The Food and Drug Administration (FDA) is asked to approve a new drug. The new drug should contain less than 25mg of the active ingredient “toxin”, which is assumed to have dangerous side effects. The FDA would like to restrict the error of “approving the drug despite its too high content of toxin” to a maximum risk of 5% (α ≤ 0.05). Let Xi : “ The content of toxin in the i-th pill [in mg].” ∼ N(µ, σ2 ) ∼ N(µ, 4). A simple random sample of n = 50 pills ( Xi ∼ i.i.d.) will be used for the test.

7. Given a significance level of α = 5% what is the highest probability of making a type II error?

8. In the sample, x¯ = 24.6. Compute the p-value. What do you conclude? [Write down the probability that you computed.]

9. Has a type I error occurred? Explain your answer.

10. Has a type II error occurred? Explain your answer.

(Answers are given but can you please work through the problem and show the steps)

Chapter 10 p2

Allied Corporation is trying to determine whether to purchase Machine A or B. It has leased the two machines for a month. A random sample of 5 employees has been taken. These employees have gone through a training session on both machines. Below you are given information on their productivity rate on both machines. (Let the difference d = Machine A - Machine B.)

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Assume the population of differences is normally distributed.

A poll was taken this year asking college students if they considered themselves overweight. A similar poll was taken five years ago. Results are summarized below. Has the proportion increased significantly? Let α = .05.

A box contains 2 identical pistols. Pistol A contains 7 live bullets and 3 blank bullets while pistol B contains 3 live bullets and 7 blanks. Following a particularly annoying question in his Stat 230 class, evil Professor Moriarty chooses a pistol at random and then fires once directly at Holmes.

a. Find the probability that Holmes survives this shot (i.e. a blank bullet is fired).

b. If Holmes survives, he takes the other pistol and fires directly at Moriarty. What is the probability that Holmes survives the first shot and Moriarty survives this second shot? (i.e. a both bullets fired are blanks.).

c. If this ill-considered game continues until one of the two is shot with a live bullet, what is the probability that the person who survives is Holmes?

(Answers are given but could you please work through the problem and show me the steps?)

Chapter 9

Understand the difference between type I and type II errors.

Establish null and alternative hypotheses

The average U.S. daily internet use at home is two hours and twenty minutes. A sample of 64 homes in Soddy-Daisy showed an average usage of two hours and 50 minutes with a standard deviation of 80 minutes. We are interested in determining whether or not the average usage in Soddy-Daisy is significantly different from the U.S. average.

The Bureau of Labor Statistics reported that the average yearly income of dentists in the year 2012 was $110,000. A sample of 81 dentists, which was taken in 2013, showed an average yearly income of $120,000. Assume the standard deviation of the population of dentists’s incomes in 2012 is $36,000.

When doing an experiment, the experimenter wants increase the chances that subjects' characteristics, that could bias the results of the experiment, and reduce the validity of the findings, are equally distributed across the treatment groups. Which of the following procedures would the experimenter utilize to accomplish this purpose?

a) random assignment b) systematic assignment c) reliable assignment d) all of the above

Assume that adults have IQ scores that are normally distributed with a mean of mu equals 100 and a standard deviation sigma equals 15. Find the probability that a randomly selected adult has an IQ less than 115.

(Answers are given but could you please work through the problem and show me the steps?)

Chapter 8

A local health center noted that in a sample of 400 patients, 80 were referred to them by the local hospital.

A simple random sample of 36 items resulted in a sample mean of 40 and a standard deviation of 12. Construct a 95% confidence interval for the population mean.

Six hundred consumers belonging to the 25-34 age group were randomly selected in a city and were asked whether they would like to purchase a domestic or a foreign automobile. Their responses are given below.

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Develop a 95% confidence interval for the proportion of all such consumers who prefer to purchase domestic automobiles.

1. Assume that a sample is used to estimate a population proportion p. Find the 99.9% confidence interval for a sample of size 159 with 40% successes. Enter your answer as an open-interval (i.e., parentheses) using decimals (not percents) accurate to three decimal places.

C.I. =

An economist with the Liquor, Hospitality and Miscellaneous Workers' Union collected data on the weekly salaries of workers in the hospitality industry in Cairns and Townsville. The union believed that the weekly salaries of employees in Cairns were higher and they were mounting a case for the equalisation of salaries between the northern cities. The researcher took samples of size 30 and 37 in Cairns and Townsville, respectively, and found that the average and standard deviation of the weekly salaries were $585.43 and $38.72 respectively in Townsville, and $616.19 and $29.13 in Cairns. Use Cairns minus Townsville.

