.39	-0.05793	0.003356
5.42	-0.02793	0.00078
5.47	0.022069	0.000487
5.63	0.182069	0.033149
5.34	-0.10793	0.011649
5.46	0.012069	0.000146
5.3	-0.14793	0.021884
5.75	0.302069	0.091246
5.68	0.232069	0.053856
5.85	0.402069	0.161659