Question

Consider the sample of scores to the​ right, arranged in increasing order. The sample mean and sample standard deviation of these scores​ are, respectively, 82.9 and 17.5. ​Chebychev's rule states that for any data set and any real number k greater than ​1, at least 100(1- 1 divided by k squared) % of the observations lie within k standard deviations to either side of the mean. Complete parts​ (a) and​ (b) below. a. Use​ Chebychev's rule to obtain a lower bound on the percentage of observations that lie within three standard deviations to either side of the mean. Determine k to be used in​ Chebychev's rule. k equals= Use k in​ Chebychev's rule to find the lower bound on the percentage of observations that lie within three standard deviations to either side of the mean.

Answer 1

(a)

Given:
Sample mean = = 82.9

Sample Standard Deviation = s = 17.5

Chebychev's rule states :

At least

                                (1)

of data from a sample must fall within k standard deviations from the mean.

To obtain a lower bound on the percentage of observations that lie within three standard deviations to either side of the mean

To determine k to be used in​ Chebychev's rule.:

k = 3

So,

value of k = 3

(b)

Use k in​ Chebychev's rule to find the lower bound on the percentage of observations that lie within three standard deviations to either side of the mean. :

Substituting k = 3 in (1), we get the lower bound on the percentage of observations that lie within three standard deviations to either side of the mean.as follows:

= 88.89%

So,

Answer is:

88.89%