Dixie Showtime Movie Theaters, Inc., owns and operates a chain of cinemas in several markets in the southern U.S. The owners would like to estimate weekly gross revenue as a function of advertising expenditures. Data for a sample of eight markets for a recent week follow.

(a) Use the data to develop an estimated regression with the amount of television advertising as the independent variable. Let x represent the amount of television advertising. If required, round your answers to three decimal places. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300)

(b) How much of the variation in the sample values of weekly gross revenue does the model in part (a) explain? If required, round your answer to two decimal places.

(c) Use the data to develop an estimated regression equation with both television advertising and newspaper advertising as the independent variables. Let x1 represent the amount of television advertising. Let x2 represent the amount of newspaper advertising. If required, round your answers to three decimal places. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300)

(d) How much of the variation in the sample values of weekly gross revenue does the model in part (c) explain? If required, round your answer to two decimal places. %

Two players (player A and player B) are playing a game against each other repeatedly until one is bankrupt. When a player wins a game they take $1 from the other player. All plays of the game are independent and identical. a) Suppose player A starts with $2 and player B starts with $1. If player A wins a game with probability p, what is the probability that player A wins all the money? b) Suppose player A starts with $6 and player B starts with $6. If player A wins a game with probability 0.5, what is the probability the game ends (someone loses all their money) on exactly the 10th play of the game?

For the random variables below, indicate whether you would expect the distribution to be best described as geometric, binomial, Poisson, exponential, uniform, or normal. For each item, give a brief explanation of your answer.

a) The number of heads in 13 tosses of a coin.

b) The number of at-bats (attempts) required for a baseball player to get his first hit.

c) The height of a randomly chosen adult female.

d) The time of day that the next major earthquake occurs in Southern California.

e) The number of automobile accidents in a town in one week.

f) The amount of time before the first score in a lacrosse game.

g) The number of times a die needs to be rolled before a 3 appears.

h) The number of particles emitted by a radioactive substance in five seconds.

The quality control manager of Ridell needs to estimate the mean breaking point of a large shipment of helmets sent to the Philadelphia Eagles. Given the production process, the known standard deviation of the population of breaking points is 15.5 lbs. A random sample of 49 helmets were selected and subjected to increasing pressure until every one of them broke. The breaking point of each helmet was recorded, and average breaking point of the sample was 150 lbs.

What are the critical values from the z distribution associated with a 95% confidence interval?

n a study designed to test the effectiveness of magnets for treating back​ pain, 3535 patients were given a treatment with magnets and also a sham treatment without magnets. Pain was measured using a scale from 0​ (no pain) to 100​ (extreme pain). After given the magnet​ treatments, the 3535 patients had pain scores with a mean of 12.0 and a standard deviation of 2.2. After being given the sham​ treatments, the 3535 patients had pain scores with a mean of 10.2 and a standard deviation of 2.6

Construct the 90​% confidence interval estimate of the mean pain score for patients given the sham treatment.

What is the confidence interval estimate of the population mean μ​?

The distribution of heights of adult men in the U.S. is approximately normal with mean 69 inches and standard deviation 2.5 inches. Use what you know about the EMPIRICAL RULE to answer the following.

a)Approximately what percent of men are taller than 69 inches?

b)Approximately what percent of men are between 64 and 66.5 inches?

Find the sample size necessary to estimate the mean IQ score of statistics students such that it can be said with 95​% confidence that the sample mean is plus or minus3 IQ points of the true mean. Assume that the standard deviation is 15 and determine the required sample size.

2)

In a survey of

2,416 adults, 1,876 reported that​ e-mails are easy to​ misinterpret, but only 1,231 reported that telephone conversations are easy to misinterpret. Complete parts​ (a) through​ (c) below.

a. Construct a​ 95% confidence interval estimate for the population proportion of adults who report that​ e-mails are easy to misinterpret.

EXERCISES ON DISCRETE DISTRIBUTIONS

6. An exam consists of 12 questions that present four possible answers each. A person, without knowledge about the subject of the exam, answers the random exam questions.
a.What is the probability that you get the right answer when answering a question?
b. Find the probability that such person does not answer any questions well
C. Calculate the probability of correcting a question.
d. Obtain the probability that you answer all the questions correctly.
e. Obtain the probability of answering more than half of the questions correctly

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?