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Statistics in Context

Barbara Blatchley

Chapter 5

VARIABILITY: THE “LAW OF LIFE” - all with Video Answers

Educators


Chapter Questions

07:32

Problem 1

Evaluate the data sets shown in Table 5.4, and make a rough guess about the amount
of variability (a great deal of variability, a moderate amount, or very little) in each set.
Then, calculate the actual range, variance, and standard deviation of the data. Were
your estimates of the amount of variability accurate?
Set A
$$
\begin{array}{|r|r|}
\hline 1 & 5.36 \\
\hline 2 & 8.20 \\
\hline 3 & 6.99 \\
\hline 4 & 5.29 \\
\hline 5 & 6.15 \\
\hline 6 & 6.54 \\
\hline 7 & 8.43 \\
\hline 8 & 8.00 \\
\hline 9 & 9.68 \\
\hline 10 & 8.50 \\
\hline 11 & 6.40 \\
\hline 12 & 9.75 \\
\hline 13 & 5.15 \\
\hline 14 & 8.87 \\
\hline 15 & 7.55 \\
\hline
\end{array}
$$
Set B
$$
\begin{array}{|c|c|}
\hline \text { ID } & x \\
\hline 1 & 7.99 \\
\hline 2 & 7.97 \\
\hline 3 & 7.07 \\
\hline 4 & 8.01 \\
\hline 5 & 8.03 \\
\hline 6 & 7.99 \\
\hline 7 & 8.00 \\
\hline 8 & 8.00 \\
\hline 9 & 7.99 \\
\hline 10 & 8.00 \\
\hline 11 & 7.06 \\
\hline 12 & 8.00 \\
\hline 13 & 8.09 \\
\hline 14 & 7.92 \\
\hline 15 & 8.12 \\
\hline
\end{array}
$$
Set C
$$
\begin{array}{|r|r|}
\hline 10 & x \\
\hline 1 & 13.13 \\
\hline 2 & 2.00 \\
\hline 3 & 4.87 \\
\hline 4 & 7.43 \\
\hline 5 & 1.59 \\
\hline 6 & 14.39 \\
\hline 7 & 10.70 \\
\hline 8 & 8.00 \\
\hline 9 & -0.34 \\
\hline 10 & 10.84 \\
\hline 11 & 6.10 \\
\hline 12 & 15.24 \\
\hline 13 & 18.22 \\
\hline 14 & 3.93 \\
\hline 15 & 5.22 \\
\hline
\end{array}
$$

Matthias Wuest
Matthias Wuest
Numerade Educator
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Problem 2

Body size (including body weight) has been linked to reproductive success and to
levels of aggression in feral cats (Natoli, Schmid, Say, & Pontier, 2007). The authors
of that study report that a large body size leads to a high rank in the dominance hierarchy and so greater access to reproductive partners. Male cats were described by
Natoli et al. as weighing approximately 20% more than females, and female domestic
shorthair cats usually weigh between 7 and 9 pounds. Find the mean, median, mode,
range, IQR, variance, and standard deviation for the data shown below. Based on
the descriptive statistics you’ve calculated, how many male animals are likely in this
sample?
$$
\begin{array}{|c|c|}
\hline 10 & \text { Weight } \\
\hline 1 & 9.9 \\
\hline 2 & 7 \\
\hline 3 & 10.2 \\
\hline 4 & 10.8 \\
\hline 5 & 6.3 \\
\hline 6 & 9.1 \\
\hline 7 & 9 \\
\hline 8 & 6.3 \\
\hline 9 & 9.1 \\
\hline 10 & 9.1 \\
\hline 11 & 11.9 \\
\hline 12 & 11 \\
\hline 13 & 10.2 \\
\hline 14 & 12.7 \\
\hline 15 & 11.1 \\
\hline 16 & 7 \\
\hline 17 & 8.2 \\
\hline 18 & 7 \\
\hline 19 & 8.1 \\
\hline 20 & 8 \\
\hline
\end{array}
$$

Bryan Meares
Bryan Meares
Numerade Educator
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Problem 3

According to Wikipedia.org, the smallest cat ever officially weighed and measured is a
domestic shorthair named “Mr. Peebles,” who weighs only 3 pounds and stands only
6 inches high (Mr. Peebles apparently has an unspecied genetic defect that has
caused him to stay so small). Compare Mr. Peebles’ body weight to the mean you
obtained from our set of 20 cats in question 2. Where is Mr. Peebles’ weight in the
distribution in Table 5.5? How far from the mean is his body weight?

