Question

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. 20 to 60, at least ? %
   
2. 15 to 65, at least ? %
   
3. 31 to 49, at least ? %
   
4. 27 to 53, at least ? %
   
5. 24 to 56, at least ? %

Answer 1

Formulas: 1) If ơ is standard deviation of the random variable X then the variance of random variable X is denoted by var(X) 2) If E(X) is mean of the random variable X then Chebychev's inequality: (PGX-E(X) ε)2 1-var(X) Given that Mean(X) E(x) 40 Standard deviation(X) σ = 5 var(X) = σ2-25

Now we have to find the percentage of the given data within each of the following ranges using chebychev's theorem. a) P(20 X 60) = P(20-40 X-E(X) 60-40) = P(-20 X-E(X) 20) var(X) 21 21-_25 2120) (20)2 15 2 16 20.9375 P(20 X 60)-atleast 93.7590-9400

b) P(15 X 65) = P(15-40 X-E(X) 65-40) = P(-25 X-E(X) 25) -PCIX-E(X)| 25) var(X) 21- _25 (25)2 24 2 25 2 0.96 P(15 X 65) = atleast 96%

c) P(31 X 49) = P(31-40 X-E(X) 49-40) var(X) 21 25 (9)2 56 81 0.6914 15 X 65) = atleast 69.14% = 69%

d) P(27 X 53) = P(27-40 X-E(X) 53-40) - P(-13 sX - E(X) s 13) -PCIX-E(X)| 13) var(x) 21- _25 144 169 2 0.8521 (13)2 P(15 X 65)-atleast 85.21%-85%