Dr. Strangelove has created a device that predicts whether or not a missile has been launched. Given that a missile has been launched, the device will correctly report that a missile has been launched 95 percent of the time. Given that a missile has not been launched, there is a 2 percent chance that the device will incorrectly report that a missile has been launched. There is a 1 percent chance that a missile will be launched. What is the probability that the device will indicate that a missile has been launched?

a. 0.293

b. 0.324

c.0.675

d. 0.971

e. 0.016

A coffee machine fills a cup automatically with a volume of coffee normally distributed with expectation 2.2dl and standard deviation 0.3dl. The cups used can take a volume of coffee with expectation value 2.5 dl and standard deviation 0.45 dl. The volume of the coffee machine drops and the volume of a cup is independent.

What is the probability of it accidentally overflowing with a cup?

You know that the definition of the variance is Var[X] = 1 n − 1 Z ∞ −∞ (Xi − µ) 2 dFX(x), so if we have a plug-in estimator S 2 = 1 n − 1 Xn i=1 (xi − X¯) 2 and we are interested in SES2 = p Var[S2] we can use the bootstrap to obtain it. Write a function in R called bootVarSE that computes a bootstrap estimate of the standard error of the sample variance estimator. Test your function using 100 standard random normal draws (that is, rnorm(100)) as data input.

5. Nineteen people move out of a neighborhood; four are minorities. Of the nineteen, eight move onto a block with new housing, and one of these eight is a minority. How likely is it that, if there were no discrimination, less than two people out of the eight people on this new block would be minorities? If the resulting probability is less than 0.05, evidence for discrimination exists. Does such evidence exist in this case? (3)

6. There is an average of four accidents per year at a particular intersection.   What is the probability of more than one accident there next month? Hint: Use 1 month = 1/12 of a year to first get the number of accidents that are expected next month.       (3)

Research Question: Is the average weight (in grams) of all Emerald Ash Borer Beetle less than 3. To help, my Uncle Superfly (who happens to be a biologist) took a random sample of n=10 beetles from my back yard and recorded their weights.

They are:

1.5, 2, 3.5, 2.5, 1.5, 1.5, 2.5, 2, 2.5, 3

1. A study of a disease reveals that there is an average of 1 case every 22 square miles. Residents of a town that has an area of 10 square miles are concerned because there are two cases in their area. The state’s Department of Health has decided to investigate further if the probability of getting two or more cases in this town is less than 0.05. Does the Department of Health investigate further?   (3)

2. A lottery is carried out by choosing five balls, without replacement, from a box of 35 balls. The lottery ticket has five numbers on it. Find the probability that exactly four of the balls that come out of the box match the numbers on the lottery ticket. (3)

3. A neighborhood has 32 households – 27 white, and 5 nonwhite. A subset of 9 of these households move to an adjacent neighborhood. What is the probability that less than two of the households in the new neighborhood are nonwhite? (3)

Let x1, x2,x3,and x4 be a random sample from population with normal distribution with mean ? and variance ?2 . Find the efficiency of T = 17 (X1+3X2+2X3 +X4) relative to

x= x/4 , Which is relatively more efficient? Why?

A researcher wants to compare the levels of recall in younger and older subjects. Her hypothesis is that if subjects are required to process verbal information (lists of words) older subjects do less processing and therefore will recall fewer words. The following data are from a group of older subjects and a group of younger subjects who were told to memorize the words presented to them so they could be recalled at a later time. The dependent variable is the number of correctly recalled items. Do these data indicate a significant difference in items recalled between younger and older adults? Use two-tailed α = .05. DATA: Younger adults: 23, 20, 16, 17, 21, 20, 20, 22, 10, 22 SS=44 M= 20. Older adults: 10, 19, 14, 5, 10, 11, 14, 15, 11, 11 SS= 126 M= 12

In a study of heart surgery, one issue was the effect of drugs called beta-blockers on the pulse rate of patients during surgery. The available subjects were divided at random into two groups of 30 patients each. One group received a beta-blocker; the other, a placebo. The pulse rate of each patient at a critical point during the operation was recorded. The treatment group had mean 65.2 and standard deviation 7.8. For the control group, the mean was 70.3 and the standard deviation was 8.3.

(a) Do beta-blockers reduce the pulse rate at the 5% level? At the 1% level? Carry out a complete test.

(b) Give a 99% confidence interval for the difference in mean pulse rates.

PLEASE SHOW THE DIAGRAM

8. The mean pulse rate for adults is 72 beats per minute (www.healthepic.com) and let’s suppose that the standard deviation is 10. Find:

a. The probability that a randomly chosen adult has a pulse rate over 78 assuming that the rates are normally distributed.

2. The probability that a random sample of 19 adults will have a mean score over 78.

PLEASE DON'T FORGET THE DIAGRAM

Particles are a major component of air pollution in many areas. It is of interest to study the sizes of contaminating particles. Let X represent the diameter, in micrometers, of a randomly chosen particle. Assume that in a certain area, the probability density function of X is inversely proportional to the volume of the particle; that is, assume that

f(x) = c /(x^3) x ≥ 1

f(x) = 0 x < 1 where c is a constant.

a. Find the value of c so that f (x) is a probability density function.

b. Find the mean particle diameter.

c. Find the cumulative distribution function of the particle diameter.

d. The term PM10 refers to particles 10 μm or less in diameter. What proportion of the contaminating particles are PM10?

e. The term PM2.5 refers to particles 2.5 μm or less in diameter. What proportion of the contaminating particles are PM2.5?

f. What proportion of the PM10 particles are PM2.5?

g. Consider the pdf of the particle diameter. The median of the particle diameter must be

(i) larger than

(ii) smaller than

(iii) approximately equal to the mean, 2 micrometers. Choose an appropriate answer and explain why.

h. Compute the first quartile of the particle diameter.

As a part of a study, a chemical plant sampled the amount of chemicals in the soil surrounding their building. Samples were taken in front of the building (sample 1) and samples were taken in the back of the building (sample 2). The results in ppm are listed below.

(a) Please test if there is a significant difference in the amount of chemicals in the soil in the front vs the back of the chemical plant. List the 5 steps of the hypothesis test.

(b) What is the 95% confidence interval for the difference in the amount of chemicals between the two soil sources?  

Crossett Trucking Company claims that the mean weight of its delivery trucks when they are fully loaded is 5,300 pounds and the standard deviation is 170 pounds. Assume that the population follows the normal distribution. Thirty trucks are randomly selected and weighed.

Within what limits will 99 percent of the sample means occur? (Round your z-value to 2 decimal places and final answers to 1 decimal place.)

Sample means ____ to _____