Question

A probability distribution has a mean of 30 and a standard deviation of 2. Use Chebychev's inequality to find a bound on the probability that an outcome of the experiment lies between the following. (Enter your answers to two decimal places.)
(a) 25 and 35

(b) 20 and 40

Answer 1

Chebyshev's inequality says that
\[P(|Y-\mu|<k\sigma)\ge1-\frac{1}{k^2}\]
where \(\mu\) is the mean of the distribution of \(Y\) and \(\sigma\) is the std. deviation.

For part (a), the desired interval is \(25<Y<35\), which can be written as
\[-5+30<Y<5+30\\
-5<Y-30<5\\
|Y-\mu|=|Y-30|<5\]
Finding \(k\) is easy enough:
\[|Y-\mu|<k\sigma~~\Rightarrow~~|Y-30|<2k=5~~\Rightarrow~~k=\frac{5}{2}\]
This means that the probability \(Y\) is less than \(2.5\sigma\) from \(\mu\) is
\[P\left(|Y-30|<2\cdot\frac{5}{2}\right)\ge\color{red}{1-\frac{1}{(5/2)^2}}=\cdots\]
The red part is your desired bound.

Answer 2

0.84

Answer 3

Right. Try to set up the second part. It's done very similarly.

Answer 4

I don't get it though... wouldn't it be the same??

Answer 5

not the same number of std. devs.

Answer 6

how many std devs from 20 to 30?

Answer 7

10?

Answer 8

no, how much is 1 std dev? think of it as a yard stick.

Answer 9

you still there?

Answer 10

hello sorry I am really confused Im looking in the book again

Answer 11

how much is 1 std dev? look in the problem statement.