Bryan Meares
Bryan Meares
Numerade Educator
02:42

Problem 4

The largest cat ever officially weighed and measured was a Tabby named “Himmie,”
who died of respiratory failure at the age of 10 (Himmie was apparently just plain overweight). Himmie weighed an astonishing 46 pounds, 15.2 ounces. Compare Himmie’s
weight to the distribution you’ve described in question 2. How far above average
weight was he? (Just FYI: Guinness World Records is no longer accepting nominations
for heaviest pet in an effort to keep pet owners from overfeeding their animals.)

Erika Bustos
Erika Bustos
Numerade Educator
08:09

Problem 5

In a study about the effects of olfactory system stimulation on the ability to concentrate at a visual search task, subjects are asked to complete a visual search task while
smelling either an unpleasant or a pleasant odor. The time needed to complete the
visual search task (in seconds) is recorded. Table 5.6 shows this hypothetical data.
Calculate the mean, median, mode, and standard deviation of the data. Then, calculate the IQR for the data, and create a box-and-whisker plot.
table cant copy

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
03:16

Problem 6

Several studies (Keinan, 2002, for example) have found evidence that belief in magic and luck increased as stress levels increased. The more stressed people were, the more likely they were to engage in magical thinking. Suppose you did your own study examining the relationship between test anxiety and belief in the power of a lucky charm. You ask one group of students to bring their own personal lucky charm to their statistics final exam. Another group is the control group and has to take their statistics exam without any lucky objects present during the test.

$$
\begin{array}{|c|c|}
\hline \text { Without Lucky Charm } & \text { With Lucky Charm } \\
\hline 82 & 64 \\
\hline 95 & 38 \\
\hline 68 & 100 \\
\hline 75 & 42 \\
\hline 70 & 77 \\
\hline 70 & 100 \\
\hline 79 & 97 \\
\hline 79 & 99 \\
\hline 76 & 95 \\
\hline
\end{array}
$$
a. Calculate the mean for each group.
b. Calculate the standard deviation for each group.
c. In your opinion, does having a lucky charm result in better performance on the
exam? How do the two groups differ? How are they similar?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:02

Problem 7

Match the histograms of the distributions on the next page with their descriptive
statistics below.
$$
\begin{array}{l|c|c|c|c}
& \text { A } & \text { B } & \text { C } & \text { D } \\
\hline \text { Mean } & 7.1579 & 5.4878 & 5.9535 & 3.3 \\
\hline \text { Median } & 8.00 & 5.00 & 6.00 & 3.00 \\
\hline \text { Mode } & 10.00 & 5.00 & 5.00 & 1.00 \\
\hline \text { Standard deviation } & 2.47 & 2.29 & 2.44 & 2.34 \\
\hline \text { Skewness } & -0.68 & +0.20 & -0.016 & +0.95 \\
\hline
\end{array}
$$
What would the skewness measurement be for a distribution that is perfectly normal in
shape?

Trinity Steen
Trinity Steen
Numerade Educator
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Problem 8

