This set of problems is due at the beginning of class, Friday, 1 March.

Note: You do not have to show the R commands you use to solve the following problems. However, you do need to print out any graphical results, and label them appropriately. Moreover, for any hypothesis test, you must state the null hypothesis you are testing, define all parameters or random variables that you use, state the value of the test statistic, state the p-value for the test, and state what conclusion you may draw from the test.

Problem from Chapter 4: 4.54 (data in corn)

Problems from Chapter 5: 5.3a, 5.24 (Note: “Develop a method for estimating $\theta$” means “Find the maximum likelihood estimate for $\theta$.” You may ignore the “Bonus” part of the question.)

And:

1. A coin is tossed 100 times, landing heads 61 times. Find exact and approximate 80% and 99% confidence intervals for the true probability that the coin will land head.
2. Suppose $X_1, X_2, \ldots, X_n$ is an iid random sample from a uniform distribution on the interval $[0, \theta]$. Let $Y = X_{(n)}$ be the maximum of $X_1, X_2, \ldots, X_n$.
   1. Use the fact that $P(Y \le y) = P(X_1 \le y, X_2 \le y, \ldots, X_n \le y)$ to find the cumulative distribution function of $Y$.
   2. Use the previous result to find the probability density function of $Y$.
   3. Is $Y$ an unbiased estimator for $\theta$? If not, find a constant $k$ such that $W = kY$ is an unbiased estimator for $Y$.
   4. Find the variance of $W$.
   5. Compare the variance of $W$ with the variance of the method of moments estimator $2\bar{X}$.