Question

a collection of sample data has 753 values. we are promised _(a)_ _(b)_ of the data is within 2 standard deviations of the mean. (a) choose the correct choice for a word in blank (a) in the above statement. a. about b. at least c. at most d. exactly (b) choose the correct choice for a percentage in blank (b) in the above statement. a. 68% b. 75% c. 89% d. 95% e. 99.7%

Explanation:

Step1: Recall Chebyshev's theorem

Chebyshev's theorem states that for any data set (regardless of the shape of the distribution), at least $1-\frac{1}{k^{2}}$ of the data lies within $k$ standard - deviations of the mean. Here $k = 2$.

Step2: Calculate the proportion for $k = 2$

Substitute $k = 2$ into the formula $1-\frac{1}{k^{2}}$. We get $1-\frac{1}{2^{2}}=1 - \frac{1}{4}=\frac{3}{4}=0.75$ or 75%. Chebyshev's theorem gives a lower - bound, so we use the term "at least".

Answer:

(a) B. at least
(b) B. 75%