Suppose you were a veterinarian responsible for the health of the animals in a laboratory colony. One very important clue to health in any animal (including humans) is
body weight. To help you maintain the health of the animals in your care, you need to
know what a typical laboratory animal weighs (in this example, the albino SpragueDawley lab rat, which is very commonly used for lab research because they are calm
and easy to handle). The body weights for a set of 30 adult, albino Sprague-Dawley laboratory rats (15 males and 15 females) are shown in Table 5.8. Find the mean, median,
mode, range, variance, and standard deviation for this set of animals.
$$
\begin{array}{c|c}
\hline \text { Females } & \text { Males } \\
\hline 229 & 510 \\
\hline 300 & 485 \\
\hline 310 & 476 \\
\hline 248 & 528 \\
\hline 304 & 448 \\
\hline 286 & 498 \\
\hline
\end{array}
$$
$$
\begin{array}{|l|l|}
\hline 291 & 468 \\
\hline 284 & 575 \\
\hline 291 & 489 \\
\hline 279 & 529 \\
\hline 289 & 411 \\
\hline 285 & 453 \\
\hline 247 & 518 \\
\hline 282 & 441 \\
\hline 248 & 466 \\
\hline
\end{array}
$$
Once you’ve found these statistics (our measure of typical), answer the following
questions:
a. In your newest shipment of rats, they’ve forgotten to tell you the sex of each animal,
and all you have to go on are the body weights shown in the chart below. Using only
the body weights provided, determine if the animals are probably male or probably
female.
b. Were there any rats that you were unable to make a prediction for? If so, why did
you have difficulty?
c. Now that we know what “typical” is, speculate how you might determine what atypical would be. How far away from the mean would an unhealthy animal (too fat or
too thin) be?
$$
\begin{array}{c|c|c|c}
\hline \begin{array}{c}
\text { Body Weight (in } \\
\text { Grams) }
\end{array} & \begin{array}{c}
\text { Check Here if Rat is } \\
\text { Probably FEMALE }
\end{array} & \begin{array}{c}
\text { Check Here if Rat is } \\
\text { Probably MALE }
\end{array} & \begin{array}{c}
\text { Check Here if you can't make a } \\
\text { Prediction About the Sex of the Animal } \\
\text { Based on Body Weight Alone }
\end{array} \\
\hline 249 & & & \\
\hline 500 & & & \\
\hline 390 & & & \\
\hline 400 & & & \\
\hline 290 & & & \\
\hline
\end{array}
$$

Shu Naito
Shu Naito
Numerade Educator

Problem 9

A relatively new area of research in personality theory focuses on how brain function
is related to personality. For example, Bickart, Wright, Dautoff, Dickerson, and Barrett
(2011) examined the size of the amygdala and related it to the size of a person’s social
network. The amygdala is a structure in the medial (toward the center of your brain)
temporal lobe (the part of cortex that is under your temples). It is part of the limbic
system, which controls emotional response. Bickart et al. found that people with larger amygdalae (plural) tended to also have a larger social network (measured as the total
number of social contacts a person maintains). The data in Table 5.9 are representative
of what Bickart et al. found. Using the descriptive statistics we’ve discussed so far, describe the volume of the amygdala and the size of the social network for each of the 20
participants in this hypothetical study.
table cant copy

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Problem 10

The “beauty is good” effect says that physically attractive individuals are perceived as
having positive qualities. In a study of this effect, Swami et al. (2010) asked collegeeducated men to rate the relative attractiveness of a series of photographs of women.
The women in the photographs were classied by Body Mass Index (BMI) into the
following categories: emaciated (BMI rating of 1 or 2), underweight (BMI rating of 3
or 4), normal weight (BMI rating of 5 or 6), overweight (BMI rating of 7 or 8), or obese (BMI rating of 9 or 10). The men were given vignettes to read that allegedly described
the personalities of the women they were being asked to rate. Some of the participants read vignettes that described positive personality traits, some read stories that
described negative personalities, and a nal group (the control group) did not receive
any information about the personalities of the women they were rating. All of the
participants were asked to rate the most attractive photograph on a 10-point scale where
1 = lowest BMI and 10 = highest BMI. The data in Table 5.10 are similar to that
obtained by Swami et al. Calculate the mean, median, mode, and standard deviation
for these data. Did the apparent personality of the women pictured affect the ratings
of attractiveness offered by the male participants in the study?
$$
\begin{array}{c|c|c}
\hline \text { Negative Vignette } & \text { Positive Vignette } & \text { Control Group } \\
\hline 3.94 & 3.39 & 5.97 \\
\hline 2.00 & 3.62 & 4.23 \\
\hline 3.56 & 3.75 & 3.75 \\
\hline 2.35 & 3.63 & 3.75 \\
\hline 3.34 & 4.23 & 3.74 \\
\hline 3.17 & 3.79 & 5.20 \\
\hline 2.61 & 5.00 & 4.74 \\
\hline 4.42 & 4.21 & 3.85 \\
\hline 4.74 & 3.22 & 2.66 \\
\hline 2.63 & 3.59 & 3.73 \\
\hline 2.88 & 4.67 & 4.75 \\
\hline 2.73 & 3.92 & 3.33 \\
\hline 3.40 & 3.22 & 4.50 \\
\hline 4.07 & 4.71 & 4.72 \\
\hline 3.88 & 4.75 & 4.35 \\
\hline
\end{array}
$$

Rashmi Sinha
Rashmi Sinha
Numerade Educator
05:29

Problem 11

Five groups of adolescents took the Stanford–Binet IQ test. The distribution of test
scores for each group (n = 25 in each group, N = 125) are shown in Figure 5.5. The
solid horizontal line shows the average score on the test (μ = 100), and the two
dashed horizontal lines indicate the standard deviation of scores on the test
( = 15). The dashed line above the mean indicates a score of 115 (the mean plus one
standard deviation), and the dashed line below the mean indicates a score of 75 (the
mean minus one standard deviation). Describe the IQ scores in the ve groups.

