137
8
8.1
(a) No: There is no fixed
n
(i.e., there is no definite upper limit on the number of defects). (b)
Yes: It is reasonable to believe that all responses are independent (ignoring any “peer pressure”),
and all have the same probability of saying “yes” since they are randomly chosen from the popu-
lation. Also, a “large city” will have a population over 1000 (10 times as big as the sample). (c) Yes:
In a “Pick 3” game, Joe’s chance of winning the lottery is the same every week, so assuming that a
year consists of 52 weeks (observations), this would be binomial.
8.2
(a) Yes: It is reasonable to assume that the results for the 50 students are independent, and each
has the same chance of passing. (b) No: Since the student receives instruction after incorrect answers,
her probability of success is likely to increase. (e) No: Temperature may affect the outcome of the test.
8.3
(a) .2637. (b) The binomial probabilities for
x
0, . . ., 5 are: .2373, .3955, .2637, .0879, .0146, .0010.
(e) The cumulative probabilities for
x
0, . . ., 5 are: .2373, .6328, .8965, .9844, .9990, 1. Compared with
Corinne’s cdf histogram, the bars in this histogram get taller, sooner. Both peak at 1 on the extreme right.
8.4 Let
X
the number of correct answers.
X
is binomial with
n
50,
p
0.5.
(a)
P
(
X
25)
1
P
(
X
24)
1
binomcdf (50, .5, 24)
1
.444
.556.
(b)
P
(
X
30)
1
P
(
X
29)
1
binomcdf (50, .5, 29)
1
.899
.101.
(c)
P
(
X
32)
1
P
(
X
31)
1
binomcdf (50, .5, 31)
1
.968
.032.
8.5
(a) Let
X
the number of correct answers.
X
is binomial with
n
10,
p
0.25. The proba-
bility of at least one correct answer is
P
(
X
1)
1
P
(
X
0)
1
binompdf (10, .25,
0)
1
.056
.944.
(b) Let
X
the number of correct answers. We can write
X
X
1
X
2
X
3
, where
X
i
the
number of correct answers on question
i
. (Note that the only possible values of
X
i
are 0 and 1,
with 0 representing an incorrect answer and 1 a correct answer.) The probability of at least one
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