Inquiries to an interactive computer system at Rhode Block...

Inquiries to an interactive computer system at Rhode Block Security are of four types and make reference to different data bases as follows: $$ \begin{array}{c|c|c|c} \hline \begin{array}{c} \text { Inquiry } \\ \text { type } \end{array} & \begin{array}{c} \text { Percent of } \\ \text { type } \end{array} & \begin{array}{c} \text { Mean reference time } \\ \text { (msec) } \end{array} & \begin{array}{c} \text { Standard deviation } \\ \text { of reference } \\ \text { time(msec) } \end{array} \\ \hline \text { A } & 40 & 100 & 80 \\ \text { B } & 30 & 120 & 30 \\ \text { C } & 20 & 80 & 40 \\ \text { D } & 10 & 40 & 20 \\ \hline \end{array} $$ For the entire collection of inquiries, what is (a) the mean reference time? (b) the variance of reference time?

Inquiries to an interactive computer system at Rhode Block Security are of four types and make reference to different data bases as follows:
$$
\begin{array}{c|c|c|c}
\hline \begin{array}{c}
\text { Inquiry } \\
\text { type }
\end{array} & \begin{array}{c}
\text { Percent of } \\
\text { type }
\end{array} & \begin{array}{c}
\text { Mean reference time } \\
\text { (msec) }
\end{array} & \begin{array}{c}
\text { Standard deviation } \\
\text { of reference } \\
\text { time(msec) }
\end{array} \\
\hline \text { A } & 40 & 100 & 80 \\
\text { B } & 30 & 120 & 30 \\
\text { C } & 20 & 80 & 40 \\
\text { D } & 10 & 40 & 20 \\
\hline
\end{array}
$$
For the entire collection of inquiries, what is
(a) the mean reference time?
(b) the variance of reference time?

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Key Concepts

Weighted Mean

The weighted mean is a key concept in statistics used to compute an overall average from several groups, each having its own mean and proportional weight. In this context, each inquiry type has its own mean reference time and a specified percentage, and by multiplying each mean by its corresponding weight (percentage) and summing the results, the overall mean reference time is obtained.

Law of Total Variance

The law of total variance is a foundational concept used when dealing with mixtures of subpopulations. It states that the overall variance is the sum of the weighted variances of each group (i.e., the expected value of the squared deviations within each group) plus the weighted squared differences between each group’s mean and the overall mean. This decomposition allows the assessment of variability stemming from both within-group variations and differences between group means.

Variance in Mixture Distributions

Calculating the variance of a mixture distribution involves two components: the average of the individual variances weighted by their probabilities, and an adjustment factor that accounts for the dispersion of the group means around the overall mean. This approach helps in understanding how the variability of individual components contributes to the variability observed in the overall data set.

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