Trent Speier
Trent Speier
Numerade Educator
02:26

Problem 12

. Find the answer to the riddle below.
Riddle: One day there was a re in a wastebasket in the office of the Dean of Students. In
rushed a physicist, a chemist, and a statistician. The physicist immediately starts to work
on how much energy would have to be removed from the re to stop the combustion.
The chemist works on which reagent would have to be added to the re to prevent oxidation. While they are doing this, the statistician is setting res to all the other wastebaskets in the office. “What are you doing?” the others demand. The statistician replies,
“Well, to solve the problem, ______ ______ ______ ______ ______ ______ ______.”
Solution to the riddle: Notice that there are seven (7) words in the solution to the
riddle. You’ll nd the words in the list below. Each of the words in the solution section is associated with a number. To determine which word you need, solve the seven
problems in the following problem section. The word next to the answer to the rst
problem is the rst word in the solution to the riddle. The word next to the answer to
the second problem is the second word in the solution, and so on. Round all answers
to two decimal places (e.g., 2.645 rounds to 2.65).
Solution to the riddle: Words
$$
\begin{array}{lllll}
\text { (3.00) we } & (7.00) \text { must } & (12.00) \text { you } & \text { (626) deviate } & \text { (7.67) obviously } \\
\text { (8) wonder } & (69.0) \text { a } & (623) \text { larger } & \text { (57) standard } & \text { (1.02) skewed } \\
\text { (100) titrate } & \text { (9) need } & (10.44) \text { sample } & \text { (3.04) population } & \text { (3.23) size }
\end{array}
$$
Problems section: Solve each problem to find a number. Match the number to a word in the word section to finish the riddle.
$\begin{array}{llllllllll}\text { Data Set: } & 3 & 5 & 6 & 7 & 7 & 7 & 9 & 10 & 15\end{array}$
1. The range of the data set is $\qquad$
2. The mean of the data set is $\qquad$
3. $n=$ $\qquad$
4. The sum of all the values in the data set $(\Sigma x)$ is $\qquad$ .
5. The sum of all the values squared $\left(\Sigma x^2\right)$ is $\qquad$ .
6. The variance is $\qquad$
7. The standard deviation is $\qquad$ .

Sheryl Ezze
Sheryl Ezze
Numerade Educator
07:32

Problem 13

Consider the following three data sets:

$$
\begin{array}{|c|c|c}
\hline \text { A } & \text { B } & \text { C } \\
\hline 9 & 10 & 1 \\
\hline 10 & 10 & 1 \\
\hline 11 & 10 & 10 \\
\hline 7 & 10 & 19 \\
\hline 13 & 10 & 20 \\
\hline
\end{array}
$$
a. Calculate the mean of each data set.
b. Calculate the range and the IQR for each data set.
c. Calculate the variance and the standard deviation of each data set.
d. Which data set has the smallest standard deviation?
e. Use the range rule to determine the standard deviation of each data set.
f. Compare the standard deviation derived using the range rule with the standard deviation you computed using the formula. How different are they?

Matthias Wuest
Matthias Wuest
Numerade Educator
00:28

Problem 14

Is it possible for a data set to have negative standard deviation? Why or why not?

Nick Johnson
Nick Johnson
Numerade Educator
01:40

Problem 15

In the Standard Deviation in Context section, we discussed the mean absolute deviation from the mean (abbreviated MD), which proposed an alternate method of calculating the variability in a sample:

