1. In words, explain what is measured by each of the
following:
a. SS
b. Variance
c. Standard deviation
2. Can SS ever have a value less than zero? Explain your
answer.
3. Is it possible to obtain a negative value for the
variance or the standard deviation?
4. What does it mean for a sample to have a standard
deviation of zero? Describe the scores in such a
sample.
5. Explain why the formulas for sample variance and
population variance are different.
6. A population has a mean of _ _ 80 and a standard
deviation of _ _ 20.
a. Would a score of X _ 70 be considered an extreme
value (out in the tail) in this sample?
b. If the standard deviation were _ _ 5, would a score
ofX _ 70 be considered an extreme value?
7. On an exam with a mean of M _ 78, you obtain a
score of X _ 84.
a. Would you prefer a standard deviation of s _ 2 or
s_ 10? (Hint: Sketch each distribution and find the
location of your score.)
b. If your score were X _ 72, would you prefer s _ 2
ors _ 10? Explain your answer.
8. A population has a mean of _ _ 30 and a standard
deviation of _ _ 5.
a. If 5 points were added to every score in the
population, what would be the new values for the
mean and standard deviation?
b. If every score in the population were multiplied by
3 what would be the new values for the mean and
standard deviation?
9. a. After 3 points have been added to every score in a
sample, the mean is found to be M _ 83 and the
standard deviation is s _ 8. What were the values
for the mean and standard deviation for the original
sample?
b. After every score in a sample has been multiplied
by 4, the mean is found to be M _ 48 and the
standard deviation is s _ 12. What were the values
for the mean and standard deviation for the original
sample?
10. A student was asked to compute the mean and
standard deviation for the following sample of n _ 5
scores: 81, 87, 89, 86, and 87. To simplify the
arithmetic, the student first subtracted 80 points from
each score to obtain a new sample consisting of 1, 7,
9, 6, and 7. The mean and standard deviation for the
new sample were then calculated to be M _ 6 and
s_ 3. What are the values of the mean and standard
deviation for the original sample?
11. For the following population of N _ 6 scores:
11, 0, 2, 9, 9, 5
a. Calculate the range and the standard deviation.
(Use either definition for the range.)
b. Add 2 points to each score and compute the range
and standard deviation again. Describe how adding
a constant to each score influences measures of
variability.
12. There are two different formulas or methods that can
be used to calculate SS.
a. Under what circumstances is the definitional
formula easy to use?
b. Under what circumstances is the computational
formula preferred?
13. Calculate the mean and SS (sum of squared deviations)
for each of the following samples. Based on the value
for the mean, you should be able to decide which SS
formula is better to use.
Sample A: 1, 4, 8, 5
Sample B: 3, 0, 9, 4
14. The range is completely determined by the two
extreme scores in a distribution. The standard
deviation, on the other hand, uses every score.
a. Compute the range (choose either definition) and
the standard deviation for the following sample of
n_ 5 scores. Note that there are three scores
clustered around the mean in the center of the
distribution, and two extreme values.
Scores: 0, 6, 7, 8, 14.
b. Now we break up the cluster in the center of the
distribution by moving two of the central scores out
to the extremes. Once again compute the range and
the standard deviation.
New scores: 0, 0, 7, 14, 14.
c. According to the range, how do the two
distributions compare in variability? How do they
compare according to the standard deviation?
15. For the data in the following sample:
8, 1, 5, 1, 5
a. Find the mean and the standard deviation.
b. Now change the score of X _ 8 to X _ 18, and find
the new mean and standard deviation.
c. Describe how one extreme score influences the
mean and standard deviation.
16. Calculate SS, variance, and standard deviation for the
following sample of n _ 4 scores: 7, 4, 2, 1. (Note:
The computational formula works well with these
scores.)
17. Calculate SS, variance, and standard deviation for the
following population of N _ 8 scores: 0, 0, 5, 0, 3, 0,
0, 4. (Note: The computational formula works well
with these scores.)
18. Calculate SS, variance, and standard deviation for the
following population of N _ 7 scores: 8, 1, 4, 3, 5,
3, 4. (Note: The definitional formula works well with
these scores.)
19. Calculate SS, variance, and standard deviation for the
following sample of n _ 5 scores: 9, 6, 2, 2, 6. (Note:
The definitional formula works well with these
scores.)
20. For the following population of N _ 6 scores:
3, 1, 4, 3, 3, 4
a. Sketch a histogram showing the population
distribution.
b. Locate the value of the population mean in your
sketch, and make an estimate of the standard
deviation (as done in Example 4.2).
c. Compute SS, variance, and standard deviation for
the population. (How well does your estimate
compare with the actual value of _?)
