1. Identify the possible values of each of the 3 variables in this dataset and describe what information each of the 3 variables tells us about the data

upper quarter. 2. During the years 1998–2012, a total of 29 earthquakes
of magnitude greater than 6.5 have occurred in Papua New Guinea. Assume that the time
spent waiting between earthquakes is
exponential. Do in R (Practice and check
with calculator) a. What is the probability that the next earthquake
occurs within the next three months?
b. Given that six months has passed without an
earwhat is the pr thquake in Papua New Guinea, obability that the next three
months will be free of earthquakes? c. What is the probability of zero
earthquakes occurring in 2014? d. What is the probability that at least two
earthquake distributed with a mean of 100 and

Let ? be the sample space of an experiment and let ℱ be a collection of subsets of ?.

a) What properties must ℱ have if we are to construct a probability measure on (?,ℱ)?

b) Assume ℱ has the properties in part (a). Let ? be a function that maps the elements of ℱ onto ℝ such that

i) ?(?) ≥ 0 , ∀ ? ∈ ℱ ii) ?(?) = 1 and iii) If ?1 , ?2 … are disjoint subsets in ℱ then ?(⋃ ??) = ∞ ?=1 ∑ ?(??) ∞ ?=1 . Show that 0 ≤ ?(?) ≤ 1, ∀? ∈ ℱ.

c) Is every subset of ? necessarily an event? Explain briefly. Rigorous definitions are not necessary.

d) Assume ℱ has the properties in part (a). Let ? and ? be any two subsets of ? that are elements of ℱ.

i) Show that (? ∩ ?) ∈ ℱ.

ii) Show that (? ∖ ?) ∈ ℱ, where (? ∖ ?) is the set of outcomes that are in ? but not in ?.

iii) Show that (? △ ?) ∈ ℱ, where (? △ ?) is the set of outcomes that are either in ? or in ? but not in both.

iv) Let ?1 , ?2 , ?3 … be elements of ℱ. Show that ⋂ ?? ∞ ?=1 ∈ ℱ

Joe can choose to take the freeway or not for going to work.

There is a 0.4 chance for him to take the freeway. If he chooses freeway, Joe is late to work with probability 0.3; if he avoids the freeway, he is late

with probability 0.1

Given that Joe was early, what is the probability that he took freeway?

A researcher believes that in recent years women have been getting taller. Ten years ago, the average height of young adult women living in his city was 63 inches. The standard deviation is not known. He randomly samples eight young adult women currently residing in the city and measures their heights.

The sample data obtained is: [64, 66, 68, 60, 62, 65, 66, 63]. Use a significance level of alpha=0.05 to test if women are now taller.

A state highway goes through a small town where the posted speed limit drops down to 40MPH, but which out of town drivers don’t observe very carefully. Based on historical data, it is known that passenger car speeds going through the city are normally distributed with a mean of 47 mph and a standard deviation of 4MPH. Truck speeds are found to be normally distributed with a mean of 45MPH and a standard deviation of 6MPH. The town installed a speed camera and wants to set a threshold for triggering the camera to issue citations. If the camera is triggered, the driver is mailed a flat $50 ticket for cars and a flat $75 for trucks. On average 100 cars and 25 trucks go through the city in a day.

1. If the town sets the camera triggering speed at 50MPH, how much revenue will it make in a month (assume a month has 30 days)
2. The town wants to set the triggering speed at a value such that the fastest 10% of truck drivers get ticketed. At what value should they set the trigger?
3. At this trigger value, what percentage of cars are ticketed and what is the monthly revenue for the city?

In a multiple choice exam, there are 7 questions and 4 choices for each question (a, b, c, d). Nancy has not studied for the exam at all and decides to randomly guess the answers. What is the probability that: (please round all answers to four decimal places)

a) the first question she gets right is question number 3?

b) she gets all of the questions right?

c) she gets at least one question right?

A) Suppose that the mean and standard deviation of the scores on a statistics exam are 89.2 and 6.49, respectively, and are approximately normally distributed. Calculate the proportion of scores below 77.

B)

When students use the bus from their dorms, they have an average commute time of 8.974 minutes with standard deviation 3.1959 minutes. Approximately 66.9% of students reported a commute time less than how many minutes? Assume the distribution is approximately normal.

C)

The revenue of 200 companies is plotted and found to follow a bell curve. The mean is $637.485 million with a standard deviation of $27.6736 million. Would it be unusual for a randomly selected company to have a revenue above $687.08 million?

Consider the following sample data:

a. Calculate the range.

b. Calculate MAD. (Round your intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.)

c. Calculate the sample variance. (Round your intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.)

d. Calculate the sample standard deviation. (Round your intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.)

a) In a small country, the probability that a person will die from a certain respiratory infection is 0.004. Let ? be the random variable representing the number of persons infected who will die from the infection. A random sample of 2000 persons with this disease is chosen.

