HW#1 - Statistics

her than a quantitative
are needed. The two
A numerical summary of a sample is called a statistic.
oportions (sometimes A numerical summary of a population is called a parameter.
; simply the number of
the frequency divided
Statistics are often used to estimate parameters.
1.2 Summary Statistics 23
.a1 combustion engine.
Exercises for Section 1.2
are classified as con- 1. True or false: For any list of numbers, half of them in a certain county is taken and the number with each
.s thicker than this are will be below the mean. score is as follows:
: of 1000 bearings, 9 10 2. Is the sample mean always the most frequently occur- Score 0 1 2 3 4 5 6 7 8 9 10
~d the frequencies and ring value? If so, explain why. If not, give an example. Number 1 3 2 4 25 35 198 367 216 131 18
of babies
3. Is the sample mean always equal to one of the val-
.e 910/1000 = 0.910,
ues in the sample? If so, explain why. If not, give an
a. Find the sample mean Apgar score.
b. Find the sample standard deviation of the Apgar
4. Is the sample median always equal to one of the val- scores.
ues in the sample? If so, explain why. If not, give an c. Find the sample median of the Apgar scores.
~unterpart. This is easy
~pulation of numerical
s in the population; the
Mle values; and so on.
5. Find a sample size for which the median will always
equal one of the values in the sample.
6. For a list of positive numbers, is it possible for the
standard deviation to be greater than the mean? If so,
give an example. If not, explain why not.
d. What is the first quartile of the scores?
e. What proportion of the scores is greater than the
mean?
f. What proportion of the scores is more than one
standard deviation greater than the mean?
g. What proportion of the scores is within one standard
deviation of the mean?
for a finite population,
values rather than the
7.. Is it possible for the standard deviation of a list of numbers
to equal O? If so, give an example. If not, explain
11. a statistics ,-lass of 30 students, the mean score
on the midterm was 72. In another class of 40 stuiance,
where we divide dents, the mean score was 79. What was the mean for
numerical summaries
)f a sample are called
~arameters. Of course,
tion parameters cannot
8. In a certain company, every worker received a $50per-week
raise. How does this affect the mean salary?
The standard deviation of the salaries?
the two classes combined?
12. Each of 16 students measured the circumference of a
tennis ball by four different methods, which were:
:imate the values of the 9. In another company, every worker received a 5% raise. Method A: Estimate the circumference by eye.
How does this affect the mean salary? The standard Method B: Measure the diameter with aruler, and then
;ample be finite. There- deviation of the salaries? compute the circumference.
nethods for computing
eters. For infinite popted
by procedures that
which involve infinite
2.
10. The Apgar score is used to rate reflexes and responses
of newborn babies. Each baby is assigned a score by a
medical professional, and the possible values are the
integers from 0 to 10. A sample of 1000 babies born
Method C: Measure the circumference with a ruler
and string.
Method D: Measure the circumference by rolling the
ball along a ruler.

24 CHAPTER 1 Sampling and Descriptive Statistics
The results (in cm) are as follows, in increasing order b. If the students remeasured the ball. using a ruler
for each method: marked in inches, would the effects on the mean,
Method A: 18.0, 18.0, 18.0, 20.0, 22.0, 22.0, 22.5,
23.0. 24.0, 24.0, 25.0, 25.0, 25.0, 25.0, 26.0, 26.4.
median, quartiles, and standard
same as in part (a)? Explain.
Method B: 18.8, 18.9, 18.9, 19.6, 20.1, 20.4, 20.4, 14. A list of lonumbers has a mean of 20, a median of 18,
20.4. 20.4, 20.5, 21.2, 22.0, 22.0, 22.0, 22.0, 23.6.
and a standard deviation of 5. The largest number on
MethodC: 20.2, 20.5, 20.5, 20.7, 20.8, 20.9, 21.0,
21.0, 21.0, 21.0, 21.0,21.5, 21.5, 21.5, 21.5,21.6.
the list is 39.27. By accid
to 392.7.
Method D: 20.0, 20.0, 20.0, 20.0, 20.2, 20.5, 20.5,
20.7, 20.7, 20.7, 21.0, 21.1, 21.5, 21.6, 22.1, 22.3.
a. What is the value of the mean after the change?
b. What is the value of the median after the change?
c. What is the value of the standard deviation after
a. Compute the mean measurement for each method. the change?
b. Compute the median measurement for each
method.
c. Compute the 20% trimmed mean measurement for
each method.
15. Quartiles divide a sample into four ne
'
In general, a sample of r '
nearly equal pieces by usil., ...- --,
for i = 1, . . . , k - 1. Consider the fuLLuwlLls ulur;lLu
d. Compute the first and third quartiles for each sample:
method. 2 18 23 41 44 46 49 61 62 74 76 79 82 89 92 95
e. Compute the standard deviation of the measure- a. Tertiles divide a sample into thirds. Find the tertiles
ments for each method. of this sample.
f. For which method is the standard deviation the b. Quintiles divide a sample into fifths. Find the quin-
largest? Why should one expect this method to tiles of this sample.
have the largest standard deviation?
or a larger standarddeviation? Or doesn't it matter? could conceivably be correct.
I
Explain. a. A rockis weighed five times. The readings in grams
are 48.5,47.2,4.91.49.5,46.3.
13. Refer to Exercise 12.
b. A sociologist samples five fam
a. If the measurements for one of the methods were
town and records their annual
converted to inches (1 inch.= 2.54 cm), how would
comes are $34,000, $S7 nM Q1
this affect the mean? The median? The quartiles?
$62,000.
The standard deviation?
I
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I