21. For the following sample of n _ 7 scores:
8, 6, 5, 2, 6, 3, 5
a. Sketch a histogram showing the sample
distribution.
b. Locate the value of the sample mean in your sketch,
and make an estimate of the standard deviation (as
done in Example 4.5).
c. Compute SS, variance, and standard deviation for
the sample. (How well does your estimate compare
with the actual value of s?)
22. In an extensive study involving thousands of British
children, Arden and Plomin (2006) found significantly
higher variance in the intelligence scores for males
than for females. Following are hypothetical data,
similar to the results obtained in the study. Note that
the scores are not regular IQ scores but have been
standardized so that the entire sample has a mean of
M _ 10 and a standard deviation of s _ 2.
a. Calculate the mean and the standard deviation for
the sample of n _ 8 females and for the sample of
n_ 8 males.
b. Based on the means and the standard deviations,
describe the differences in intelligence scores for
males and females.
Female Male
9 8
11 10
10 11
13 12
8 6
9 10
11 14
9 9
23. In the Preview section at the beginning of this chapter
we reported a study by Wegesin and Stern (2004) that
found greater consistency (less variability) in the
memory performance scores for younger women than
for older women. The following data represent
memory scores obtained for two women, one older
and one younger, over a series of memory trials.
a. Calculate the variance of the scores for each
woman.
b. Are the scores for the younger woman more
consistent (less variable)?
Younger Older
8 7
6 5
6 8
7 5
8 7
7 6
8 8
8 5
1. What information is provided by the sign (_/–) of a
z-score? What information is provided by the
numerical value of the z-score?
2. A distribution has a standard deviation of _ _ 12.
Find the z-score for each of the following locations
in the distribution.
a. Above the mean by 3 points.
b. Above the mean by 12 points.
c. Below the mean by 24 points.
d. Below the mean by 18 points.
3. A distribution has a standard deviation of _ _ 6.
Describe the location of each of the following z-scores
in terms of position relative to the mean. For example,
z__1.00 is a location that is 6 points above the
mean.
a. z__2.00
b. z__0.50
c. z _ –2.00
d. z _ –0.50
4. For a population with _ _ 50 and _ _ 8,
a. Find the z-score for each of the following X values.
(Note: You should be able to find these values
using the definition of a z-score. You should not
need to use a formula or do any serious
calculations.)
X _ 54 X _ 62 X _ 52
X _ 42 X _ 48 X _ 34
b. Find the score (X value) that corresponds to each of
the following z-scores. (Again, you should be able
to find these values without any formula or serious
calculations.)
z_ 1.00 z _ 0.75 z _ 1.50
z_ –0.50 z _ –0.25 z _ –1.50
5. For a population with _ _ 40 and _ _ 7, find the
z-score for each of the following X values. (Note: You
probably will need to use a formula and a calculator to
find these values.)
X _ 45 X _ 51 X _ 41
X _ 30 X _ 25 X _ 38
6. For a population with a mean of _ _ 100 and a
standard deviation of _ _ 12,
a. Find the z-score for each of the following X values.
X _ 106 X _ 115 X _ 130
X _ 91 X _ 88 X _ 64
b. Find the score (X value) that corresponds to each of
the following z-scores.
z_ –1.00 z _ –0.50 z _ 2.00
z_ 0.75 z _ 1.50 z _ –1.25
7. A population has a mean of _ _ 40 and a standard
deviation of _ _ 8.
a. For this population, find the z-score for each of the
followingX values.
X _ 44 X _ 50 X _ 52
X _ 34 X _ 28 X _ 64
b. For the same population, find the score (X value)
that corresponds to each of the following z-scores.
z_ 0.75 z _ 1.50 z _ –2.00
z_ –0.25 z _ –0.50 z _ 1.25
8. A sample has a mean of M _ 40 and a standard
deviation of s _ 6. Find the z-score for each of the
followingX values from this sample.
X _ 44 X _ 42 X _ 46
X _ 28 X _ 50 X _ 37
9. A sample has a mean of M _ 80 and a standard
deviation of s _ 10. For this sample, find the X value
corresponding to each of the following z-scores.
z_ 0.80 z _ 1.20 z _ 2.00
z_ –0.40 z _ –0.60 z _ –1.80
10. Find the z-score corresponding to a score of X _ 60
for each of the following distributions.
a. _ _ 50 and _ _ 20
b. _ _ 50 and _ _ 10
c. _ _ 50 and _ _ 5
d. _ _ 50 and _ _ 2
11. Find the X value corresponding to z _ 0.25 for each of
the following distributions.
a. _ _ 40 and _ _ 4
b. _ _ 40 and _ _ 8
c. _ _ 40 and _ _ 12
d. _ _ 40 and _ _ 20
12. A score that is 6 points below the mean corresponds to
az-score of z _ –0.50. What is the population
standard deviation?
13. A score that is 12 points above the mean corresponds
to a z-score of z _ 3.00. What is the population
standard deviation?