(i) Determine the exact distribution of ? and state TWO reasons why it was chosen? [4 marks]

(ii) State the values of ?(?) and ???(?). [2 marks]

(iii) Using a suitable approximate distribution, find the probability that fewer than 5 persons will die from the infection. (Do not use the exact distribution in part (i)). [4 marks]

The Wilson family had 9 children. Assuming that the probability of a child being a girl is 0.5, find the probability that the Wilson family had: at least 2 girls? at most 3 girls? Round your answers to 4 decimal places.

A bag contains 12 balls of the same shape and size.

Of these, 9 balls are blue, and the remaining 3 balls are red.

Suppose that you do the following iterative random experiment: In each iteration, 5 balls are removed randomly (without replacement) from the bag, in such a way that any 5 balls in the bag are equally likely to be the 5 balls that are removed. After doing this, you check whether among the 5 removed balls there are exactly 2 red balls. If so, then you STOP. Otherwise, you replace the 5 balls back into the bag, shake the bag up (to make sure it is randomly mixed again), and repeat the same experiment: random sample 5 balls from the bag, and check whether you have taken out exactly 2 red balls.

You repeat this until the process STOPs (i.e., when the 5 removed balls in some iteration contain exactly 2 red balls among them).

What is the expected number of times that you will sample 5 balls from this bag, in the above random experiment?

Here are summary statistics for randomly selected weights of newborn​ girls:

nequals=221

x overbar x=28.8

​hg,

s=7.9

hg. Construct a confidence interval estimate of the mean. Use a

99%

confidence level. Are these results very different from the confidence interval

25.0

hg less than<muμless than<31.4

hg with only

12

sample​ values,

x overbarxequals=28.2

​hg, and

sequals=3.6

​hg?

What is the confidence interval for the population mean

muμ​?

nothing

hgless than<muμless than<nothing

hg ​(Round to one decimal place as​ needed.)

Are the results between the two confidence intervals very​ different?

A.

​No, because the confidence interval limits are similar.

B.

​Yes, because the confidence interval limits are not similar.

C.

​No, because each confidence interval contains the mean of the other confidence interval.

D.

​Yes, because one confidence interval does not contain the mean of the other confidence interval.

For the remaining models, set up and solve LP models in EXCEL. 4) Optimized Cookie Production for a BCS Party. A friend was bringing small bags of cookies to sell at a fairly large BCS Championship Game Watch Party (there were no TCU fans present, however). Three kinds of cookies were sold: Stars (sold for $1 per bag), Circles (sold for $0.75 per bag), and Stars and Stripes (sold for $1.50 per bag). He was to bring the cookies to the Watch Party in three large boxes. (The boxes did not have to be full, but he could not bring more than three large boxes of cookies). By volume, it is a known fact that one of the large boxes can hold 100 bags of Stars, 120 bags of Circles, or 80 bags of Stars and Stripes (or a corresponding mix of cookies). HINT: Don’t concern yourself with what each box held; view this as an aggregate limit in the numbers of cookies. Previous parties had given him some hints on the demand for cookies – he knew that for the sake of variety, he needed to make at least 45 bags of each type of cookie. As he was planning his cookie composition, he also realized he was constrained by time in putting together the cookie bags. Circle cookies and Stars cookies took 1 minute per bag to finish; because Stars and Stripes had more icing, it took 2 minutes to finish each bag. He allocated 420 minutes (7 hours) to put the bags together. Can you determine how many of each of the three cookie types your friend should make to maximize sales (a surrogate for profit)? To quote POTUS: “Yes, you can!”

•Practice problem #4 out of Module 6. Solve using the Solver in EXCEL.

•There is one modification (Circles in box size) and the following clarification.

•

•2nd paragraph – consider that there are 3 boxes and that you don’t have to fill up the boxes (but you cannot exceed that space).

Each bag of Stars takes up 0.01 (1/100th) of a box

Each bag of Circles takes up 0.008 (1/125th) of a box

Each bag of Stars/Stripes takes up 0.0125 (1/80th) of a box

Use this information to help you in creating the LP Model.

1. The effectiveness of a blood-pressure drug is being investigated. An experimenter finds that, on average, the reduction in systolic blood pressure is 44 for a sample of size 22 and standard deviation 6. Estimate how much the drug will lower a typical patient's systolic blood pressure (using a 95% confidence level).
Give your answers to one decimal place and provide the point estimate with its margin of error. __________________ ± ________________________

2. In a survey, 31 people were asked how much they spent on their child's last birthday gift. The results were roughly bell-shaped with a mean of $42 and standard deviation of $10. Estimate how much a typical parent would spend on their child's birthday gift (use a 95% confidence level).
Give your answers to one decimal place. Provide the point estimate and margin or error. _______________ ± _________________

3. You must estimate the mean temperature (in degrees Fahrenheit) with the following sample temperatures:

Find the 80% confidence interval. Enter your answer as an open-interval (i.e., parentheses) accurate to two decimal places (because the sample data are reported accurate to one decimal place). 80% C.I. = ___________________
Answer should be obtained without any preliminary rounding.

4. The effectiveness of a blood-pressure drug is being investigated. An experimenter finds that, on average, the reduction in systolic blood pressure is 38.9 for a sample of size 339 and standard deviation 13.4. Estimate how much the drug will lower a typical patient's systolic blood pressure (using a 80% confidence level).
Enter your answer as a tri-linear inequality accurate to one decimal place (because the sample statistics are reported accurate to one decimal place).
_________< μ < __________

5. Assume that a sample is used to estimate a population mean μμ. Find the margin of error M.E. that corresponds to a sample of size 7 with a mean of 52.6 and a standard deviation of 14.3 at a confidence level of 99.9%. Report ME accurate to one decimal place because the sample statistics are presented with this accuracy.
M.E. = ________________    Answer should be obtained without any preliminary rounding. However, the critical value may be rounded to 3 decimal places.

6. Express the confidence interval (337.6,540.6)(337.6,540.6) in the form of ¯x ± ME.

¯x ± ME= ___________ + ______________

7. Assume that a sample is used to estimate a population mean μμ. Find the 95% confidence interval for a sample of size 66 with a mean of 27.9 and a standard deviation of 20.8. Enter your answer as an open-interval (i.e., parentheses) accurate to one decimal place (because the sample statistics are reported accurate to one decimal place).

95% C.I. = _____________
Answer should be obtained without any preliminary rounding. However, the critical value may be rounded to 3 decimal places.

8. Assume that a sample is used to estimate a population proportion μμ. Find the margin of error M.E. that corresponds to a sample of size 411 with a mean of 20.9 and a standard deviation of 20.6 at a confidence level of 90%.

Report ME accurate to one decimal place because the sample statistics are presented with this accuracy.
M.E. =__________________
Answer should be obtained without any preliminary rounding. However, the critical value may be rounded to 3 decimal places.

9. In a survey, 16 people were asked how much they spent on their child's last birthday gift. The results were roughly bell-shaped with a mean of $38 and standard deviation of $7. Estimate how much a typical parent would spend on their child's birthday gift (use a 95% confidence level).
Give your answers to one decimal place. Provide the point estimate and margin or error. _____________ ± _______________

10. Assume that a sample is used to estimate a population mean μμ. Find the 90% confidence interval for a sample of size 56 with a mean of 59.2 and a standard deviation of 13.2. Enter your answer as an open-interval (i.e., parentheses) accurate to one decimal place (because the sample statistics are reported accurate to one decimal place).
90% C.I. = _________________
Answer should be obtained without any preliminary rounding. However, the critical value may be rounded to 3 decimal places.

11. Assume that a sample is used to estimate a population mean μμ. Find the margin of error M.E. that corresponds to a sample of size 21 with a mean of 58.7 and a standard deviation of 17.9 at a confidence level of 80%.
Report ME accurate to one decimal place because the sample statistics are presented with this accuracy.
M.E. = _______________
Answer should be obtained without any preliminary rounding. However, the critical value may be rounded to 3 decimal places.

12. Express the confidence interval (174.8,273.6) in the form of ¯x ± ME.

¯x ± ME=_________________

13. Assume that a sample is used to estimate a population proportion μμ. Find the margin of error M.E. that corresponds to a sample of size 45 with a mean of 15.3 and a standard deviation of 6.6 at a confidence level of 99%.
Report ME accurate to one decimal place because the sample statistics are presented with this accuracy.
M.E. = ________________
Answer should be obtained without any preliminary rounding. However, the critical value may be rounded to 3 decimal places.

14. You must estimate the mean temperature (in degrees Fahrenheit) with the following sample temperatures:

Find the 90% confidence interval. Enter your answer as an open-interval (i.e., parentheses) accurate to two decimal places (because the sample data are reported accurate to one decimal place).
90% C.I. = _________________

15. The effectiveness of a blood-pressure drug is being investigated. An experimenter finds that, on average, the reduction in systolic blood pressure is 30.9 for a sample of size 889 and standard deviation 11.3. Estimate how much the drug will lower a typical patient's systolic blood pressure (using a 95% confidence level).
Enter your answer as a tri-linear inequality accurate to one decimal place (because the sample statistics are reported accurate to one decimal place).
___________< μ < ____________________

16. You must estimate the mean temperature (in degrees Fahrenheit) with the following sample temperatures:

Find the 80% confidence interval. Enter your answer as an open-interval (i.e., parentheses) accurate to two decimal places (because the sample data are reported accurate to one decimal place). 80% C.I. = _____________________

17. The effectiveness of a blood-pressure drug is being investigated. An experimenter finds that, on average, the reduction in systolic blood pressure is 50 for a sample of size 19 and standard deviation 8. Estimate how much the drug will lower a typical patient's systolic blood pressure (using a 90% confidence level).
Give your answers to one decimal place and provide the point estimate with its margin of error.
______________ ± _______________