$$
\left(M D=\frac{\sum(|x-\bar{X}|)}{n}\right)
$$

Let's play with this a bit. The mean, the median, and the mode are all "averages," and they are all measures of center. Can you use another measure of central tendency as your anchor point when you're calculating the standard deviation? Suppose we calculated the mean absolute deviation from the median instead of the mean. What would happen to our measure of variability? Use the data shown in Table 5.10a to answer the following questions:
a. Calculate both the standard deviation (using the "traditional" formula) and the mean absolute deviation (using the formula for MD shown above) using the mean of the data set. Record your answers in Table 5.10b.
b. Calculate both the standard deviation and mean absolute deviation using the median of the data set. Record your answers in Table 5.10b.
c. How different are the measures of variability calculated using the mean from those
calculated using the median?
d. Now try calculating both SD and MD using the mode as your center point. Record
your answers in Table 5.10b.
e. Compare the results you got using the three measures of center to calculate SD and
MD. Which measure of center gave you the smallest measure of variability?
$$
\begin{gathered}
x \\
2 \\
\hline 2 \\
\hline 3 \\
\hline 4 \\
\hline 14 \\
\hline
\end{gathered}
$$
$$
\begin{array}{l|l}
\text { Using the mean } & M D= \\
S D= & M D= \\
\hline \begin{array}{l}
\text { Using the median } \\
S D=
\end{array} & M D= \\
\hline \begin{array}{l}
\text { Using the mode } \\
S D=
\end{array} \\
\hline
\end{array}
$$

Tyler Moulton
Tyler Moulton
Numerade Educator
03:02

Problem 16

. The distribution in Table 5.10a is skewed (in fact, it’s positively skewed, with most of
the observations at the low end of the distribution and one very high outlier). Given
what you know about how skew in the distribution affects the measures of center, do
you think you would get different values for SD and MD using the three different measures of center (as you did in question 15) if the distribution we were working with
was perfectly normal in shape? Why or why not?

Trinity Steen
Trinity Steen
Numerade Educator
05:17

Problem 17

Consider the data shown in Table 5.11.
$$
\begin{array}{c|c|c}
\hline x & f & x+10 \\
\hline 25 & 1 & \\
\hline 33 & 1 & \\
\hline 41 & 1 & \\
\hline 42 & 2 & \\
\hline 43 & 4 & \\
\hline
\end{array}
$$$$
\begin{array}{l|l|l}
\hline 45 & 2 & \\
\hline 52 & 1 & \\
\hline 54 & 1 & \\
\hline 65 & 1 & \\
\hline
\end{array}
$$
a. Calculate the mean and standard deviation of the data shown in Table 5.11.
b. Now add a constant of 10 to each observation in the data set, and record your answers in the column labeled " $x+10$."
c. WITHOUT RECALCULATING, what will the mean of the data in column " $x+10$ " be? (Check the box next to the best answer.)
The same mean as before a constant was added to each data point.
The original mean +10 .
Unable to determine without recalculation.
d. WITHOUT RECALCULATING, what will the standard deviation of the data in column " $x+10$ " be? (Check the box next to the best answer.)
The same standard deviation as before a constant was added to each data point.
The standard deviation before the constant was added +10 .
Unable to determine without recalculation.
e. In your own words, define the mean and the standard deviation of a set of observations. Using these definitions, describe the effect of adding a constant to the mean and standard deviation. Did adding a constant affect one of these descriptive statistics and not the other? Why?

James Kiss
James Kiss
Numerade Educator
00:37

Problem 18

You and your roommate are embroiled in a spirited discussion about statistics. Your roommate says that the size of the standard deviation depends on the value (size) of the mean. Select a statement from the list of possible answers below, and defend it to your roomie.
a. Yes! The higher the mean, the higher the standard deviation will be.
b. Yes! You have to know the mean to calculate the standard deviation.
c. No! Standard deviation measures the average deviation of scores from the mean no matter what the mean is.
d. No! The standard deviation measures only how values differ from each other, not how they differ from the mean.

Deborah Ferry
Deborah Ferry
Numerade Educator
02:12

Problem 19

A paperboy wants to know all about the dogs in the neighborhoods on his newspaper delivery route-after all, forewarned is forearmed. He asks the dog owners for the weights of the dogs on his route. He then calculates the mean and standard deviation for the sample of 10 dogs he successfully obtained weights for. His standard deviation is extremely high. What do you know about the distribution of weights in this sample? Indicate whether the following statements are true (T) or false ( F ):
T F a. All of the dogs are very big (they weigh a lot). Their high weights are pulling the standard deviation upward.
T F b. The weights of the dogs may vary a great deal. Some are small dogs, and some are very large, making the standard deviation quite large.
T F c. A few dogs in the sample could be quite large. These outliers might be creating positive skew in the data.
T F d. A few dogs in the sample could be quite small. These outliers might be creating positive skew in the data.

