CHAPTER 2 │ DESCRIPTIVE STATISTICS 75
55a. Male: 1643x=
x
x
x−
1520 123
1750 107
1992 249
8
xx
n
∑− ==
341.3s=
Female: 1709.1x
≈
x
x
x−
1785 75.9
1507 202.1
1963 245.4
8
xx
n
∑− =≈
302.7s=
76 CHAPTER 2 │ DESCRIPTIVE STATISTICS
b. Team A: 0.2993x≈
x
−
x
x
0.295 0.0043
0.310 0.0107
0.325 0.0257
0.2833 0.0315
9
xx
n
∑− =≈
0.0435s=
Team B: 0.2989x≈
x
−
x
x
0.285 0.0139
0.305 0.0061
0.315 0.0161
0.270 0.0289
0.1789 0.0199
9
xx
n
∑− =≈
0.0242s=
The mean absolute deviation is less than the sample standard deviation.
56. 2
22
1111
1 0.99 1 0.99 10
0.01 0.01
kk
kk
−= ⇒− =⇒= ⇒= =
At least 99% of the data in any data set lie within 10 standard deviations of the mean.
CHAPTER 2 │ DESCRIPTIVE STATISTICS 77
2.5 MEASURES OF POSITION
2.5 Try It Yourself Solutions
1a. 35, 36, 42, 43, 44, 46, 47, 49, 51, 51, 53, 53, 54, 54, 56, 57, 58, 59, 60, 61, 61, 63, 64, 65, 65, 66,
66, 67, 68, 69, 69, 72, 73, 73, 73, 76, 77, 78, 78, 80, 81, 82, 83, 83, 85, 86, 86, 87, 89, 89
2a. (Enter the data)
b. 123
17, 23 28.5QQQ===
c. One quarter of the tuition costs is $17,000 or less, one half is $23,000 or less, and three quarters is
$28,500 or less.
4a. Min = 35, 12 3
54, 65.5, 78,QQ Q== = Max = 89
b,c.
d. It appears that half of the ages are between 54 and 78.
78 CHAPTER 2 │ DESCRIPTIVE STATISTICS
7a. Best actor: 43.7, 8.7µσ==
2.5 EXERCISE SOLUTIONS
1. The soccer team scored fewer points per game than 75% of the teams in the league.
4. The child’s IQ is higher than 93% of the other children in the same age group.
5. The interquartile range of a data set can be used to identify outliers because data values that are
greater than
()
31.5 IQRQ+ or less than
()
11.5 IQRQ− are considered outliers.
6. Quartiles are special cases of percentiles. 1
Q is the 25th percentile, 2
Q is the 50th percentile, and
3
Q is the 75th percentile.
7. False. The median of a data set is a fractile, but the mean may or may not be fractile depending on
the distribution of the data.
11. False. The 50th percentile is equivalent to 2
Q.
12. False. Any score equal to the mean will have a corresponding z-score of zero.
13. False. A z-score of 2.5− is considered unusual.
CHAPTER 2 │ DESCRIPTIVE STATISTICS 79
b. IQR = 31
QQ−= 270 – 130 = 140
17a. Min = 900, 1
Q
=
1250, 2
Q=1500, 3
Q
=
1950, Max = 2100
b. IQR = 31
QQ−= 1950 – 1250 = 700
21a.
Min = 24, 1
Q
=
28, 2
Q=35, 3
Q
=
41, Max = 60
22a.
Min = 150, 1
Q
=
172, 2
Q=177, 3
Q
=
180, Max = 182
80 CHAPTER 2 │ DESCRIPTIVE STATISTICS
b.
25. None. The Data are not skewed or symmetric.
26. Skewed right. Most of the data lie to the left in the box-and-whisker plot.
27. Skewed left. Most of the data lie to the right in the box–and-whisker plot.
31a. 1
Q
=
2, 2
Q=4, 3
Q
=
5
b.
CHAPTER 2 │ DESCRIPTIVE STATISTICS 81
b.
35a. 5 b. 50% c. 25%
36a. $17.65 b. 50% c. 50%
37. A1.43z⇒=−
B 0z⇒=
C 2.14z⇒=
The z-score 2.14 is unusual because it is so large.
