Exam 3: Numerical Methods for Describing Data Distributions

The interquartile range is resistant to the effect of outliers.

Calculate the values of these numerical summaries for the walking task without the device: a) The mean _____________ b) The standard deviation _____________ c) The median _______________ d) The interquartile range ____________________ Floor ND 4.5 5.0 4.9 3.7 8.2 6.4 10.0 6.5 4.6 4.4 4.2 7.1 5.6 Floor D 4.3 4.7 5.5 3.9 7.8 6.5 7.3 6.5 4.3 4.0 4.3 5.9 6.0

If there are no outliers, a boxplot and modified boxplot can differ in the length of the box, but not in the whisker lengths.

Multiple sclerosis (MS) is an autoimmune disease that affects the brain and spinal cord. MS affects women more than men, and is most commonly diagnosed between the ages of 20 and 40. MS sufferers typically have problems of coordination and walking. The standard care for some sufferers is a plastic device affixed to the lower leg and ankle. In a recent test of this device, volunteers with MS were randomly assigned to two experimental groups. The volunteers in one group used the plastic device; the volunteers in the other group did not use the plastic device. Each individual was timed as he or she climbed a set of stairs. Times, measured without the device (ND) and with the device (D), are presented below. Stairs ND 9.3 8.4 8.6 6.9 17.7 12.4 18.6 14.2 7.9 6.9 8.4 9.4 12.7 Stairs D 7.9 8.2 9.0 7.2 15.3 13.5 16.6 15.7 7.7 7.5 8.4 9.5 12.7 -Calculate the values of these numerical summaries for the stair-climbing task without the device: a) The mean __________ b) The standard deviation __________ c) The median___________ d) The interquartile range ______________ Note: The table above is reproduced below so that you need not flip back and forth. Stairs ND 9.3 8.4 8.6 6.9 17.7 12.4 18.6 14.2 7.9 6.9 8.4 9.4 12.7 Stairs D 7.9 8.2 9.0 7.2 15.3 13.5 16.6 15.7 7.7 7.5 8.4 9.5 12.7

Approximately 68% of observations are within 2 standard deviations of the mean if the distribution is approximately normal.

Multiple sclerosis (MS) is an autoimmune disease that affects the brain and spinal cord. MS affects women more than men, and is most commonly diagnosed between the ages of 20 and 40. MS sufferers typically have problems of coordination and walking. The standard care for some sufferers is a plastic device affixed to the lower leg and ankle. In a recent test of this device, volunteers with MS were randomly assigned to two experimental groups. The volunteers in one group used the plastic device; the volunteers in the other group did not use the plastic device. Each individual was timed as he or she climbed a set of stairs. Times, measured without the device (ND) and with the device (D), are presented below. Stairs ND 9.3 8.4 8.6 6.9 17.7 12.4 18.6 14.2 7.9 6.9 8.4 9.4 12.7 Stairs D 7.9 8.2 9.0 7.2 15.3 13.5 16.6 15.7 7.7 7.5 8.4 9.5 12.7 -Does it appear from your graphs above that the device helps the MS sufferers in the stair-climbing test? Explain your reasoning in a few sentences.

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. First Year Swarm \# Number of Drones Number of Workers Second Year Swarm \# Number of Drones Number of Workers 1 598 597 1 355 339 2 24 21 2 691 673 3 567 557 3 278 252 4 371 325 4 719 669 5 279 260 5 152 129 6 44 35 6 156 140 7 42 41 7 41 35 8 126 107 9 108 101 10 79 64 -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 $0.3 \mathrm {~cm}$ . Palate Width - Grey Kangaroos 1\mid0=1.0=124 1.\mid 1\mid 1\mid 1\mid77 1*\mid89 2.\mid01111111111 2\mid222222222223333333333333 2\mid444444444444444455555555555555 2\mid6666666666667777777 2*\mid88888888888889999999999999 * \mid0000011 3\mid223 3\mid 3\mid 3*\mid (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?

The measurements in the table at right are from coins minted in Rome during first three centuries AD. Historians believe that different mints reveal themselves in different trace element profiles in the coins. Atomic Absorption Spectrometry was used to estimate the \% by weight of gold; these data are presented here. Roman Mint: Gold Content (\% by weight) 0.22 0.38 0.24 0.38 0.20 0.43 0.24 0.36 0.21 0.32 0.23 0.42 0.18 0.47 0.15 0.50 0.17 0.28 0.17 (a) Construct a box plot for these data. (b) Would you say this distribution is skewed or approximately symmetric? Justify your response using appropriate statistical terminology. the coins minted in Rome. Some of the summary statistics are given in the table on the right. (a) Describe a procedure that uses some or all o f these summary statistics to determine whether outliers are

The mean is the middle value of an ordered data set.

s is a measure of the variability of a population.

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. First Year Swarm \# Number of Drones Number of Workers Second Year Swarm \# Number of Drones Number of Workers 1 598 597 1 355 339 2 24 21 2 691 673 3 567 557 3 278 252 4 371 325 4 719 669 5 279 260 5 152 129 6 44 35 6 156 140 7 42 41 7 41 35 8 126 107 9 108 101 10 79 64 -Using your box plots from part (a), compare the distributions of first and second year drones. Justify your comparisons by appealing to specific aspects of the box plots constructed in Question 2.

A wide variety of oak trees grow in the United States. In one Acorn Statistics study, a sample of acorns was collected from different locations, and their volumes, in $m ^ { 3 }$ , were recorded. The table at right presents summary statistics for these data. Acorn Statistics Statistic Value 38 Mean 3.0 Median 1.8 St. Dev. 2.6 Minimum 0.3 Maximum 10.5 1st Quartile 1.1 3rd Quartile 4.3 (a) Describe a procedure that uses some or all of these summary statistics to determine whether outliers are present in the data. (b) Using your procedure from part (a), determine if there are outliers in these data. 3 Engineers initially surveyed the Territory of Iowa in the 1830 's, and they were very careful to take note of the trees and vegetation. The sample of hickory tree diameters from the original survey of what is now Linn County, is presented in the stem and leaf plot below. The display uses five lines for each stem. Thus, "1t|" is the stem for diameters of 12 and 13 , "1f|" for 14 and 15 , "1s|" for 16 and 17, and so on. (The "t" then stands for leaves that are twos and threes, the " $\mathrm { f }$ " for leaves of fours and fives, etc.) The mean diameter of the 73 hickory trees in this sample is $11.849$ inches, and the standard deviation is $3.995$ inches. Linn County Trees in 1830 Hickory Diameters 1\mid0=10 inches N=73 0.\mid 0t\mid 0f\mid45 0s\mid666666677 0*\mid88888888999 1.\mid00000000 1t\mid222222222222222 1f\mid4444444444445 1\mid666 1*\mid88888888888 2.\mid0 2\mid 2 2\mid 2*\mid (a) What is the approximate diameter of a hickory tree at the 20th percentile in this distribution? (b) According to the Empirical Rule, approximately 68% of hickory tree diameters are between what two values?