STAT2910_Chap1&2_Tutorial_W24_Solution
pdf
keyboard_arrow_up
School
University of Windsor *
*We aren’t endorsed by this school
Course
2910
Subject
Mathematics
Date
Feb 20, 2024
Type
Pages
10
Uploaded by LieutenantWaspMaster985
Department of Mathematics and Statistics
STAT2910-01: Statistics for Sciences
Faculty of Science
University of Windsor
Tutorial for Chapter 1 and 2
Questions for Chapter 1
Exercise 1.
Identify each of the following variables as either quantitative or qualitative
a.
the brands of ice cream that you purchase
b.
the daily high temperature for the past four weeks
c.
the amount of sugar consumed by Canadians in one year
d.
the species of fish in the zoo
e.
the lengths of time children wait for the school bus
f.
your favourite professional football team
Solution 1.
a.
qualitative
b.
quantitative
c.
quantitative
d.
qualitative
e.
quantitative
f.
qualitative
Exercise 2.
Identify each of the following variables as qualitative or quantitative.
a.
rating of the effectiveness of a new cold remedy (not effective, effective)
b.
amount of time spent assembling a five-shelf bookcase
c.
number of children in a beginners’ swimming class
d.
university where a student is enrolled
e.
color preference for a nursery
f.
rating the Canadian foreign policy in the Middle East (fair, biased)
1
Solution 2.
a.
qualitative
b.
quantitative
c.
quantitative
d.
qualitative
e.
qualitative
f.
qualitative
Exercise 3.
Identify each of the following quantitative variables as discrete or continuous.
a.
average monthly temperature for a particular city
b.
number of employees of a statistical consulting firm who own laptop computers
c.
flight time between two cities
d.
number of puppies enrolled in an obedience class
e.
number of persons on a flight from Chicago to Calgary
f.
amount of gas purchased at a gas station
Solution 3.
a.
continuous
b.
discrete
c.
continuous
d.
discrete
e.
discrete
f.
continuous
Exercise 4.
Classify the following variables, first as qualitative or quantitative and second, if quantitative, as discrete
or continuous:
a.
the colors of cars at an auction
b.
the amount of money spent on building a new school
c.
the genders of members of parliament
d.
the styles of houses (1-story, 2-story, split level, etc.)
e.
the letter grades of students in a statistics exam (A, B, C, D, F)
f.
the number of credit cards owned by customers
2
Solution 4.
a.
qualitative
b.
quantitative, continuous
c.
qualitative
d.
qualitative
e.
qualitative
f.
quantitative, discrete
Exercise 5.
The length of time (in months) between the onset of a particular disease and its recurrence was recorded
for
n
= 48:
0.1
2.1
4.4
1.6
2.7
9.9
9.0
2.0
6.6
3.9
14.7
9.6
16.7
7.4
8.2
19.2
6.9
4.3
3.3
1.2
4.1
18.4
0.2
6.1
13.5
7.4
0.2
8.3
0.3
1.3
14.1
1.0
2.4
2.4
18.0
8.7
24.0
1.4
8.2
5.8
1.6
3.5
11.4
3.7
12.6
23.1
5.6
0.4
a.
Construct a relative frequency histogram for the data (use number of class = 8).
b.
Would you describe the shape as roughly symmetric, skewed to the right, or skewed to the left?
c.
Give the fraction of recurrence times less than 15.1 months.
d.
Construct a steam-and-leaf plot with unit 0.1.
Solution 5.
a.
The range of the data
R
= 24
.
0
−
0
.
1 = 23
.
9.
We choose to use eight class intervals of width 3
(23
.
9
/
8 = 2
.
9875, which when rounded to the next largest integer is 3). The subintervals 0.1 to
<
3.1,
3.1 to
<
6.1, 6.1 to
<
9.1, and so on, are convenient and the tally is shown below.
Class
Class
Tally
f
i
Relative Frequency
i
Boundaries
f
i
/n
1
0.1 to 3.1
11111 11111 11111 1
16
16/48
2
3.1 to 6.1
11111 1111
9
9/48
3
6.1 to 9.1
11111 11111
10
10/48
4
9.1 to 12.1
111
3
3/48
5
12.1 to 15.1
1111
4
4/48
6
15.1 to 18.1
11
2
3/48
7
18.1 to 21.1
11
2
2/48
8
21.1 to 24.1
11
2
2/48
The relative frequency histogram is shown below.
3
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help
Relative frequency histogram
Length of time in months
Relative Frequency
0.1
3.1
6.1
9.1
12.1
15.1
18.1
21.1
24.1
0
2
48
3
48
4
48
9
48
10
48
16
48
Figure 1:
b.
