Question

According to Chebyshev's theorem, what percent of the values will fall between 100 and 124 for a data set that has a mean of 112 and a standard deviation of 4?

Answer 1

Answer:

More than 88.889% of the values will fall between 100 and 124 for the data set that has a mean of 112 and a standard deviation of 4.

Step-by-step explanation:

The Chebyshev's theorem states that the probability of any random variable ''X'' assuming a value between a range of ''k times'' the standard deviation is at least $1-\frac{1}{k^{2}}$

We can write mathematically this as :

P( μ - kσ < X < μ + kσ) $\geq 1-\frac{1}{k^{2}}$ (I)

Where μ is the mean and σ is the standard deviation.

In this exercise :

μ = 112

σ = 4

If we replace this values in the equation (I) :

$P(112-k(4)$

The percent of the values falling between 100 and 124 can be written as :

$P(100$ (II)

This probability must be equal to $P(112-k(4)$ (III)

Therefore if we work with (II) and (III) ⇒

(II) = (III) ⇒

$P(100$ ⇒

$100=112-4k$

$124=112+4k$

⇒ In any of the equations we find that

$12=4k$

$k=3$

Finally, we can write that

$P(112-3(4)$

$P(100$ ≅ 88.889%

According to Chebyshev's theorem, more than 88.889% of the values will fall between 100 and 124 for the data set.