Download Applying the Concepts 3-4 - Parkway C-2

150
Chapter 3 Data Description
actually in the sixth grade and had special permission to attend the class. This student
might have scored extremely low on that particular exam on that day. Fourth, the data
value might be a legitimate value that occurred by chance (although the probability is
extremely small).
There are no hard-and-fast rules on what to do with outliers, nor is there complete
agreement among statisticians on ways to identify them. Obviously, if they occurred as a
result of an error, an attempt should be made to correct the error or else the data value
should be omitted entirely. When they occur naturally by chance, the statistician must
make a decision about whether to include them in the data set.
When a distribution is normal or bell-shaped, data values that are beyond 3 standard
deviations of the mean can be considered suspected outliers.
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a
Applying the Concepts 3-4
13.
Determining Dosages
In an attempt to determine necessary dosages of a new drug (HDL) used to control sepsis,
assume you administer varying amounts of HDL to 40 mice. You create four groups and label
them low dosage, moderate dosage, large dosage, and very large dosage. The dosages also
vary within each group. After the mice are injected with the HDL and the sepsis bacteria, the
time until the onset of sepsis is recorded. Your job as a statistician is to effectively
communicate the results of the study.
14,
I. Which measures of position could be used to help describe the data results?
2. If 40% of the mice in the top quartile survived after the injection, how many mice would
that be?
15
3. What information can be given from using percentiles?
4. What information can be given from using quartiles?
5. What information can be given from using standard scores?
See page 176 for the answers.
--Exercises
3-4
1. What is a z score? A z score tells how many standard
deviations the data value is above or below the mean.
2. Define percentile
rank. A percentile rank indicates the
percentage of data values that fall below the specific rank.
3. What is the difference between a percentage and a
percentile?
A percentile is a relative measurement of position;
a percentage is an absolute measure of the part to the total.
4. Define quartile.
A quartile is a relative measure of position
obtained by dividing the data set into quarters.
5. What is the relationship between quartiles and
percentiles? Q, = P2,;Q2= P,,,;Q, = P7,
6. What is a decile? A decile is a relative measure of position
obtained by dividing the data set into tenths.
7. How are deciles related to percentiles?
D, = PIO; D2 = P20; D, = P,,,; etc.
8. To which percentile, quartile, and decile does the
median correspond? Pso;Q2;Ds
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1
9. Value of Major League Teams If the mean value of
major league teams is $127 million and the standard
deviation is $9 million, find the corresponding z score
for each team's value.
d. 113.5 -1.5
a. 136 I
e. 133 0.67
b. 109-2
c.
104.5 -2.5
10. Reaction Time The reaction time to a stimulus for a
certain test has a mean of 2.5 seconds and a standard
deviation of 0.3 second. Find the corresponding z score
for each reaction time.
a.
b.
c.
d.
e.
2.7
3.9
2.8
3.1
2.2
0.67
4.67
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2
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Section ~
11. Exam Scores A final examination for a psychology
course has a mean of 84 and a standard deviation of 4.
Find the corresponding z score for each raw score.
a. 87 0.75
d. 76 2
b. 79 -1.25
e. 82 -0.5
c. 93
2.25
12. Aptitude Test Scores An aptitude test has a mean of
220 and a standard deviation of 10. Find the
corresponding z score for each exam score.
d. 212 -0.8
a. 200-2
e. 225 0.5
b. 232 1.2
c. 218 -0.2
13. Exam Scores Which of the following exam scores has
a better relative position?
a. A score of 42 on an exam with X = 39 and
s = 4. 0.75
b.
A score of 76 on an exam with
s
= 3.
1.67
X=
71 and
The score for part h is higher.
14. Test Scores A student scores 60 on a mathematics test
that has a mean of 54 and a standard deviation of 3, and
she scores 80 on a history test with a mean of 75 and a
standard deviation of 2. On which test did she perform
better?
2.0: 2.5: history tcst
15. Which score indicates the highest relative position?
a. A score of 3.2 on a test with
s
X = 4.6
b. A score of 630 on a test with X = 800 and
s = 200. -0.85
c. A score of 43 on a test with X = 50 and s = 5.
- 1.4: score in part h is highest
16. Weights of Fifth-Grade Boys This distribution
represents the data for weights of fifth-grade boys.
Find the approximate weights corresponding to each
percentile given by constructing a percentile graph.