14. For a population with a standard deviation of _ _ 8, a
score of X _ 44 corresponds to z _ –0.50. What is the
population mean?
15. For a sample with a standard deviation of s _ 10, a
score of X _ 65 corresponds to z _ 1.50. What is the
sample mean?
16. For a sample with a mean of _ _ 45, a score of
X _ 59 corresponds to z _ 2.00. What is the sample
standard deviation?
17. For a population with a mean of _ _ 70, a score of
X _ 62 corresponds to z _ –2.00. What is the
population standard deviation?
18. In a population of exam scores, a score of X _ 48
corresponds to z__1.00 and a score of X _ 36
corresponds to z _ –0.50. Find the mean and standard
deviation for the population. (Hint: Sketch the
distribution and locate the two scores on your sketch.)
19. In a distribution of scores, X _ 64 corresponds to
z_ 1.00, and X _ 67 corresponds to z _ 2.00. Find
the mean and standard deviation for the distribution.
20. For each of the following populations, would a score
ofX _ 50 be considered a central score (near the
middle of the distribution) or an extreme score (far out
in the tail of the distribution)?
a. _ _ 45 and _ _ 10
b. _ _ 45 and _ _ 2
c. _ _ 90 and _ _ 20
d. _ _ 60 and _ _ 20
21. A distribution of exam scores has a mean of _ _ 80.
a. If your score is X _ 86, which standard deviation
would give you a better grade: _ _ 4 _ _ 8?
b. If your score is X _ 74, which standard deviation
would give you a better grade: _ _ 4 or _ _ 8?
22. For each of the following, identify the exam score that
should lead to the better grade. In each case, explain
your answer.
a. A score of X _ 56, on an exam with _ _ 50 and
_ _ 4; or a score of X _ 60 on an exam with
_ _ 50 and _ _ 20.
b. A score of X _ 40, on an exam with _ _ 45 and
_ _ 2; or a score of X _ 60 on an exam with
_ _ 70 and _ _ 20.
c. A score of X _ 62, on an exam with _ _ 50 and
_ _ 8; or a score of X _ 23 on an exam with
_ _ 20 and _ _ 2.
23. A distribution with a mean of _ _ 62 and a standard
deviation of _ _ 8 is transformed into a standardized
distribution with _ _ 100 and _ _ 20. Find the new,
standardized score for each of the following values
from the original population.
a. X _ 60
b. X _ 54
c. X _ 72
d. X _ 66
24. A distribution with a mean of _ _ 56 and a standard
deviation of _ _ 20 is transformed into a standardized
distribution with _ _ 50 and _ _ 10. Find the new,
standardized score for each of the following values
from the original population.
a. X _ 46
b. X _ 76
c. X _ 40
d. X _ 80
25. A population consists of the following N _ 5 scores:
0, 6, 4, 3, and 12.
a. Compute _ and _ for the population.
b. Find the z-score for each score in the population.
c. Transform the original population into a new
population of N _ 5 scores with a mean of
_ _ 100 and a standard deviation of _ _ 20.
26. A sample consists of the following n _ 6 scores: 2, 7,
4, 6, 4, and 7.
a. Compute the mean and standard deviation for the
sample.
b. Find the z-score for each score in the sample.
c. Transform the original sample into a new sample
with a mean of M _ 50 and s _ 10.
1. A local hardware store has a “Savings Wheel” at
the checkout. Customers get to spin the wheel and,
when the wheel stops, a pointer indicates how much
they will save. The wheel can stop in any one of
50 sections. Of the sections, 10 produce 0% off,
20 sections are for 10% off, 10 sections for 20%,
5 for 30%, 3 for 40%, 1 for 50%, and 1 for 100%
off. Assuming that all 50 sections are equally likely,
a. What is the probability that a customer’s purchase
will be free (100% off)?
b. What is the probability that a customer will get no
savings from the wheel (0% off)?
c. What is the probability that a customer will get at
least 20% off?
2. A psychology class consists of 14 males and 36 females.
If the professor selects names from the class list using
random sampling,
a. What is the probability that the first student
selected will be a female?
b. If a random sample of n _ 3 students is selected
and the first two are both females, what is the
probability that the third student selected will be
a male?
3. What are the two requirements that must be satisfied
for a random sample?
