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Chapter 14 Solutions
14.1. (a)
The standard deviation is
σ/
√
840
.
=
2
.
0702.
(b)
The missing number is
2
√
840
.
=
4
.
1404.
(c)
The 95% confdence interval is
x
±
2
√
840
.
=
268 to 276.
14.2.
Shown below are sample output screens For (a) 10 and (b) 1000 SRSs. In 99.4%
oF all repetitions oF part (a), students should see between 5 and 10 hits (that is,
at least 5 oF the 10 SRSs capture the true mean
µ
). Out oF 1000 80% confdence
intervals, nearly all students will observe between 76% and 84% capturing the mean.
14.3.
Search Table A For 0.0125 (halF oF the 2.5% that is
not
included in a 97.5% confdence interval). This area
corresponds to
z
∗
=
2
.
24. SoFtware gives
z
∗
=
2
.
2414.
z
*
–
z
*0
Standard
Normal curve
Probability
= 0.0125
Probability
= 0.0125
14.4.
State:
What is the true conductivity oF this liquid?
Plan:
We will estimate the true conductivity
µ
, the mean oF all measurements oF its
conductivity, by giving a 90% confdence interval.
Solve:
The statement oF the problem in the text suggests that the conditions For inFerence
should be satisfed. The mean oF the sample is
x
=
4
.
988
3 microsiemens per centimeter
(
µ
S
/
cm). ±or 90% confdence, the critical value is
z
∗
=
1
.
645. A 90% confdence interval
For
µ
is thereFore
x
±
z
∗
µ
σ
√
n
¶
=
4
.
988
3
±
1
.
645
µ
0
.
2
√
6
¶
=
4
.
988
3
±
0
.
1343
=
4
.
8540 to 5
.
1226
µ
S
/
cm
.
Conclude:
We are 90% confdent that the true conductivity is between 4.8540 and
5.1226
µ
S
/
cm.
169
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Chapter 14
Introduction to Inference
14.5. (a)
The two low scores (72 and 74) are both possible outliers, but
there are no other apparent deviations from Normality.
(b)
State:
What
is the mean IQ
µ
of all seventhgrade girls in this school district?
Plan:
We will estimate
µ
by giving a 99% conFdence interval.
Solve:
The problem states that these girls are an SRS of the popula
tion. In part (a), we saw that the scores appear to come from a Normal
distribution. With
x
.
=
105
.
84, our 99% conFdence interval for
µ
is
105
.
84
±
2
.
576
µ
15
√
31
¶
.
=
105
.
84
±
6
.
94
=
98
.
90 to 112
.
78 IQ points.
Conclude:
We are 99% conFdent that the mean IQ of seventhgrade
girls in this district is between 98.90 and 112.78.
7
24
7
8
8
69
9
13
9
68
10
023334
10
578
11
11222444
11
89
12
0
12
8
13
02
14.6. (a)
If
µ
=
115, the distribution is approximately
Normal with mean
µ
=
115 and standard deviation
σ/
√
25
=
6.
(b)
The actual result lies out toward the
high tail of the curve, while 118.6 is fairly close to the
middle. If
µ
=
115, observing a value similar to 118.6
would not be too surprising, but 125.8 is less likely, and it therefore provides some evidence
that
µ>
115.
121
127
133
109
103
97
115
125.8
118.6
14.7. (a)
If the claim is true, the sampling distribution
of
x
is Normal with mean 5
µ
S
/
cm and standard
deviation
√
6
.
=
0
.
0816
µ
S
/
cm.
(b)
4.98 is less than
0.25 standard deviations below the presumed mean,
while 4.7 is about 3.67 standard deviations below. If
µ
=
5, observing a value similar to 4.98 would not be too surprising, but 4.7 is less likely,
and it therefore provides evidence that
µ
is different from 5. (SpeciFcally, it suggests that
µ<
5.)
5.08
5.16
5.24
4.92
4.84
4.76
5
4.7
4.98
14.8.
H
0
:
µ
=
115 vs.
H
a
:
115. (Because the teacher suspects that older students have a
higher mean, we use a onesided alternative.)
14.9.
H
0
:
µ
=
5 vs.
H
a
:
µ
6=
5. (We are concerned about deviation from 5 in either direction,
so we use a twosided alternative.)
14.10.
H
0
:
µ
=
50 vs.
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This note was uploaded on 11/23/2010 for the course STAT 2325151 taught by Professor T during the Spring '10 term at Waters College.
 Spring '10
 T
 Standard Deviation

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