Consider the Following Sample of Exam Scores, Arranged in Increasing $\begin{array} { l l l l l l } 22 & 35 & 44 & 55 & 67 & 70 \end{array}$

Consider the following sample of exam scores, arranged in increasing order: $\begin{array} { l l l l l l } 22 & 35 & 44 & 55 & 67 & 70 \end{array}$
$\begin{array} { l l l l l l } 78 & 81 & 81 & 82 & 84 & 88 \end{array}$
$\begin{array} { l l l l l l } 88 & 89 & 90 & 90 & 91 & 92 \end{array}$
$\begin{array} { l l l l l l } 92 & 93 & 93 & 94 & 94 & 95 \\ 95 & 96 & 96 & 97 & 99 & 100 \end{array}$
$\begin{array} { l l l l l l } 95 & 96 & 96 & 97 & 99 & 100 \end{array}$
Note: $\bar { x } = 82.4$ and $s = 19.5$

A) (i) Use Chebychev's rule to obtain a lower bound on the percentage of observations that lie within 2 standard deviations to either side of the mean.
(ii) Use the data to obtain the exact percentage of observations that lie within $< b >$ standard deviations to either side of the mean. Compare this answer with part (i) .
(i) Chebychev's rules says that $75 \%$ of the data values will lie between $82.4 - 2 ( 19.5 )$ and $82.4 + 2 ( 19.5 )$ , or between $43.4$ and $121.4$ ;
(ii) 26 of the thirty observations (87\%) lie within the interval $43.4$ to $121.4$ . "At least $75 \%$ " means $75 \%$ or more, and $87 \%$ is more than $75 \%$ , so Chebychev's rule does apply to these data.
B) (i) Chebychev's rules says that $75 \%$ of the data values will lie between $82.4 - 2 ( 19.5 )$ and $82.4 + 2 ( 19.5 )$ , or between $40.4$ and $123.4$ ;
(ii) 27 of the thirty observations ( $90 \%$ ) lie within the interval $40.4$ to $123.4$ . "At least $75 \%$ " means $75 \%$ or more, and $90 \%$ is more than $75 \%$ , so Chebychev's rule does apply to these data.
C) (i) Chebychev's rules says that $75 \%$ of the data values will lie between $82.4 - 2 ( 19.5 )$ and $82.4 + 2 ( 19.5 )$ , or between $43.4$ and $121.4$ ;
(ii) 28 of the thirty observations ( $93 \%$ ) lie within the interval $43.4$ to $121.4$ . "At least $75 \% "$ means $75 \%$ or more, and $93 \%$ is more than $75 \%$ , so Chebychev's rule does apply to these data.
D) (i) Chebychev's rules says that $75 \%$ of the data values will lie between $82.4 - 2 ( 19.5 )$ and $82.4 + 2 ( 19.5 )$ , or between $53.4$ and $116.4$ ;
(ii) 28 of the thirty observations ( $93 \%$ ) lie within the interval $53.4$ to $116.4$ . "At least $75 \%$ " means $75 \%$ or more, and $93 \%$ is more than $75 \%$ , so Chebychev's rule does apply to these data.