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**Non-Devore problem #1** (for ch. 6): If random
variable *X* has the binomial
distribution,
then an estimator of *p* is
.

(a) What is the bias of
for estimating *p*?

(b) What is the variance of
?

(c) What is the standard error of
?

(d) What is the mean squared error (MSE) of
for estimating *p*?

(e) A researcher wants to estimate the unknown *p* in such a way
that the standard error of
is at most 0.10. What value
of *n* should be chosen? Keep in mind that *p* is unknown, so
you answer cannot depend on *p*. And *X* has not yet been
observed (no data), so your answer cannot depend on *X* either. A
further hint: You can find how large *n* has to be assuming
*p* is known, and then find the *p* that maximizes that function.

**Non-Devore problem #2** (for ch. 7):
We continue to consider the estimator
from the previous
problem. Remember that if *n* is large enough,
*X* is approximately Gaussianly distributed, using the central limit
theorem.

(a) Explain why

has approximately the standard normal distribution if *n* is large
enough.

(b) Use part (a) to show that

is a
%
confidence interval for the
unknown *p* if *n* is large enough.

(c) You will note that the result in part (b) gives the lower and upper
endpoints of the interval as a function of *p*, which is
unknown. Instead, we usually use

as a
%
confidence interval for the unknown *p*
if *n* is large enough. Note that we have replaced *p* with its
estimator
.
Now, do problem 7.20 in Devore.

**Problems from Devore**:

- Problem 6.4
- Problem 6.8
- Problem 6.10
- Problem 6.12
- Problem 6.14
- Problem 6.16
- Problem 6.19
- Problem 6.34
- Problem 7.2
- Problem 7.12
- Problem 7.14
- Problem 7.32
- Problem 7.55