Statistics 20: Quiz 5 Solutions
November 29, 2006
1. Suppose
X
is a Normal random variable with SD 1 and unknown mean. Your null hypothesis
is that its mean
E
[
X
] is 0, and the alternative is that
E
[
X
]
6
= 0. If you want to be 95%
conﬁdent in your results, for what values of
X
will you reject the null hypothesis?
We reject the null hypothesis when
X <

1
.
96 or
X >
1
.
96. This ensures that the probability
of a Type I error under the null hypothesis is no greater than 0.05.
2. You suspect that a casino has unfairly weighted a die. You want to test that the probability
of obtaining an ace is fair. You roll the die 600 times and observe 81 aces. Is the die fair?
How conﬁdent can you be in your conclusion?
Let
X
1
,...,X
n
be independent random variables that take the value 1 if an ace is thrown
and 0 otherwise. If the die is fair,
p
=
P
(
X
i
= 1) = 1
/
6 for all rolls. Let
X
=
∑
n
i
=1
X
i
represent the total number of aces. In
n
= 600 rolls of a fair die, we expect to see
E
[
X
] =
np
= 600(1
/
6) = 100 aces.
Null Hypothesis:
E
[
X
] = 100. That is, we believe the die is fair and wish to accumulate
evidence to disprove this notion.
Alternative Hypothesis:
E
[
X
]
6
= 100. This is a twosided alternative, so we will reject the null
hypothesis if the observed number of aces is too high or too low.
By default, we set
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 Spring '08
 Staff
 Statistics, Normal Distribution, Null hypothesis, Probability theory, Type I and type II errors

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