189
12
12.1 (a) Population: the 175 residents of Tonya’s dorm;
p
is the proportion who like the food.
(b)
±
0.28.
12.2 (a) The population is the 2400 students at Glen’s college, and
p
is the proportion who believe
tuition is too high. (b)
±
0.76.
12.3 (a) The population is the 15,000 alumni, and
p
is the proportion who support the president’s
decision. (b)
±
0.38.
12.4 (a) No—the population is not large enough relative to the sample. (b) Yes—we have an SRS,
the population is 48 times as large as the sample, and the success count (38) and failure count (12)
are both greater than 10. (c) No—there were only 5 or 6 “successes” in the sample.
12.5 (a) No—
np
0
and
n
(1
²
p
0
) are less than 10 (they both equal 5). (b) No—the expected number
of failures is less than 10 (
n
(1
²
p
0
)
±
2). (c) Yes—we have an SRS, the population is more than 10
times as large as the sample, and
np
0
±
n
(1
²
p
0
)
±
10.
12.6 (a)
so the 95% confidence interval is 0.54
³
(1.96)(0.01561)
±
0.51 to 0.57. The margin of error is about 3%, as stated.
(b) We weren’t given sample sizes for each gender. (However, students who know enough
algebra can get a good estimate of those numbers by solving the system
x
´
y
±
1019 and
0.65
x
´
0.43
y
±
550: approximately 508 men and 511 women.)
(c) The margin of error for women alone would be greater than 0.03 since the sample size is
smaller.
12.7 (a) The methods can be used here, since we assume we have an SRS from a large population,
and all relevant counts are more than 10. For TVs in rooms:
and
±
so the 95% confidence interval is
to 0.689. For preferring Fox:
and
±
so the 95% confidence interval is
to 0.203.
(b) In both cases, the margin of error for a 95% confidence interval (“19 cases out of 20”) was
(no more than) 3%.
(c) We test
H
0
:
p
±
0.5 versus
H
a
p
µ
0.5. The test statistic is
which gives very strong evidence against
H
0
(
P
¶
0.0002); we conclude that more than half
of teenagers have TVs in their rooms. (Additionally, the interval from (a) does not include
0.50 or less.) With the TI83,
z
±
10.379 and
P
±
1.577
·
10
²
25
.
12.8 (a)
and since
and
are both greater than 10, the confidence
interval
based
on
z
can
be
used.
The
95%
confidence
interval
for
p
is
or 0.59435 to 0.72565.
.66
³
1
1.96
2
1
11
.66
21
.34
2>
200
2
±
.66
³
0.06565,
n
1
1
²
ˆ
p
2
±
68
n
ˆ
p
±
132
ˆ
p
±
.66,
z
±
1
0.66
²
0.50
3
1
0.5
0.5
2
1048
±
10.36,
0.18
³
1
1.96
0.01187
2
±
0.157
0.01187,
SE
p
ˆ
±
2
1
0.18
0.82
1048
ˆ
p
2
±
0.18
±
0.631
0.66
³
1
1.96
0.01463
2
2
1
0.66
0.34
1048
±
0.01463,
SE
p
ˆ
ˆ
p
1
±
0.66
SE
p
ˆ
±
2
1
0.54
0.46
1019
±
0.01561,
ˆ
p
ˆ
p
ˆ
p
6851F_ch12_189_193
17/9/02
20:05
Page 189
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Chapter 12
(b) Yes; the 95% confidence interval contains
only
values that are less than 0.73, so it is
likely that for this particular population,
p
differs from 0.73 (specifically, is less than 0.73).
12.9 (a)
and
(b) Checking conditions,
and
are both at least 10. Provided that there are at least (84) (
p
) = 840 applicants
in the population of interest, we are safe constructing the confidence inerval.
0.1098 to 0.2473.
12.10
—use
n
5
356. With
so the true mar
gin of error is (1.645)(0.0265)
5
0.0436.
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 Fall '08
 O
 Normal Distribution, Statistical hypothesis testing, Shaq

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