James Kiss
James Kiss
Numerade Educator
01:45

Problem 20

Motor control researchers were interested in seeing how the performance of a welllearned task was altered by small perturbations. Perturbations are deviations from normal movement caused by an external source. In the case of this experiment, five participants were asked to point at a target. They performed six repetitions of the task. During three of their repetitions, a mechanical device bumped their arm. The distance from the target (error) was measured in millimeters. The data each participant's unperturbed (control) and perturbed repetitions are shown in Table 5.12.
$$
\begin{array}{ccc|ccc}
\hline \text { ID } & \text { Condition } & \text { Error } & \text { ID } & \text { Condition } & \text { Error } \\
\hline 1 & \text { Control } & 2.8 & 1 & \text { Perturbed } & 4.3 \\
\hline 1 & \text { Control } & 1.7 & 1 & \text { Perturbed } & 4.0 \\
\hline 1 & \text { Control } & 3.8 & 1 & \text { Perturbed } & 3.7 \\
\hline 2 & \text { Control } & 4.0 & 2 & \text { Perturbed } & 4.5 \\
\hline 2 & \text { Control } & 3.4 & 2 & \text { Perturbed } & 3.0 \\
\hline 2 & \text { Control } & 2.3 & 2 & \text { Perturbed } & 2.7 \\
\hline 3 & \text { Control } & 3.8 & 3 & \text { Perturbed } & 3.9 \\
\hline 3 & \text { Control } & 1.5 & 3 & \text { Perturbed } & 2.7 \\
\hline 3 & \text { Control } & 3.4 & 3 & \text { Perturbed } & 4.5 \\
\hline 4 & \text { Control } & 2.5 & 4 & \text { Perturbed } & 2.2 \\
\hline 4 & \text { Control } & 1.8 & 4 & \text { Perturbed } & 3.7 \\
\hline 4 & \text { Control } & 3.0 & 4 & \text { Perturbed } & 1.7 \\
\hline 5 & \text { Control } & 2.7 & 5 & \text { Perturbed } & 4.4 \\
\hline 5 & \text { Control } & 3.1 & 5 & \text { Perturbed } & 1.7 \\
\hline 5 & \text { Control } & 1.6 & 5 & \text { Perturbed } & 2.9 \\
\hline
\end{array}
$$
a. Why did the researchers have the participants perform both tasks three times each?
b. Calculate the mean, standard deviation, and variance for the tasks performed under
the control and perturbed conditions.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:51

Problem 21

You’ve been hired by a local professional baseball team to help the manager select the
pitchers for an upcoming series. The team has provided you with some useful data: the earned run averages (ERAs) of three different pitchers against the 10 teams in
the league. Earned run average is the average number of runs a pitcher gives up in a
nine-inning game, so lower is better.
$$
\begin{array}{ccccccccc}
\hline \text { Pitcher } & \text { Opponent } & \text { ERA } & \text { Pitcher } & \text { Opponent } & \text { ERA } & \text { Pitcher } & \text { Opponent } & \text { ERA } \\
\hline 1 & \text { Team A } & 3.87 & 2 & \text { Team A } & 2.63 & 3 & \text { Team A } & 1.45 \\
\hline 1 & \text { Team B } & 4.11 & 2 & \text { Team B } & 4.59 & 3 & \text { Team B } & 1.86 \\
\hline 1 & \text { Team C } & 4.08 & 2 & \text { Team C } & 2.68 & 3 & \text { Team C } & 1.9 \\
\hline 1 & \text { Team D } & 3.9 & 2 & \text { Team D } & 6.19 & 3 & \text { Team D } & 1.82 \\
\hline 1 & \text { Team E } & 4.08 & 2 & \text { Team E } & 4.49 & 3 & \text { Team E } & 1.55 \\
\hline 1 & \text { Team F } & 3.92 & 2 & \text { Team F } & 2.11 & 3 & \text { Team F } & 1.35 \\
\hline 1 & \text { Team G } & 3.70 & 2 & \text { Team G } & 6.51 & 3 & \text { Team G } & 9.02 \\
\hline 1 & \text { Team H } & 4.21 & 2 & \text { Team H } & 2.69 & 3 & \text { Team H } & 2.08 \\
\hline 1 & \text { Team I } & 4.28 & 2 & \text { Team I } & 1.71 & 3 & \text { Team I } & 2.02 \\
\hline 1 & \text { Team J } & 4.28 & 2 & \text { Team J } & 7.40 & 3 & \text { Team J } & 1.93 \\
\hline
\end{array}
$$
a. Find the mean ERA and standard deviation for each pitcher. Which pitcher is best,
based on ERA?
b. Which pitcher is the most consistent?
c. Which pitcher is the least reliable?
d. If your team were playing the nal game of the championship series against Team G,
which pitcher would you NOT start?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
10:03

