Chapter 3: Descriptive Measures

Objects such as asteroids and comets that come into proximity with the Earth are called near-Earth objects (NEOs). The National Aeronautics and Space Administration (NASA) tracks and catalogues all NEOs that are at least $1$kilometer wide. Data on NEOs can be found on the NASA website; The following table gives the relative velocities in kilometers per second , arranged in increasing order, for the NEO close approaches to the earth during June $2013$.

The sample mean and sample standard deviation of these velocities are $11.9$and $6.5$ respectively.
a. Compare the percentage of the observations that actually lie within two standard deviations to either side of the mean with that given by Chebyshev's rule with$k=2$

b. Repeat part (a) with$k=3$

c. Interpret your results from parts (a) and (b).

Early-Onset Dementia. Dementia is the loss of intellectual and social abilities severe enough to interfere with judgment, behavior, and daily functioning. Alzheimer's disease is the most common type of dementia. In the article "Living with Early Onset Dementia: Exploring the Experience and Developing Evidence-Based Guidelines for Practice" (Alsheimer's Care Quarterly), P. Harris and J. Keady explored the experience and struggles of people diagnosed with dementia and their families. The ages of a simple random sample of $21$people with early-onset dementia have a mean of $52.5$years and a standard deviation of $6.5$years.

a. construct a graph
b. Apply Chebyshev's rule with $k=2$ to make pertinent statements about the observations in the sample.
c. repeat part (b) with $k=3$

Iron is essential to most life forms and to normal human physiology. It is an integral part of many proteins and enzymes that maintain good health. Recommendations for iron are provided in Dietary Reference Intakes, developed by the Institute of Medicine of the National Academy of Sciences, The recommended dietary allowance (RDA) of iron for adult females under the age of $51$is $18$milligrams (mg) per day. The iron intakes during a $24$-hour period for a random sample of$45$ adult females under the age of $51$have a mean of $14.7mg$and standard deviation of $3.1mg$.

a. Construct a graph .
b. Apply Chebyshev's rule with $k=2$ to make pertinent statements about the observations in the sample.
c. Repeat part (b) with $k=3$.

Body mass index (BMI) is a measure of body fat based on height and weight. According to the document Dietary Guidelines for Americans published by the U.S. Department of Agriculture and the U.S. Department of Health and Human Services, for adults, a BMI of greater than $25$indicates an above healthy weight (i.e., overweight or obese). The BMIs of $75$randomly selected U.S. adults have a mean of $26.0$and a standard deviation of $5.0$

a. Construct a graph
b. Apply Chebyshev's rule with $k=2$ to make pertinent statements about the observations in the sample.
c. repeat part (b) with $k=3$

A study by researchers at the University of Maryland addressed the question of whether the mean body temperature of humans is $98.6^{°}F$The results of the study by P. Mackowiak et al. appeared in the article "A Critical Appraisal of $98.6°F$the Upper Limit of the Normal Body Temperature, and Other Legacies of Carl Reinhold August Wunderlich" (Journal of the American Medical Association ). Among other data, the researchers obtained the body temperatures of $93$healthy humans. The temperatures had a mean of $98.1°F$and a standard deviation of $0.65°F$

a. Construct a graph
b. Apply Chebyshev's rule with $k=2$ to make pertinent statements about the observations in the sample.
c. Repeat part (b) with $k=3$

The cheetah (Acinonyx jubatus) is the fastest land mammal and is highly specialized to run down prey. The cheetah often exceeds speeds of $60$miles per hour (mph) and. according to the online document "Cheetah Conservation in Southern Africa" by J. Urbaniak, the cheetah is capable of speeds up to $72mph$The following table gives the top speeds, in miles per hour, arranged in increasing order, for a sample of $35$cheetahs.

The sample mean and sample standard deviation of these speeds are $59.53mph$and $4.27mph$, respectively. A histogram of the speeds is bell shaped.
a. Is it reasonable to apply the empirical rule to estimate the percentages of observations that lie within one, two, and three standard deviations to either side of the mean?
b. Use the empirical rule to estimate the percentages of observations that lie within one, two, and three standard deviations to either side of the mean.
c. Use the data to obtain the exact percentages of observations that lie within one, two, and three standard deviations to either side of the mean.
d. Compare your answers in parts (b) and (c).

Malnutrition and Poverty. R. Reifen et al. studied various nutritional measures of Ethiopian school children and published their findings in the paper "Ethiopian-Born and Native Israeli School Children Have Different Growth Patterns" . The study, conducted in Azezo, North West Ethiopia, found that malnutrition is prevalent in primary and secondary school children because of economic poverty. The weights, in kilograms (kg), of $60$randomly selected male Ethiopian-born school children aged $12-15$ years old are presented in increasing order in the following table.
The sample mean and sample standard deviation of these weights are $45.30$kg and $4.16$kg, respectively.

a. Is it reasonable to apply the empirical rule to estimate the percentages of observations that lie within one, two, and three standard deviations to either side of the mean?
b. Use the empirical rule to estimate the percentages of observations that lie within one, two, and three standard deviations to either side of the mean.
c. Use the data to obtain the exact percentages of observations that lie within one, two, and three standard deviations to either side of the mean.
d. Compare your answers in parts (b) and (c).

We have provided simple data set for you to practices the basics of finding measures of center. For each data set determine the:

a) Mean

b)Median

c) Mode.

The given data set is:$4,2,0,2,2$.

Giant Tarantulas. One of the larger species of tarantulas is the Grammostola nollicoma, whose common name is the Brazilian giant tawny red. A tarantula has two body parts. The anterior part of the body is covered above by a shell, or carapace, F. Costa and F. Perez-Miles discussed the carapace length of the adult male G. mollicoma in the article "Reproductive Biology of Uruguayan Theraphosids" . The carapace lengths of a random sample of $50$adult male G. mollicoma have a mean of $18.14mm$and a standard deviation of $1.76mm$

a. Construct a graph
b. Apply Property $1$of the empirical rule to make pertinent statements about the observations in the sample.
c. Repeat part (b) for Property $2$of the empirical rule.
d. Repeat part (b) for Property $3$of the empirical rule.

Student-to-Faculty Ratio. A sample of fifth-grade classes was studied in the journal article "Predicting Feelings of School Safety for Lower, Middle, and Upper School Students" by R. Bachman et al. One of the variables collected was the class size in terms of student-to-faculty ratio. The student-to-faculty ratios of the $81$fifth grade classes sampled have a mean of $15.83$and a standard deviation of $1.74$

a. Construct a graph
b. Apply Property $1$ of the empirical rule to make pertinent statements about the observations in the sample.
c. Repeat part (b) for Property $2$ of the empirical rule.
d. Repeat part (b) for Property $3$ of the empirical rule.