Let X1, ..., Xn be a sample from a binomial random variable with known parameter n and unknown parameter p. (a) Find the maximum likelihood estimator for p. (b) Is the estimator unbiased? (c) Is the estimator consistent?

If X is a negative binomial rv, then Y= r+Xis the total number of trials necessary to obtainr S’s. Obtain the mgf of Y and then its mean valueand variance. Are the mean and variance intuitively consistent with the expressions for E(X)and V(X)? Explain

Let X1, X2, X3, ..., X, be a random sample from a distribution with known variance Var(X,) = o², and unknown mean EX, = 0. Find a (1 – a) confidence interval for 0. Assume that n is large.

You may need to use the appropriate technology to answer this question. The following data are from a completely randomized design. In the following calculations, use a = 0.05. Treatment Treatment Treatment 3 64 82 69 47 71 53 54 87 60 35 64 46 50 76 57 148.67 108.67 96.67 (a) Use analysis of variance to test for a significant difference among the means of the three treatments.

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'Melanoma' is a form of skin cancer and each year 17% of the patients who suffer from the disease die. A random sample of 10,000 melanoma patients is formed at the start of a year. Let Y denote the number of patients in this sample who will die during the year? (a) What is the expected value of Y? That is, compute E(Y). Answer: [Select] (b) What is the variance of Y? That is, compute var(Y). Answer: [Select] (c) What is the probability that Y exceeds 1,800? That is, compute P(Y> 1,800). Answer: [Select]

Let Y₁, Y₂, ..., Y₁ denote a random sample from a normal distribution with unknown mean and unknown variance o². For testing Ho: o² = 0 vs. Ha: o²o, show that the a-level Likelihood Ratio Test (LRT) is equivalent to the x² test

Independent random samples, each containing 70 observations, were selected from two populations. The

Executives of a supermarket chain are interested in the amount of time that customers spend in the stores during shopping trips. The mean shopping time,  μ , spent by customers at the supermarkets has been reported to be  36  minutes, but executives hire a statistical consultant and ask her to determine whether it can be concluded that  μ  is greater than  36  minutes. The consultant plans to do a statistical test and collects a random sample of  50  shopping times at the supermarkets. Suppose that the population of shopping times at the supermarkets has a standard deviation of  16  minutes and that the consultant performs her hypothesis test using the  0.05  level of significance.