4. What is sampling with replacement, and why is it used?
5. Draw a vertical line through a normal distribution for
each of the following z-score locations. Determine
whether the tail is on the right or left side of the line
and find the proportion in the tail.
a. z _ 2.00
b. z _ 0.60
c. z _ –1.30
d. z _ –0.30
6. Draw a vertical line through a normal distribution for
each of the following z-score locations. Determine
whether the body is on the right or left side of the line
and find the proportion in the body.
a. z _ 2.20
b. z _ 1.60
c. z _ –1.50
d. z _ –0.70
7. Find each of the following probabilities for a normal
distribution.
a. p(z _ 0.25)
b. p(z _ –0.75)
c. p(z _ 1.20)
d. p(z _ –1.20)
8. What proportion of a normal distribution is located
between each of the following z-score boundaries?
a. z _ –0.50 and z__0.50
b. z _ –0.90 and z__0.90
c. z _ –1.50 and z__1.50
9. Find each of the following probabilities for a normal
distribution.
a. p(–0.25 _ z _ 0.25)
b. p(–2.00 _ z _ 2.00)
c. p(–0.30 _ z _ 1.00)
d. p(–1.25 _ z _ 0.25)
10. Find the z-score location of a vertical line that
separates a normal distribution as described in
each of the following.
a. 20% in the tail on the left
b. 40% in the tail on the right
c. 75% in the body on the left
d. 99% in the body on the right
11. Find the z-score boundaries that separate a normal
distribution as described in each of the following.
a. The middle 20% from the 80% in the tails.
b. The middle 50% from the 50% in the tails.
c. The middle 95% from the 5% in the tails.
d. The middle 99% from the 1% in the tails.
12. For a normal distribution with a mean of μ _ 80 and
a standard deviation of _ _ 20, find the proportion of
the population corresponding to each of the following
scores.
a. Scores greater than 85.
b. Scores less than 100.
c. Scores between 70 and 90.
13. A normal distribution has a mean of μ _ 50 and
a standard deviation of _ _ 12. For each of the
following scores, indicate whether the tail is to the
right or left of the score and find the proportion of
the distribution located in the tail.
a. X _ 53
b. X _ 44
c. X _ 68
d. X _ 38
14. IQ test scores are standardized to produce a normal
distribution with a mean of μ _ 100 and a standard
deviation of _ _15. Find the proportion of the
population in each of the following IQ categories.
a. Genius or near genius: IQ greater than 140
b. Very superior intelligence: IQ between 120
and 140
c. Average or normal intelligence: IQ between 90
and 109
15. The distribution of scores on the SAT is approximately
normal with a mean of μ _ 500 and a standard
deviation of _ _ 100. For the population of students
who have taken the SAT,
a. What proportion have SAT scores greater than 700?
b. What proportion have SAT scores greater than 550?
c. What is the minimum SAT score needed to be in
the highest 10% of the population?
d. If the state college only accepts students from the
top 60% of the SAT distribution, what is the
minimum SAT score needed to be accepted?
16. The distribution of SAT scores is normal with μ _ 500
and _ _ 100.
a. What SAT score, X value, separates the top 15% of
the distribution from the rest?
b. What SAT score, X value, separates the top 10% of
the distribution from the rest?
c. What SAT score, X value, separates the top 2% of
the distribution from the rest?
17. A recent newspaper article reported the results
of a survey of well-educated suburban parents.
The responses to one question indicated that by
age 2, children were watching an average of
μ _ 60 minutes of television each day. Assuming
that the distribution of television-watching times is
normal with a standard deviation of _ _ 20 minutes,
find each of the following proportions.
a. What proportion of 2-year-old children watch more
than 90 minutes of television each day?
b. What proportion of 2-year-old children watch less
than 20 minutes a day?
18. Information from the Department of Motor Vehicles
indicates that the average age of licensed drivers is
μ _ 45.7 years with a standard deviation of _ _ 12.5
years. Assuming that the distribution of drivers’ ages
is approximately normal,
a. What proportion of licensed drivers are older than
50 years old?
b. What proportion of licensed drivers are younger
than 30 years old?
19. A consumer survey indicates that the average
household spends μ _ $185 on groceries each
week. The distribution of spending amounts is
approximately normal with a standard deviation
of _ _ $25. Based on this distribution,
a. What proportion of the population spends more
than $200 per week on groceries?
b. What is the probability of randomly selecting a
family that spends less than $150 per week on
groceries?
c. How much money do you need to spend on
groceries each week to be in the top 20% of
the distribution?
20. Over the past 10 years, the local school district has
measured physical fitness for all high school freshmen.
During that time, the average score on a treadmill
endurance task has been μ _ 19.8 minutes with a
standard deviation of _ _ 7.2 minutes. Assuming that
the distribution is approximately normal, find each of
the following probabilities.
a. What is the probability of randomly selecting
a student with a treadmill time greater than
25 minutes? In symbols, p(X _ 25) _ ?
b. What is the probability of randomly selecting a
student with a time greater than 30 minutes? In
symbols, p(X _ 30) _ ?
c. If the school required a minimum time of 10 minutes
for students to pass the physical education course,
what proportion of the freshmen would fail?
21. Rochester, New York, averages μ _ 21.9 inches of
snow for the month of December. The distribution
of snowfall amounts is approximately normal with
a standard deviation of _ _ 6.5 inches. This year, a
local jewelry store is advertising a refund of 50%