82 CHAPTER 2 │ DESCRIPTIVE STATISTICS
41a. Statistics: 78 63
78 2.14
7
x
xz
µ
σ
−−
=⇒= = ≈
Biology: 29 23
29 1.54
3.9
x
xz
µ
σ
−−
=⇒= = ≈
b. The student had a better score on the statistics test.
3.9
σ
b. The student performed equally well on the two tests.
43a. 34,000 35,000
34,000 0.44
2,250
x
xz
µ
σ
−−
=⇒== ≈−
37, 250 35,000
37,250 1
2,250
x
xz
µ
σ
−−
=⇒== =⇒
84th percentile
35,000 35,000
35,000 0
2,250
x
xz
µ
σ
−−
=⇒== =⇒
50th percentile
b. 29 33
29 1
4
x
xz
µ
σ
−−
=⇒= = =−⇒
16th percentile
41 33
41 2
4
x
xz
µ
σ
−−
=⇒= = =⇒
97.5th percentile
25 33
25 2
4
x
xz
µ
σ
−−
=⇒= = =−⇒
2.5th percentile
CHAPTER 2 │ DESCRIPTIVE STATISTICS 83
45. 72 inches
60% of the heights are below 72 inches.
46. 98th percentile
98% of the heights are below 77 inches
48. 70 69.9
70 0.03
3.0
x
xz
µ
σ
−−
=⇒= = ≈
66 69.9
66 1.3
3.0
x
xz
µ
σ
−−
=⇒= = =−
68 69.9
68 0.63
3.0
x
xz
µ
σ
−−
=⇒= = ≈−
49. 71.1 71.1
71.1 0.0
3.0
x
xz
µ
σ
−−
=⇒= = =
Approximately the 50th percentile.
b.
c. Half of the executives are between 42 and 56 years old.
84 CHAPTER 2 │ DESCRIPTIVE STATISTICS
52.
Midquartile 13
37 5
22
QQ++
===
54.
Midquartile 13
8.9 13.05 10.975
22
QQ++
== =
56a. Concert 1: Symmetric
Concert 2: Skewed right
Concert 1 has less variation.
b. Concert 2 is more likely to have outliers because its distribution is wider.
c. Concert 1, because 68% of the data should be between 16.3± of the mean.
d. No, you do not know the number of songs played at either concert or the actual lengths of the
songs.
CHAPTER 2 │ DESCRIPTIVE STATISTICS 85
58. Percentile Number of data values less than 100
Total number of data values
3100 4th percentile
73
x
=⋅
=⋅ ≈
60a.
1
Q
=
9, 2
Q=11, 3
Q
=
13
IQR 31
13 9 4QQ=−=−=
1.5 IQR 6×=
()
()
1
3
1.5 IQR 9 6 3
1.5 IQR 13 6 19
Q
Q
−× =−=
+× =+=
Any values less than 6 or greater than 19 is an outlier. So, 2 and 24 are outliers.
86 CHAPTER 2 │ DESCRIPTIVE STATISTICS
62a. Summary statistics:
Column Min
1
Q Median 3
QMax
Speed (in miles per hour) 52 65 70 72 88
63a. Summary statistics:
Column Min
1
Q Median 3
Q Max
CHAPTER 2 REVIEW EXERCISE SOLUTIONS
1. Max Min 30 8
Class width = 4.4 5
Number of classes 5
−−
==⇒
Class Midpoint, x Boundaries Frequency, f Relative
frequency
Cumulative
frequency
8-12 10 7.5-12.5 2 0.10 2
13-17 15 12.5-17.5 10 0.50 12
CHAPTER 2 │ DESCRIPTIVE STATISTICS 87
2.
3. Max Min 12.10 11.86
Class width = 0.03 0.04
Number of classes 7
−−
=≈⇒
Class Midpoint Frequency, f Relative
frequency
11.86-11.89 11.875 3 0.125
11.90-11.93 11.915 5 0.208
11.94-11.97 11.955 8 0.333
4. Max Min 12.10 11.86
Class width = 0.03 0.04
Number of classes 7
−−
=≈⇒
Class Midpoint Frequency, f Relative
frequency
11.86-11.89 11.875 3 0.125
11.90-11.93 11.915 5 0.208
11.94-11.97 11.955 8 0.333
88 CHAPTER 2 │ DESCRIPTIVE STATISTICS
5.
Class Midpoint Frequency, fCumulative
frequency
79-93 86 9 9
94-108 101 12 21
32f=
∑
7. 10 0
002552
Key: 10 10=