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Non-Devore problem #1 (for ch. 6): If random
variable X has the binomial
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
(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: