I have two different problems involving Tchebysheff's Theorem. Hopefully there isn't a rule about asking two different questions in one post.(adsbygoogle = window.adsbygoogle || []).push({});

Number 1

1. The problem statement, all variables and given/known data

The US mint produces dimes with an average diameter of .5 inch and a standard deviation of .01. Using Tchebysheff's theorem, find a lower bound for the number of coins in a lot of 400 coins that are expected to have a diameter between .48 and .52.

2. Relevant equations

μ=.5

σ=.01

k= number of standard deviations from the mean= 2

Tchebysheff's theorem:

P(|Y-μ|<kσ)>= 1- [itex]\frac{1}{k^{2}}[/itex]

3. The attempt at a solution

Plugging everything in, I get P(|Y-μ|<(2)(.01))>= 1-[itex]\frac{1}{2^{2}}[/itex]

Simplifying, it becomes P(|Y-μ|<.02)>= .75

The lower bound of the probability of a coin being under two standard deviations from the mean is .75. What's throwing me is the lot of 400 coins. I don't know how to take the result of Tchebysheff's theorem and apply it to the sample. Or am I just overthinking it and I just need to multiply 400*.75?

Question 2

1. The problem statement, all variables and given/known data

Let Y be a random variable such that

p(-1)= [itex]\frac{1}{18}[/itex]

p(0)= [itex]\frac{16}{18}[/itex]

p(1)= [itex]\frac{1}{18}[/itex]

a) show that E(Y)=0 and V(Y)=[itex]\frac{1}{9}[/itex]

b) Use the probability distribution of Y to calculate P(|Y-μ|>=3σ). Compare this exact probability with the upper bound provided by Tchebysheff's theorem to see that the bound provided by Tchebysheff's theorem is actually attained when k = 3.

2. Relevant equations

Tchebysheff's Theorem

P(|Y-μ|<kσ)>=1-[itex]\frac{1}{k^{2}}[/itex] or P{|Y-μ|>=kσ)<=[itex]\frac{1}{k^{2}}[/itex]

3. The attempt at a solution

I thought I could just plug numbers in to find E(Y) and V(Y). However, in order to get E(Y)= 0, I'd need k=0, which gives me an undefined fraction. Instead, using p(0)=[itex]\frac{16}{18}[/itex], I found k= [itex]\frac{3\sqrt{2}}{4}[/itex]. But I don't know if this number is even useful.

Thanks in advance for any help I get.

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# Tchebysheff's Theorem questions (statistics)

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