The Forces That Determine the Size of Groups ("Swarms") of Social

question 27

Essay

The forces that determine the size of groups ("swarms") of social insects and the rates at
which they grow are not well understood. Biologists have observed large variability in the size of swarms across species. In a study of the social wasp, Polybia occidentalis,
investigators dismantled a nest of these insects and marked a few insects for future
identification in new swarms. Twenty-five days after dismantling the original swarm they
had located new swarms of wasps from the original colonies. The data below present the
numbers of adults and workers in the new swarms in two different years. "Drones" are
fertile males that mate with the queen. "Workers" are infertile females that labor in the nest
and defend the colony. $\begin{array}{|c|r|r|r|r|r|}\hline \begin{array}{c}\text { First } \\\text { Year } \\\text { Swarm } \\\#\end{array} & \begin{array}{c}\text { Number } \\\text { of } \\\text { Drones }\end{array} & \begin{array}{c}\text { Number } \\\text { of } \\\text { Workers }\end{array} & \begin{array}{c}\text { Second } \\\text { Year } \\\text { Swarm } \\\#\end{array} & \begin{array}{c}\text { Number } \\\text { of } \\\text { Drones }\end{array} & \begin{array}{c}\text { Number } \\\text { of } \\\text { Workers }\end{array} \\\hline 1 & 598 & 597 & 1 & 355 & 339 \\\hline 2 & 24 & 21 & 2 & 691 & 673 \\\hline 3 & 567 & 557 & 3 & 278 & 252 \\\hline 4 & 371 & 325 & 4 & 719 & 669 \\\hline 5 & 279 & 260 & 5 & 152 & 129 \\\hline 6 & 44 & 35 & 6 & 156 & 140 \\\hline 7 & 42 & 41 & 7 & 41 & 35 \\\hline 8 & 126 & 107 & & & \\\hline 9 & 108 & 101 & & & \\\hline 10 & 79 & 64 & & & \\\hline\end{array}$
-Grey Kangaroos are large social marsupials, indigenous to Australia. As part of a study of these creatures, biologists measured various aspects of their skeletal structure. Data on palate width from a sample of 124 grey kangaroos are presented in the stem-and-leaf display below. The display uses five lines for each stem. Thus, $" 2 \mathrm { t } \mid "$ is the stem for palate widths of 22 and 23 , "2f|" for 24 and $25 , " 2 \mathrm {~s} \mid$ " for 26 and 27 , and so on. (The " $t$ " then stands for leaves that are twos and threes, the " " for $^ { \prime \prime }$ leaves of fours and fives, etc. " 2 ." Indicates leaves that are zeros and ones; " $2 * "$ indicates leaves that are eights and nines.) The mean palate width of this sample is $2.6$ $\mathrm { cm }$ , and the standard deviation is $0.3 \mathrm {~cm}$ .
$\begin{array} { l } \text { Palate Width - Grey Kangaroos } \\1 \mid 0 = 1.0 \mathrm {~cm} \quad \mathrm {~N} = 124 \\1 . \mid \\1 \mathrm { t } \mid \\1 \mathrm { f } \mid \\1 \mathrm {~s} \mid 77 \\1 * \mid 89 \\2 . \mid 01111111111 \\2 \mathrm { t } \mid 222222222223333333333333 \\2 \mathrm { f } \mid 444444444444444455555555555555 \\2 \mathrm {~s} \mid 6666666666667777777 \\2 * \mid 88888888888889999999999999 \\\text { * } \mid 0000011 \\3 \mathrm { t } \mid 223 \\3 \mathrm { f } \mid \\3 \mathrm {~s} \mid \\3 * \mid\end{array}$
(a) Approximately what percent of palate widths in this sample exceed 2.9 cm?
(b) What is the approximate percentile of a palate width that is 2.0 cm?

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An annuity in which the payment interval does not equal the compounding interval, and payments are made at the end of each payment interval.

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Payments made twice a year, often in the context of loans, investments, or insurance premiums.

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The interest rate used in calculating the present value of future cash flows.

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The monetary worth of an asset, service, or company based on its ability to generate income or its utility.