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Video Self-Test Tutorial Consider a sample with a mean of 40 and a standard deviation of...

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Self-Test Tutorial

Consider a sample with a mean of 40 and a standard deviation of 5. Use Chebyshev's theorem to determine the percentage of the data within each of the following ranges (to the nearest whole number).

  1. 30 to 50, at least _____ %
  2. 25 to 55, at least _____ %
  3. 31 to 49, at least _____ %
  4. 28 to 52, at least _____ %
  5. 22 to 58, at least _____ %

Solutions

Expert Solution

Chebyshev’s theorem states that for any set of data, the proportion of values that lie within k standard deviations (k>1) of the mean is at least (1-1/k2).      
a)      
(30-40)/5 =   -2  
(50-40)/5 =   2  
30-50 is within k=2 standard deviations from mean.      
1-1/k2 = 1-1/2^2 =       0.75
at least 75% of data lies between 30 and 50      
      
b)      
(25-40)/5 =   -3  
(55-40)/5 =   3  
25-55 is within k=3 standard deviations from mean.      
1-1/k2 = 1-1/3^2 =       0.888888889
at least 89% of data lies between 25 and 55      
      
c)      
(31-40)/5 =   -1.8  
(49-40)/5 =   1.8  
31-49 is within k=1.8 standard deviations from mean.      
1-1/k2 = 1-1/1.8^2 =       0.691358025
at least 69% of data lies between 31 and 49      
      
d)      
(28-40)/5 =   -2.4  
(52-40)/5 =   2.4  
28-52 is within k=2.4 standard deviations from mean.      
1-1/k2 = 1-1/2.4^2 =       0.826388889
at least 83% of data lies between 28 and 52      
      
e)      
(22-40)/5 =   -3.6  
(58-40)/5 =   3.6  
22-58 is within k=3.6 standard deviations from mean.      
1-1/k2 = 1-1/3.6^2 =       0.922839506
at least 92% of data lies between 22 and 58     


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