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In Exercise 4.2.27, in finding a confidence interval for the ratio of the variances of two normal distributions, we used a statistic  which has an Fdistribution when those two variances are equal. If we denote that statistic by F,  Exercise 4.2.27 Let  be two independent random samples from the respective normal distributions  where the four parameters are unknown. […]

Define the setsA certain hypothesis assigns probabilities pi0 to these sets Ai in accordance with This hypothesis (concerning the multinomial pdf with k = 8) is to be tested, at the 5% level of significance, by a chi-square test. If the observed frequencies of the sets Ai, i = 1, 2,…, 8, are, respectively, 60,

A die was cast n = 120 independent times and the following data resulted: If we use a chi-square test, for what values of b would the hypothesis that the die is unbiased be rejected at the 0.025 significance level?

Let the result of a random experiment be classified as one of the mutually exclusive and exhaustive ways and also as one of the mutually exhaustive ways . Say that 180 independent trials of the experiment result in the following frequencies: where k is one of the integers 0, 1, 2, 3, 4, 5. What

Kloke and McKean (2014) present a data set concerning crime and alcoholism. The data they discuss is in Table 4.7.1. It contains the frequencies of criminals who committed certain crimes and whether or not they are alcoholics. The data are also in the file crimealk.rda. (a) Using code similar to that given in Exercise 4.7.6,

Recall that logHence, by using a uniform(0, 1) generator, approximate log 2. Obtain an error of estimation in terms of a large sample 95% confidence interval. Write an R function for the estimate and the error of estimation. Obtain your estimate for 10,000 simulations and compare it to the true value.

Prove the converse of Theorem MCT. That is, let X be a random variable with a continuous cdf  Assume that is strictly increasing on the space of X. Consider the random variable  Show that Z has a uniform distribution on the interval (0, 1).

It is proposed to fit the Poisson distribution to the following data: (a) Compute the corresponding chi-square goodness-of-fit statistic. (b) How many degrees of freedom are associated with this chi-square? (c) Do these data result in the rejection of the Poisson model at the α = 0.05 significance level?

Similar to Exercise 4.8.2 but now approximate Exercise 4.8.2 Recall that logHence, by using a uniform(0, 1) generator, approximate log 2. Obtain an error of estimation in terms of a large sample 95% confidence interval. Write an R function for the estimate and the error of estimation. Obtain your estimate for 10,000 simulations and compare

1. Suppose X is a random variable with the pdf  where b > 0. Suppose we can generate observations from f(z). Explain how we can generate observations from . 2. Determine a method to generate random observations for the logistic pdf, (4.4.11). Write an R function that returns a random sample of observations from a logistic distribution.