Weight (pounds)
52.5-55.5
55.5-58.5
58.5-61.5
61.5-64.5
64.5-67.5
Frequency
9
12
17
22
15
a. 25th 58
b. 60th 62.8
c. 80th 64.5
d. 95th 67.1
17. For the data in Exercise 16, find the approximate
percentile ranks of the following weights.
a.
b.
c.
d.
57 pounds
62 pounds
64 pounds
59 pounds
21st
58th
77th
33rd
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18. Achievement Test Scores (ans) The data shown
represent the scores on a national achievement test for a
group of tenth-grade students. Find the approximate
percentile ranks of these scores by constructing a
percentile graph.
Q. 220 7
d. 280 76
b. 245 25
e. 300 93
c. 276 64
Score
Frequency
196.5-217.5
217.5-238.5
238.5-259.5
259.5-280.5
280.5-301.5
301.5-322.5
5
17
22
48
22
6
I
19. For the data in Exercise 18, find the approximate scores
that correspond to these percentiles.
Q. 15th 235
d. 65th 275
b. 29th 255
e. 80th 283
c. 43rd 261
20. Airplane Speeds (ans) The airborne speeds in miles
per hour of 21 planes are shown. Find the approximate
values that correspond to the given percentiles by
constructing a percentile graph.
Class
and
= 1.5. -0.93
Measures of Position
Frequency
366-386
387-407
408-428
429-449
450-470
471-491
492-512
513-533
4
2
3
2
I
2
3
4
21
Q. 9th 376
b. 20th 389
c. 45th 432
d. 60th 473
e. 75th 498
Source: The World Almanac and Book of Facls.
21. Using the data in Exercise 20, find the approximate
percentile ranks of the following miles per hour (mph).
Q. 380 mph 17th
b. 425 mph 39th
c. 455 mph 53rd
o
d. 505 mph 79th
e. 525 mph 91st
22. Weights Find the percentile ranks of each weight
in the data set. The weights are in pounds.
78, 82, 86, 88, 92, 97 8th:25th:42nd:58th:75th:92nd
23. In Exercise 22, what value corresponds to the 30th
percentile? 82
3-51
152
Chapter 3 Data Description
O
24. Test Scores Find the percentile rank for each test
score
in the data set.
7th; 21st; 36th; 50th; 64th; 79th; 93rd
0
30. Using the procedure shown in Example 3-37,
check each data set for outliers.
a. 16,18,22,19,3,21,17,20
3
b. 24,32,54,31,16,18,19,14,17,20
c. 321,343,350,327,200
None
d 88,72,97,84,86,85,100
None
e. 145,119,122,118,125,116
145
12,28,35,42,47,49,50
25. In Exercise 24, what value corresponds to the 60th
percentile? 47
26. Hurricane Damage Find the percentile rank for
each value in the data set. The data represent the
values in billions of dollars of the damage of
10 hurricanes. 5; 15;25;35;45;55;65; 75;85;95
f
14, 16,27,18, 13, 19,36,15,20
54
None
O
31. Another measure of average is called the
midquartile; it is the numerical value halfway
between QI and Q3' and the formula is
1.1, 1.7, 1.9,2.1,2.2,2.5,3.3,6.2,6.8,20.3
Source: Insurance Services Office.
Midquartile
27. What value in Exercise 26 corresponds to the 40th
percentile? 2.1
O
28. Test Scores Find the percentile rank for each test
score
in the data set.
8th; 25th; 42nd; 58th; 75th; 92nd
= Q, +2
Q3
Using this formula and other formulas, find Q" Q2'
Q3' the midquartile, and the interquartile range for each
data set.
a. 5, 12, 16,25,32,38
12;20.5;32;22; 20
b. 53, 62, 78, 94, 96, 99, 103 62;94; 99; 80.5;37
5, 12, 15, 16,20,21
29. What test score in Exercise 28 corresponds to the 33rd
percentile? 12
Technology
Step by Step
MINITAB
StepbyStep
Calculate
Descriptive
Statistics
from Data
Example MT3-1
1. Enter the data from Example3-23 into C1 of MINITAB.Name the columnAutoSales.
2. SelectStabBasic Statistics>DisplayDescriptiveStatistics.
3. The cursor will be blinkingin the Variablestext box. Double-clickC1 AutoSales.
4. Click [Statistics] to view the statisticsthat can be calculatedwith this command.
a) Check the boxes for Mean, Standard deviation, Variance, Coefficient of variation,
Median, Minimum,Maximum,and N nonmissing.
Descriptive Statistics - Statistics
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b) Remove the checks from other options.
3-52
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