Homework Assignment #7

Statistics 220, Fall 2005

Due November 2nd

Show your work! Partial credit will be given on the basis of your work; writing down the answer without showing your work will get you no credit.

Non-Devore problem #1 (for ch. 6): If random variable X has the binomial $(n,p)$ distribution, then an estimator of p is $\hat{p} = X/n$ .
(a) What is the bias of $\hat{p}$ for estimating p?
(b) What is the variance of $\hat{p}$ ?
(c) What is the standard error of $\hat{p}$ ?
(d) What is the mean squared error (MSE) of $\hat{p}$ for estimating p?
(e) A researcher wants to estimate the unknown p in such a way that the standard error of $\hat{p}$ 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 $\hat{p}$ from the previous problem. Remember that if n is large enough, X is approximately Gaussianly distributed, using the central limit theorem.
(a) Explain why
\[ \frac{\hat{p} - p}{\sqrt{p(1-p)/n}} \]
has approximately the standard normal distribution if n is large enough.
(b) Use part (a) to show that
\[ \left( \hat{p} - z_{\alpha/2} \sqrt{\frac{p\left(1-p\right)}{n}}, \:\:\: \hat{p} + z_{\alpha/2} \sqrt{\frac{p\left(1-p\right)}{n}} \right) \]
is a $100(1-\alpha)$ % 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
\[ \left( \hat{p} - z_{\alpha/2} \sqrt{\frac{\hat{p}\left(1-\hat{p}\right)}{n}}, \:\:\: \hat{p} + z_{\alpha/2} \sqrt{\frac{\hat{p}\left(1-\hat{p}\right)}{n}} \right) \]
as a $100(1-\alpha)$ % confidence interval for the unknown p if n is large enough. Note that we have replaced p with its estimator $\hat{p}$ . Now, do problem 7.20 in Devore.

Problems from Devore:

  1. Problem 6.4
  2. Problem 6.8
  3. Problem 6.10
  4. Problem 6.12
  5. Problem 6.14
  6. Problem 6.16
  7. Problem 6.19
  8. Problem 6.34
  9. Problem 7.2
  10. Problem 7.12
  11. Problem 7.14
  12. Problem 7.32
  13. Problem 7.55