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According to the Bureau of Crime Statistics and Research of Australia, the mean length of imprisonment for motor-vehicle-theft offenders in Australia is 16.7 months. A group of researchers would like to perform a hypothesis test to decide whether the mean length of imprisonment for motor-vehicle-theft offenders in Sydney differs from the national mean in Australia. They have found out that Sydney population standard deviation is 6.0. They have also decided to choose a random sample of size 100 and perform the test at the significance level of 0.05. Suppose that, in reality, the mean length of imprisonment in Sydney is 15.5 months.

a. State the null and alternative hypotheses.

b. Determine the probability of a Type I error.

c. Determine the probability of a Type II error.

d. Simulate 1,000 samples, each of size 100. e. Determine the mean of each sample in part (d).

f. For the 1,000 samples obtained in part (d), about how many would you expect to lead to nonrejection of the null hypothesis? Explain your answer.

g. For the 1,000 samples obtained in part (d), determine the number that lead to nonrejection of the null hypothesis.

h. Compare your answers from parts (f) and (g), and comment on any observed difference.

i. Plot the power curve for the range of true between 14 to 19. Interpret your plot.