Question

Consider the sample of exam scores to the​ right, arranged in increasing order. The sample mean and sample standard deviation of these exam scores​ are, respectively, 83.0 and 16.2. ​Chebychev's rule states that for any data set and any real number kgreater than​1, at least 100 left parenthesis 1 minus 1 divided by k squared right parenthesis​ % of the observations lie within k standard deviations to either side of the mean.

Sample-
28 52 57 60 63 73
76 78 81 82 86 87
88 88 89 89 90 91
91 92 92 93 93 93
93 95 96 97 98 99
Use​ Chebychev's rule to obtain a lower bound on the percentage of observations that lie within two standard deviations to either side of the mean.

Answer 1

Using Chebyshev's rule with k = 2, at least 75% of the observations in a data set will lie within two standard deviations to either side of the mean, regardless of the shape of the data's distribution.

Step-by-step explanation:

The student is asking for the application of Chebyshev's rule to determine a lower bound on the percentage of exam scores that fall within two standard deviations from the mean.

Chebyshev's rule is a statistical theorem that provides a way to estimate the minimum proportion of observations within a certain number of standard deviations from the mean, without making any assumptions about the underlying distribution of the data.

To calculate the lower percentage bound using Chebyshev's rule for k standard deviations from the mean:

Determine the value of k. In this case, k = 2.

Use the formula 1 - (1 / k2) to find the proportion (as a percentage) of values that fall within k standard deviations from the mean.

Substitute the value of k into the formula to calculate the lower bound.

For k = 2:

1 - (1 / 22) = 1 - (1 / 4) = 1 - 0.25 = 0.75 or 75%

Therefore, according to Chebyshev's rule, at least 75% of the observations fall within two standard deviations of the mean.