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

question 125

Multiple Choice

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.

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Definitions:

Negatively Correlated

A relationship between two variables in which one variable increases as the other decreases.

Grades

Marks or scores assigned to a student's work to assess academic performance.

Hours

Units of time equivalent to 60 minutes, often used to measure the duration of events or the operational times of businesses and services.

Cause-and-effect

The principle that an action or event will produce a certain response or outcome in the natural or human-made world.