Problem 22

A cereal company runs periodic checks on how well the machines in its factory package it’s product. The company want to make sure the machines aren’t putting too
much or too little cereal (based on weight) in each box. Table 5.14 shows the weight
in kilograms of 30 sampled packages. The company will reject any box that is either
two standard deviations too heavy or two standard deviations too light, based on the
mean of the sample. Are there any boxes in the sample that need to be rejected?
$$
\begin{array}{cc|cc|cl}
\hline \text { ID } & \text { Weight } & \text { ID } & \text { Weight } & \text { ID } & \text { Weight } \\
\hline 1 & 0.355 & 11 & 0.361 & 21 & 0.352 \\
\hline 2 & 0.347 & 12 & 0.355 & 22 & 0.353 \\
\hline 3 & 0.347 & 13 & 0.349 & 23 & 0.351 \\
\hline 4 & 0.354 & 14 & 0.353 & 24 & 0.346 \\
\hline 5 & 0.351 & 15 & 0.352 & 25 & 0.354 \\
\hline
\end{array}
$$
$$
\begin{array}{rr|cc|cc}
\hline 6 & 0.349 & 16 & 0.349 & 26 & 0.348 \\
\hline 7 & 0.349 & 17 & 0.351 & 27 & 0.345 \\
\hline 8 & 0.348 & 18 & 0.354 & 28 & 0.351 \\
\hline 9 & 0.351 & 19 & 0.353 & 29 & 0.347 \\
\hline 10 & 0.348 & 20 & 0.348 & 30 & 0.341 \\
\hline
\end{array}
$$

Madhumathi R
Madhumathi R
Numerade Educator
02:59

Problem 23

Consider the following data set:

$$
6.8,7.3,1.5,2.3,7.6,1.2,7.9,8.4,9.7,6.9
$$

Now calculate:
a. The mean.
b. The standard deviation.
c. The mean absolute deviation.

Why is the standard deviation preferred over the mean absolute deviation?

Jerrah Biggerstaff
Jerrah Biggerstaff
Numerade Educator
05:48

Problem 24

You've been tasked by the International Olympic Committee to detect if any of the sprinters in the 100-meter event cheated in the final race by anticipating the start gun. You know that the mean reaction time for sprinters in this event is 215 milliseconds (ms), and that the variance is $115 \mathrm{~ms}^2$. If a sprinter is more than one standard deviation faster than the mean reaction time, they should be disqualified under rules of the International Association of Athletics Federation. Based on the reaction times listed below, which athlete(s), if any, should be disqualified?
$$
\begin{array}{c|c}
\hline \text { Lane } & \text { Start RT } \\
\hline 1 & 211 \\
\hline 2 & 205 \\
\hline 3 & 208 \\
\hline 4 & 221 \\
\hline 5 & 207 \\
\hline 6 & 206 \\
\hline 7 & 208 \\
\hline 8 & 207 \\
\hline
\end{array}
$$

Erin Moser
Erin Moser
Numerade Educator
00:41

Problem 25

Table 5.16 shows the data for a small sample of patients investigated by a researcher
specializing in bone health and osteoporosis. All of the patients had presented with a
low-trauma fracture at a regional fracture clinic. The T-score variable is a measure of bone density given as the standard deviation away from that of a healthy male adult.
Calculate the mean, median, mode, range, IQR, variance, and standard deviation for
each of the variables in the data set.
table cant copy

Abhishek Kumar
Abhishek Kumar
Numerade Educator
03:27

Problem 26

Using the same bone health data, calculate the mean and standard deviation for the
different education levels. From these descriptive statistics, how would you describe
the groups relative to each other.

Carly Stoner
Carly Stoner
Numerade Educator
18:39

Problem 27

From the bone health data, construct a box-and-whisker plot for both female and male
patients. How do the genders compare?

Evelyn Cunningham
Evelyn Cunningham
Numerade Educator