Data set, Chebyshev's Theorem

Question

Consider a data set that has a mean of 18.1 and a standard deviation of 6.4. According to Chebyshev's Theorem, give an interval that has at least 88.9% of the data.

__________ to ____________

Answer 1

Hi Eva,

Chebychev’s Inequality states:

P(mu-k*sigma) <= x <= mu + k*sigma >= 1-(1/k²)

We were given:

mu=18.1

sigma=6.4

We need k. To find it, we use the probability we were given, 88.9%, and convert to a decimal. This value is the right side of Chebychev’s Inequality:

1-(1/k²) = 0.889

Subtract 1 from both sides:

-(1/k²) = -0.111

Divide out the negative and multiply by k^2:

1=0.111k²

k²=9.00

k= +/- 3.00

Now that we know k=3, we can substitute in on the left side of the inequality and get our range:

[18.1-3(6.4)] <= x <= [18.1 + 3(6.4)]

-1.1 <= x <= 37.3, or

-1.1 to 37.3

I hope this helps.

Answer 2

Chebyshev’s Theorem Application

Chebyshev’s Theorem is a statistical rule that applies to any data set, regardless of its distribution shape. The theorem is particularly useful for estimating the minimum proportion of observations within a specified number of standard deviations from the mean.

Case Analysis

In the given data set, we have a mean of 18.1 and a standard deviation of 6.4. To find an interval that contains at least 88.9% of the data according to Chebyshev’s Theorem, we use the following steps:

Step 1: Calculate k

First, we calculate the value of k using the theorem's formula where 88.9% (0.889) of the data values fall within k standard deviations from the mean. The formula is: 

1 - 1/k² ≥ 0.889

Solving for k gives us a value, which in this case is approximately 3.00.

Step 2: Calculate the Interval

Using the calculated k value, we determine the interval by the formula: 

Interval = mean ± k × standard deviation

This results in an interval from approximately -1.11 to 37.31.

Conclusion

Therefore, according to Chebyshev’s Theorem, at least 88.9% of the data in this data set lies within the interval -1.11 to 37.31.