*Please show work and include explanations** Problem #1 In India, both average clearing time and the variance of clearing times are large

*****Please show work and include explanations******

Problem #1

In India, both average clearing time and the variance of clearing times are large ("clearing time" is time for funds to be transferred when a payment is authorized). You are a consultant who has been hired by India to improve efficiency of the banking system by reducing both the average and the variance. You install new hardware and software, and reorganize work flows.

Before your reforms, mean clearing time between any pair of large or medium-sized cities was 3.9 days, and the population standard deviation of check-clearing times was 1.5 days. (These values are known.)

Now you want to demonstrate the impact of your reforms on average clearing time. You take a random sample of 41 clearing times since your reforms were implemented. In this sample, the mean clearing time is 3.5 days, and the sample standard deviation of clearing times is 1.1 days. A normal plot indicates that clearing times are approximately normally distributed.

Test the null hypothesis that mean clearing time is at least 3.9 days, against the alternative that it is less, using a 5% significance level. What rejection region should be used to control maximum probability of Type I error at 5%?
Test the null hypothesis that the population variance of clearing times is at least 1.52 days, against the alternative that it is less, at significance level 0.05. What rejection region should be used to control maximum probability of Type I error at 5%?

Problem #2

It is known that life expectancy (population mean age at death) of males in urban areas of a less developed country is 55 years. A demographer suspects that life expectancy in rural areas of this country is less than 55 years. She identifies a random sample of 196 recent deaths of males in rural areas. In this sample, the mean age at death is 51.5 years, and the standard deviation of ages at death is 21 years. (This problem does not reflect the actuarial definition of life expectancy, which is the expected number of years until death, computed at birth, using the age-specific mortality rates prevailing at the time of birth.)

Test the null hypothesis that the population mean is at least 55 years, against the alternative hypothesis that it is less, controlling the probability of Type I error at 10%.

Problem #3

Suppose the military has designed a new missile-guidance system. With the old system, 80% of missiles landed within thirty meters of the target at a range of five hundred miles. The military claims that the new system is better (that more than 80% of missiles will land within thirty meters at five hundred miles). Ten randomly-chosen, identical missiles will be launched using the new guidance system in order to test the null hypothesis that the accuracy rate of the new system is no better than that of the old system against the alternative hypothesis that the accuracy rate of the new system is better.

State the null and alternative hypotheses in terms of the parameter of a probability model for the sample observations.
What is the form of the rejection region?
Is a sample of size ten sufficiently large for the maximum probability of Type I error to be controlled at 10%? Show why or why not. What is the minimum sample size that permits the maximum probability of Type I error to be controlled at 10%?
The military decides to increase the sample size to fifteen. What critical value should be used to control the maximum probability of Type I error at 10%?
What is the power of this test (the hypothesis test using the rejection region you just calculated) if the true accuracy rate is 90%?

Problem #4

In order to shrink its labor force, a large company (such as IBM) is considering offering an early retirement package to its employees who are over fifty-five and who have been on the payroll at least ten years. However, this plan will not be cost-effective if more than 25% of such employees choose to accept this package. A random sample of 100 eligible employees indicates that 30 of them would accept the early retirement package, if it were offered. Using a 1% level of significance, test the null hypothesis that the fraction of all eligible employers who would accept the package is no greater than 0.25, against the alternative that it is greater.