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Problem 1
The breaking strength of a rivet has a mean value of 10,000 psi and a standard deviation of 500 psi.
(a) What is the probability that the mean breaking strength for a random sample of 40 rivets would be less than
9,900 psi?
Since n = 40 is large enough, we can count on the sampling distribution to be approximately normal no matter
what the population distribution may look like.
So X
̅
bar ~ normal with mean = 10,000 and standard deviation = 500 / sqrt(40)
≈
79.06 psi.
( )
( )
1038
.
0
26
.
1
40
500
000
,
10
900
,
9
900
,
9
=

<
≈

<
=
<
Z
P
Z
P
X
P
(b) If the sample size had been 15 rather than 40, could the probability requested in part (a) be calculated from
the given information? Explain why or why not.
According to the guideline given in your text,
n
should be greater than 30 in order to apply the central limit
theorem for calculations on the sampling distribution of a sample mean when sigma is known. Therefore, using
the same procedure for
n
= 15 as was used for
n
= 40 would not be appropriate.
Problem 2
Let
μ
denote the true average radioactivity level of a water source (picocuries per liter). The value 5 pCi/L is
considered the dividing line between safe and unsafe water.
You recommend running a hypothesis test to verify
that the water source is safe. That is, the water would be considered unsafe unless significant evidence would
show that it was safe.
(a) State your null and alternative hypotheses, making sure to use proper math notations.
0
:
5
H
=
versus
:
5
a
H
<
(b) Explain in plain English what a type I error and a type II error would be in this specific context.
A type I error would involve deciding that the water is safe (rejecting
0
H
) when it isn’t (
0
H
is in fact true).
[Note that this would be a very serious error, so we may want to use a very small significance level
α
to make
sure that we are highly unlikely to commit this error].
A type II error would involve deciding that the water unsafe (failing to reject
0
H
)
when it is actually safe (
0
H
is in fact not true). [Note that, though an annoying error, this is less serious than the type I error in this context].
Problem 3
If the number of events (often called arrivals) in a time period
t
is Poisson distributed with parameter
λ
=
α
.
t
,
then what distribution must the time between successive arrivals follow? [Make sure to name the distribution and
give its parameter(s).]
If the number of events (often called arrivals) in a time period
α
.
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 Spring '11
 Guthrie,E

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