Practice exercises for exam 1
•
If you are asked for a probability that cannot be calculated by hand, you can express
your answer either in terms of normal probabilities (e.g.
P
(
Z < .
..
),
P
(
Z > .
..
)), or
in terms of R commands (e.g.
pnorm(.
..)
).
1. Suppose our goal is to estimate the expected value
μ
of a population, and to provide a
95% conﬁdence interval around our estimate. The standard deviation of the population
is 2. If we aim to have a conﬁdence interval that is around 0.25 units wide, what sample
size is required?
2. Suppose we have a test statistic
T
that is standardized under the null hypothesis. We
observe a test statistic value of
T
= 2
.
2. What is the twosided pvalue for our data?
3. Suppose we observe data
X
1
,...,X
n
, and use it to form a 95% conﬁdence interval for
the expected value of the population. Rather than using the sample standard deviation
to form the interval, we use the “nominal value”
σ
= 1
.
5 that is based on previously
collected data of a similar type. However the truth is that
σ
= 2. What is the actual
coverage probability of our conﬁdence interval?
4. Suppose we observe
X
1
,...,X
n
from a population with mean
μ
and standard deviation
1
/
2. For every pair of distinct observations
X
i
,X
j
, the correlation coeﬃcient between
the observations is cor(
X
i
,X
j
) = 0
.
4.
(a) What is the covariance between each pair of distinct observations?
(b) What is the variance of the sample mean of these data?
(c) Now suppose we are able to obtain an independent sample from a population with
the same mean and standard deviation as this one. What is the variance of the
sample mean in this case?
5. Suppose we are studying a quantity
X
that follows a normal population with mean
μ
and variance 1. However, we are sampling in such a way that negative values can
never be included in the sample (apart from this, the sample is representative of the
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 Winter '12
 KerbyShedden
 Normal Distribution, Probability, Standard Deviation, Variance, Probability theory, researcher, Researcher B

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