statistics problems really need you help

1. During the morning rush hour customers arrive at a busy Tim Horton’s drive through at a rate of 15 in a 20 minute period.

a. What is the probability that a customer will arrive within the next 3 minutes?

b. What is the probability that the time between arrivals will be 12 minutes or more?

c. What is the probability that the next customer will arrive within 4 and 6 minutes?

d. What is the probability that only 1 customer will arrive during a 20 minute period?

2. Government standards limit the airborne amount of a known cancer causing gas. A company using a known cancer causing gas in its production process limits the exposure of its employees to the gas by shutting down production when the amount of the gas reaches 3 ppm with a standard deviation of 0.5 ppm.

a. If you were the plant manager and you are told management wants as little down time as possible, would you choose a larger or a smaller value of alpha to test whether the amount of gas is dangerous? Explain.

b. If you were an employee in the plant, would you want the choice of alpha to be larger or smaller? Explain.

c. A random sample of 50 air specimens from various sensors in the plant produced a mean of 3.1 ppm. Is this sample data sufficient to shut down production? Use alpha = 0.01.

d. Describe both Type I and Type II errors in the context of this problem.

e. Calculate the probability of Type II error if in fact the true mean plant level if 3.1 ppm. Use alpha = 0.01.

f. What is the value of the Power of the test in this case?

g. Calculate the probability of Type II error if in fact the true mean plant level if 3.1 ppm when alpha = 0.05.

h. What is the Power of the test now?

i. What do the findings of parts e, f , h , and i confirm about the relationships between alpha, beta and the Power of the test?

3. Drug use among athletes remains a problem at Olympic competitions. Detection is not foolproof due to the inaccuracy of tests. Suppose a test is carried out for a particular drug on a population of 1,000 athletes of which there are 100 drug users. 50 of these drug users would test positive. Of the non-drug users, 9 would test positive.

a. The sensitivity of the test is measured by the probability that a drug user will test positive. Calculate this probability.

b. The specificity of the drug test is measured by the probability that a non-drug user will test negative. Calculate this probability.

c. Find the probability that if an athlete tests positive, the athlete really is a drug user.

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4. Answer the following.

a. Babies are born at a rate of 2 per hour at the local hospital. What is the probability that none would be born in a sixty minute period?

b. In a total of 10 births what is the probability that 6 will be boys if the probability of a boy is 0.51?

5. The average time for all Economics students to complete a 120 minute midterm is 78 minutes with a standard deviation of 12 minutes.

a. What proportion of the students finish the midterm in 72 minutes or less?

b. What proportion of students finish in 82 minutes or more?

c. What proportion of students finish in between 100 and 110 minutes?

d. How many minutes would 90% of students finish in?

6. McFarland Insurance Company knows that the average cost to process a claim for their company is currently $60. Given the economic conditions they initiated cost-cutting measures. To evaluate the effectiveness of these measures after six months, McFarland selected at random 26 claims and determined the cost to process them. The sample information is as follows:

$45 49 62 40 43 61

48 53 67 63 78 64

48 54 51 56 63 69

58 51 58 59 56 57

38 76

a. Calculate the mean and standard deviation of the sample data.

b. Derive a 99% confidence interval for the average cost of processing a claim after the cost cutting measures were implemented.

c. Perform the hypothesis test to determine if the measures reduced the cost of processing the claims. Use alpha = 0.01.

d. Could you have drawn the conclusion you reached in part c using the information from part b? Explain why or why not. (You will need to think about this one!)

e. What assumption is necessary to conduct these tests?

7. A company is considering monitoring their employee Internet usage, but not before informing them of the decision to do so. They will go ahead with the plan if more than 50% of their employees respond that they agree as long as they are informed of the new practice. In a sample of 405 employees of the company, 223 replied that they were not opposed to the new action.

a. Using a level of significance of 0.05, test the hypothesis that the majority of employees have no objection to the monitoring.

b. What assumptions must hold for this test to be performed?

c. Calculate the probability of Type II error if in fact 55% were not opposed to the new action.

d. What is the value of the Power of the test in this case?

e. Describe Type I and Type II error, and the Power of the Test in general terms AND in the context of the question.