I am trying to teach myself statistics and am reading DeGroot's book. Anyways, I'm in Chebychev's Inequality and am confused by this example:
I understand the second part about if we know the distribution beforehand, we get a smaller required equal to 15. However, I am unable to replicate the first part based on the definition of Chebychev's inequality.
I tried the following. Pr(0.4 <= Xbar <= 0.6) should be the same thing as Pr(abs(Xbar - μ) <= 0.1) and so:
Pr(abs(Xbar - μ) >= 0.1) <= σ^2 / (n * 0.1^2)
We know the variance of the distribution is n * (0.5) * (1 - 0.5) since its binomial. However, when I plug that variance in, obviously the ns cancel out and we are unable to find the necessary sample size. I am not sure what I am doing wrong. I also don't understand his explanation and why he approaches the problem the way he does. I understand the left hand side of his answer and realize he plugs in T/n as the sample mean. I don't get how he got the right hand side.