1. Determine a point estimate for the value of the difference in average weekly salary between the two groups (in dollars to 2 decimal places).

2. Calculate the standard error for the difference between the means assuming that the workers' salaries in both locations are normally distributed and have the same population variance (in dollars to 2 decimal places).   

3. Use Kaddstat to determine a 95% confidence interval for the difference between the average weekly salaries in Cairns and Townsville. lower limit   
upper limit   (in dollars to 2 decimal places

W represent the wingspan of an airplane and V represents the velocity which are 2 random variables.
W has a normal distribution with mean 10 and standard deviation 4.
Also V = 0.5.W + U, where U is a random variable (we might call error). Assume that U has a standard normal distribution and is independent of W. Derive each element of the variance-covariance matrix for W and V using properties of Variance and Covariance.

Historical data indicated that the time required to service the conveyor belt at Coca-Cola Amatil's Richlands operations can be modelled as a normal distribution with a standard deviation of 20 minutes. A random sample of 20 services revealed a mean service duration of 119.9 minutes. Determine a 90% confidence interval for the mean service time in minutes. State the lower bound of this interval correct to two decimal places.

The question in the textbook says:

"One study mentioned in an article that 90% of the students scored above the national average on standardized tests. Using your knowledge on mean, median and mode, explain why the school reports are incorrect. Does your analysis change if the term "average" refers to mean? To median? Explain what effect this misinformation might have on the perception of the nation's schools."

I really do not know how to answer this question, please help.

1. A toxicologist was interested in the effects of an insecticide on the mortality of a certain type of beetle. Six groups of beetles were exposed to various concentrations (C) of the insecticide. At the end of the experiment dead beetles in each group were counted.

Consider the logistic regression model

logit [P(death at concentration C)] = β0 + β1 ×C

1. Write down the contribution to the likelihood function from the first group (that is, the group with death=15, size=50 and C = 10.8).
2. Re-write the above for the log-likelihood function.
3. Write down the contribution to the likelihood function from the last group (that is, the group with death=29, size=49 and C = 13.5).
4. Re-write the above for the log-likelihood function.
5. Use proc fcmp to write a SAS program that computes the log-likelihood function for the above application. In other words, your function accepts two numbers as values of β0 and β1, and outputs the log-likelihood function evaluated at the prescribed input values. Use the testing values β0 = 2 and β1 = 3 to make sure that you function works properly (it should give you a value of −5795.3).
6. Use SAS proc logistic to find the ML estimates and the corresponding maximized log-likelihood function value. Use your own function developed in part (5) to verify the SAS results.
7. By adding or subtracting 10% from the ML estimates, you can create four pairs of (β0,β1). Compute the log-likelihood function for all these 4 pairs of parameter values. Can you find a larger log-likelihood function value than that evaluated at the ML estimates? Why?

2) Revisit the toxicology example: six groups of beetles were exposed to various concentrations

(C) of the insecticide. At the end of the experiment dead beetles in each group were counted.

Consider the logistic regression model

logit[P(death at concentration C)] = β0 + β1 X C

1. Under the assumption β1 = 0, write down the log-likelihood function (of _0 alone).
2. Write a SAS program to draw the above log-likelihood function.
3. Use SAS to find the ML estimate β0 under the assumption β1 = 0.
4. Use your own SAS program to find the maximized log-likelihood function value at the above ML estimate of _0. Compare your results with what SAS reports.
5. Using results from Homework 2, calculate the likelihood ratio test statistics for testing H0 : β1 = 0 vs. HA : β1 ≠0.
6. What is the p-value of the above?
7. Calculate the OR comparing two concentrations, one of which is 0:4 units higher than the other.
8. Calculate the estimated probability of beetle death at concentration C = 12:4.
9. Use SAS to draw a graph relating probability of beetle death to concentration.

A pawn shop is open to purchasing goods that may not have been acquired through honest means. Every time the shop makes a questionable transaction there is a probability that the sale gets busted by the local police. The owner estimates that the probability that any single transaction will be busted is 2% independent from all others. He also estimates that the net profit he makes on one transaction is well-described by a normal distribution with average $50 with standard deviation of $15.

i. How much profit can they expect to see before the operation is busted?

ii. Find the standard deviation for total profit.

iii. A local police chief, for a monthly fee of $100, can make sure that they are not bothered as often, effectively reducing probability of a bust on each transaction to 0.5%. Suppose that they do 1 questionable transaction a day. Estimate whether they can expect to make more or less profit if they pay the bribe?