The data is skewed right.
c.
The fraction of recurrence time less than 15.1 is given by
16 + 9 + 10 + 3 + 4
48
= 0
.
875
d.
The Stem and Leaf plot is shown below
4
The decimal point is at the |
0 | 12234
1 | 023466
2 | 01447
3 | 3579
4 | 134
5 | 68
6 | 169
7 | 44
8 | 2237
9 | 069
10 | 11 | 4
12 | 6
13 | 5
14 | 17
15 | 16 | 7
17 | 18 | 04
19 | 2
20 | 21 | 22 | 23 | 1
24 | 0
Figure 2: Stem and Leaf plot
Exercise 6.
Construct a stem-and-leaf plot for the following set of data.
28
13
26
12
20
14
21
16
17
22
17
25
13
30
13
22
15
21
18
18
16
21
18
31
15
19
Solution 6.
The stem-and-leaf plot with code 1
|
2 = 12, is
5
1
2 3 3 3 4 5 6 6 7 7 8 8 9
2
0 1 1 1 2 2 5 6 8
3
0 1
Exercise 7.
The calcium (Ca) content of a powdered mineral substance was analyzed ten times with the following
percent compositions recorded:
0
.
0271
,
0
.
0282
,
0
.
0279
,
0
.
0281
,
0
.
0268
,
0
.
0271
,
0
.
0281
,
0
.
0269
,
0
.
0275
,
0
.
0276
.
a.
Draw a stem and leaf plot for the data. Use the numbers in the hundredths and thousandths places
as the stem.
b.
Are any of the measurements inconsistent with the other measurements, indicating that the technician
may have made an error in the analysis?
Solution 7.
a.
The stem-and-leaf plot with code 26
|
8 = 0
.
0268, is
26
8 9
27
1 1 5 6 9
28
1 1 2
b.
There is no unusual measurement.
Questions for Chapter 2
Exercise 8.
The following data represent the number of small cracks per bar for a sample of eight steel bars: 4, 6, 10,
1, 3, 1, 25, and 8.
a.
What is the average number of small cracks per bar?
b.
Find the standard deviation for the number of small cracks per bar.
c.
Which, if any, of the observations appear to be outliers? Justify your answer.
Solution 8.
a.
The mean is given by
¯
x
=
1
n
n
X
i
=1
x
i
= 7
.
25
b.
To find the standard deviation for the number of small cracks per bar, first we calculate the variance
s
2
=
1
n
−
1
n
X
i
=1
(
x
i
−
¯
x
)
2
= 61
.
642
.
Then,
s
=
√
61
.
642 = 7
.
85
6
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help
c.
The value 25 has a
z
−
score of 2
.
26 making it a suspect outlier.
Exercise 9.
Twenty-eight applicants interested in working in community services took an examination designed to
measure their aptitude for social work. A stem-and-leaf plot of the 28 scores appears below, in which the
first column is the count per “branch,” the second column is the stem value, and the remaining digits are
the leaves.
4
6
5
9
6
3688
7
026799
8
145667788
9
1234788
a.
What is the range of these data?
b.
What is the median score?
c.
What is the sample mean for this data set?
d.
What is the value of the sample standard deviation?
e.
Should the Empirical Rule be applied to this data set?
f.
Use the range approximation to determine an approximate value for the standard deviation. Is this a
good approximation?
g.
What is the value of the first and third quartiles?
h.
What is the interquartile range?
Solution 9.
a
The range is
R
= 98
−
46 = 54
b
To find the median, first we find its position, which is given by
p
= 0
.
5(
n
+ 1) = 14
.
5
,
Then the median is
m
=
84+85
2
= 84
.
5.
c
The sample mean is
¯
x
=
1
n
n
X
i
=1
x
i
= 80
.
64
d
The standard deviation is given by
s
=
v
u
u
t
1
n
−
1
n
X
i
=1
(
x
i
−
¯
x
)
2
= 12
.
85
.
e
No. The data do not appear to be mound-shaped
7
f
s
≈
R/
4 = 52
/
4 = 13. This approximation is very close to the actual value of
s
= 12
.
85.
g
Position of first quartile
p
1
= 0
.
25(29) = 7
.
25, then
Q
1
= 70 + 0
.
25(2) = 70
.
5 Position of third quartile
p
3
= 0
.
75(29) = 21
.
75, then
Q
3
= 88 + 0
.
75(3) = 90
.
25.
h
The interquartile range is
IQR
=
Q
3
−
Q
1
= 90
.
25
−
70
.
5 = 19
.
75
Exercise 10.
