Question

Suppose the mean and the standard deviation of a distribution are as follows: population mean and standards deviation are 60 and 5, respectively. At least what proportion of the observations lie between 45 and 75?

Answer 1

Solution:

Given: the mean and the standard deviation of a distribution are as follows: population mean and standards deviation are 60 and 5, respectively.

That is: and

Since population distribution is unknown , we will use Chebeshev's theorem which states that: At least proportion of the data lies within k standard deviation from the mean.

That is:

Thus comparing lower and upper limits with given range of values, we get:

That is: k = 3

and

Thus 45 and 75 are k = 3 standard deviations from the mean.

Thus

Thus at least 0.8889 proportion of the observations lie between 45 and 75

or in % form we can say: at least 88.89% of the observations lie between 45 and 75