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

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

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

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

116.4

. "At least

75 \%

" means

75 \%

or more, and

93 \%

is more than

75 \%

, so Chebychev's rule does apply to these data.

Which Score Has a Higher Relative Position, a Score of 43.5

Multiple Choice

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