A new manufacturing plant has 20 job openings. To select the best 20 applicants from among the 1000
job seekers, the plant’s personnel office administers a written aptitude test to all applicants. The average
score on the aptitude test is 150 points, with a standard deviation of 10 points. Assume the distribution
of test scores is approximately mound-shaped.
a.
What percentage of the test scores will fall between 130 and 160 points?
b.
How many applicants will score between 130 and 160 points?
c.
One of the applicants scored 192 points on the test. What might you conclude about this test score?
Solution 10.
a.
Approximately 81
.
5% of the test scores will fall between 130 and 160 points, represented by the blue
area
0.00
0.01
0.02
0.03
0.04
125
150
175
density
probability
A:0.0228
B:0.8186
C:0.1587
Figure 3:
8
b.
Approximately 815 applicants will score between 130 and 160 points
c.
The score should be regarded as an outlier; the score should be double-checked to see if it was recorded
correctly.
Exercise 11.
Consider the following set of measurements:
5
.
4
,
5
.
9
,
3
.
5
,
4
.
1
,
4
.
6
,
2
.
5
,
4
.
7
,
6
.
0
,
5
.
4
,
4
.
6
,
4
.
9
,
4
.
6
,
4
.
1
,
3
.
4
,
2
.
2
.
a.
Find the mean, median, variance and standard deviation of this sample
b.
Find the 25th, 50th, and 75th percentiles
c.
What is the value of the interquartile range?
Solution 11.
Consider the following set of measurements:
5
.
4
,
5
.
9
,
3
.
5
,
4
.
1
,
4
.
6
,
2
.
5
,
4
.
7
,
6
.
0
,
5
.
4
,
4
.
6
,
4
.
9
,
4
.
6
,
4
.
1
,
3
.
4
,
2
.
2
.
a.
The requested quantities are given by
The mean
¯
x
=
1
n
n
X
i
=1
x
i
≈
4
.
4
For the median the position is
p
= 0
.
5
×
(
n
+ 1) = 0
.
5
×
16 = 8, then the median is
m
= 4
.
6
The standard deviation is
s
=
v
u
u
t
1
n
−
1
n
X
i
=1
(
x
i
−
¯
x
)
2
= 1
.
125
b.
25th percentile =
Q
1
= 3
.
5; 50th percentile =
Q
2
= 4
.
6; 75th percentile=
Q
3
= 5
.
4
c.
IQR =
Q
3
−
Q
1
= 5
.
4
−
3
.
5 = 1
.
9
Exercise 12.
The following data represent the scores for a sample of 10 students on a 20-point chemistry quiz: 16, 14,
2, 8, 12, 12, 9, 10, 15, and 13. Calculate the z-score for the smallest and largest observations. Is either of
these observations unusually large or unusually small?
Solution 12.
For
x
= 2,
z
−
score =
2–11
.
1
4
.
095
= –2
.
22
.
For
x
= 16,
z
−
score =
16–11
.
1
4
.
095
= 1
.
197
.
Since the z-score for the smallest observation exceeds 2 in absolute value, the smallest observation is
unusually small. However, the largest observation is not unusually large.
9
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help
Exercise 13.
Two students are enrolled in different sections of an introductory statistics class at a local university. The
first student, enrolled in the morning section, earns a score of 76 on a midterm exam where the class
mean was 64 with a standard deviation of 8. The second student, enrolled in the afternoon section, earns
a score of 72 on a midterm exam where the class mean was 60 with a standard deviation of 7.5. If the
scores on the midterm exams are normally distributed, which student scored better relative to his or her
classmates?
Solution 13.
given that
z
1
= (76–64)
/
8 = 1
.
5
,
and
z
2
= (72–60)
/
7
.
5 = 1
.
6
,
the student in the afternoon section scored better relative to her classmates since her z-score is larger.
Exercise 14.
a.
If the 90th and 91st observations in a set of 100 data values are 158 and 167, respectively, what is the
90th percentile value?
b.
If the 18th and 19th observations in a set of 25 data values are 42.6 and 43.8, what is the 70th percentile
value?
Solution 14.
a.
The position of the 90th percentile is = 0
.
9
×
(
n
+ 1) = 0
.
9
×
101 = 90
.
9. Hence, the 90th percentile
p
value is given by.
p
= 158 + 0
.
9
×
(167
−
158) = 166
.
1
b.
The position of the 70th percentile is = 0
.
7
×
(
n
+ 1) = 0
.
7
×
26 = 18
.
2. Hence, the 70th percentile
p
value is given by.
p
= 42
.
6 + 0
.
2
×
(43
.
8
−
42
.
6) = 42
.
84
10