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.

​

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?

A repairman has 20 jobs that need to be completed today. Usually, 70% of all jobs are fairly straightforward, so that the time it takes to complete them is well-described by normal distribution with average 10 minutes and standard deviation 1 minute. The rest are challenging jobs, so that completion time is well-modeled with exponential distribution with average 1 hour.

i. If she starts at 8am, what is the expected time when she will finish all already assigned jobs?

ii. When asked when she will finish the jobs, she wants to give a safe answer by overestimating it by one standard deviation. Find the standard deviation for the time needed to finish all jobs.

A study examined parental influence on the decisions of teenagers from a certain large region to smoke. A randomly selected group of​ students, from the​ region, who had never smoked were questioned about their​ parents' attitudes toward smoking. These students were questioned again two years later to see if they had started smoking. The researchers found​ that, among the

263

students who indicated that their parents disapproved of kids​ smoking,

53

had become established smokers. Among the

43

students who initially said their parents were lenient about​ smoking,

18

became smokers. Do these data provide strong evidence that parental attitude influences​ teenagers' decisions about​ smoking? Complete parts a through i below.

​a) What kind of design did the researchers​ use?

A prospective observational study

Your answer is correct.

An experimental study

A retrospective observational study

​b) Write the appropriate hypotheses. Let

p1

be the proportion of students whose parents disapproved of smoking who became smokers. Let

p2

be the proportion of students whose parents were lenient about smoking who became smokers.

Choose the correct answer below.

A.

H0​:

p1minus

p2equals

0

HA​:

p1minus

p2greater than

0

B.

H0​:

p1minus

p2equals

0

HA​:

p1minus

p2not equals

0

Your answer is correct.

C.

H0​:

p1minus

p2not equals

0

HA​:

p1minus

p2equals

0

D.

H0​:

p1minus

p2greater than

0

HA​:

p1minus

p2equals

0

​c) Are the assumptions and conditions necessary for inference​ satisfied?

A.

​No, because the Independent Groups Assumption is not satisfied.

B.

​Yes, all of the assumptions and conditions are satisfied.

Your answer is correct.

C.

​No, because the​ Success/Failure Condition is not satisfied.

D.

​No, because the​ 10% Condition is not satisfied.

E.

​No, because the Randomization Condition is not satisfied.

​d) Test the hypothesis and state the conclusion.

Determine the test statistic.

zequals

negative 3.13

​(Round to two decimal places as​ needed.)

Find the​ P-value.

Pequals

. 002

​(Round to three decimal places as​ needed.)

State the conclusion. Use a significance level of

alpha

equals0.10

.

Choose the correct answer below.

A.

Do not reject

the null hypothesis. There

is

sufficient evidence that parental attitude influences​ teenagers' decisions about smoking.

B.

Do not reject

the null hypothesis. There

is not

sufficient evidence that parental attitude influences​ teenagers' decisions about smoking.

C.

Reject

the null hypothesis. There

is not

sufficient evidence that parental attitude influences​ teenagers' decisions about smoking.

D.

Reject

the null hypothesis. There

is

sufficient evidence that parental attitude influences​ teenagers' decisions about smoking.

Your answer is correct.

​e) Explain in this context what your​ P-value, P, means. Choose the correct answer below.

A.

If the observed difference is the true​ difference, then there is about a

​(100 times

​P)%

chance that there is no difference in the proportions.

B.

If there is no difference in the​ proportions, there is about a

​(100 times

​P)%

chance of seeing the observed difference or larger by natural sampling variation.Your answer is correct.

C.

There is about a

​(100 times

​P)%

chance that there is no difference in the proportions.

D.

There is about a

​(100 times

​P)%

chance that there is a difference in the proportions.

​f) If that conclusion is actually​ wrong, which type of error was​ committed?

A.

A Type

II

error was committed because the null hypothesis is

false

​,

but was

not

rejected.

B.

A Type

Upper I

error was committed because the null hypothesis is

true

​,

but was

mistakenly

rejected.

Your answer is correct.

​g) Create a 90​%CI for the difference of two​ proportions,p 1 minus p 2

. I am having trouble with G.)

[Counting and Probability] Consider the experiment of flipping a coin four times.

a. Using a tree, determine the probability of one or two tails, with a biased coin with P(H) = 2/3. Compare to the probability with an unbiased coin

b. [Bayes’ Rule] Using the results of the part a suppose we have two coins, one unbiased, and a biased coin with P(H) = 2/3. We select a coin at random, flip it three times, and observe either one or two tails. What is the probability we started with the biased coin?
[Hint: use a two-level tree, with the second level using the probabilities from a.]

Define and discuss the difference between linear regression and multiple regression. Are there any assumptions which must be met before using multiple regression?