Question

recall that chebyshevs theorem states that for any set of data and for any constant k greater than 1, the proportion of the data that must lie within k standard deviations on either side of the mean is at least 1 - \\(\frac{1}{k^{2}}\\). therefore, for sample data with mean \\(\bar{x}\\) and standard deviation s, at least 1 - \\(\frac{1}{k^{2}}\\) of data must fall between \\(\bar{x}-ks\\) and \\(\bar{x}+ks\\). when k = 2, we have the following. 1 - \\(\frac{1}{k^{2}}\\)=1 - \\(\frac{1}{2^{2}}\\) = 1 - \\(\frac{1}{4}\\) = \\(\frac{3}{4}\\) or blank %. therefore, for any set of data, at least 75% of the data must fall between \\(\bar{x}-2s\\) and \\(\bar{x}+(blank)s, or, in other words, within blank standard deviations of the mean.

Explanation:

Step1: Substitute k = 2 into the formula

$1-\frac{1}{k^{2}}=1 - \frac{1}{2^{2}}=1-\frac{1}{4}$

Step2: Calculate the proportion

$1-\frac{1}{4}=\frac{3}{4}$

Step3: Convert to percentage

$\frac{3}{4}=75\%$

Step4: Fill in the blanks for the interval

For any set of data, at least 75% of the data must fall between $\bar{x}-2s$ and $\bar{x}+(2)s$, within 2 standard - deviations of the mean.

Answer:

The blanks should be filled as follows: $\frac{1}{4}$, 75, 2, 2