1.3 Graphical Summaries 39
itterplots can also be Exercises for Section 1.3
ore than two values.
~ts. The article "Ad-
)perties of Titanium
Journal, 2001 : 126sgth
characteristics of
Figure 1.17a is a plot
rersus carbon content
rength (in ksi) versus
1. As part of a quality-control study aimed at improving
aproduction line, the weights (in ounces) of 50 bars of
soap are measured. The results are as follows, sorted
from smallest to largest.
11.6 12.6 12.7 12.8 13.1 13.3 13.6 13.7 13.8 14.1
14.3 14.3 14.6 14.8 15.1 15.2 15.6 15.6 15.7 15.8
15.8 15.9 15.9 16.1 16.2 16.2 16.3 16.4 16.5 16.5
16.5 16.6 17.0 17.1 17.3 17.3 17.4 17.4 17.4 17.6
17.7 18.1 18.3 18.3 18.3 18.5 18.5 18.8 19.2 20.3
Method 1
0.32
0.35
0.37
0.39
0.42
0.47
0.5 1
0.58
Method 2
0.25
0.40
0.48
0.55
0.56
0.58
0.60
0.65
a. Construct a stem-and-leaf plot for these data.
0.60
0.62
0.70
0.76
b. Construct a histogram for these data. 0.65 0.80
I
c. Construct a dotplot for these data.
d. Construct a boxplot for these data. Does the box-
0.68
0.75
0.91
0.99
plot show any outliers?
2. Following is a list of the number of hazardous waste
sites in each of the 50 states of the United States as
of April 1995. The data are taken from The World
Almanac and Book of Facts 1996 (World Almanac
Books, Mahwah, NJ, 1996). The list has been sorted
into numerical order.
a. Construct a histogram for the results of each
method.
b. Construct comparative boxplots for the two
methods.
c. Using the boxplots, what differences can be seen
between the results of the two methods?
1 2 3 4 4 5 6 8 8 9
10 10 10 11 11 11 12 12 12 12
13 13 14 15 16 17 17 18 18 19
19 20 22 23 24 25 29 30 33 37
38 39 40 55 58 77 81 96 102 107
5. Each of 32 students, comprised of two lab sections
of 16 students each, estimated the circumference of
a tennis ball by eye. The results, in centimeters, are
given here. (The results for the first group of students
also appear in Exercise 12 of Section 1.2.)
tent (%)
a. Construct a stem-and-leaf plot for these data. Group 1 Group 2
arbon content and yield
gher nitrogen content is
b. Construct a histogram for these data.
c. Construct a dotplot for these data.
d. Construct a boxplot for these data. Does the box-
18.0
18.0
18.0
15.0
18.0
18.0
plot show any outliers? 20.0
22.0
19.0
19.0
3. Refer toTable 1.2 (page 20). Construct a stem-and-leaf 22.0 19.0
L7b) shows some clear
.to upper right. In this
d yield strength: Welds
s scatterplot might lead
plot with the ones digit as the stem (for values greater
than or equal to 10 the stem will have two digits) and
the tenths digit as the leaf. How many stems are there
(be sure to include leafless stems)? What are some ad-
22.5
23.0
24.0
24.0
25 .0
19.0
19.5
20.0
20.0
20.0
try to increase nitrogen
p on a scatterplot does
liscuss in Section 7.1 .)
erplot of yield strength
ship between these two
vantages and disadvantages of this plot, compared to
the one in Figure 1.6 (page 26)?
4. Two methods were studied for the recovery of protein.
Thirteen runs were made using each method, and the
fraction of protein recovered was recorded for each
25 .O
25 .O
25 .O
26.0
26.4
20.0
20.0
22.0
24.0
25 .O
ying to predict strength run. The results are as follows: