Find the odds for and the odds against the event rolling a fair die and getting a 6 or a 2.

a. The odds for the event are __ to __ ​(Simplify your​ answers.)

b. The odds against the event are __ to __ . ​(Simplify your​ answers.)

The liquid chlorine added to swimming pools to combat algae has a relatively short shelf life before it losesits effectiveness. Records indicate that the mean shelf life of a jug of chlorine is 2180 hours. An experiment was conducted in which an additive, 'Holdlonger' was added to the jugs of chlorine in an attempt to increase its shelf life. The following data represent the shelf life (in hours) of nine jugs of chlorine with the Holdlonger added:

2172
2153
2157
2158
2179
2164
2154
2178
2168

(a)What assumption is required in order to proceed with a hypothesis test with this data? (Check all that apply)

A. Is dependent
B. Comes from a standard normal population
C. Comes from a normal population
D. Comes from a random sample

(b) Will the data follow a normal distribution?  ? Yes No
Report the P-value of the normality test:
(c)State the hypotheses which would be tested in this experiment.

A. H0:μ=2180H0:μ=2180, HA:μ>2180HA:μ>2180
B. H0:μ≥2180H0:μ≥2180, HA:μ<2180HA:μ<2180
C. H0:μ=2180H0:μ=2180, HA:μ≠2180HA:μ≠2180
D. H0:μ=2180H0:μ=2180, HA:μ<2180HA:μ<2180
(d) Reject the null hypothesis (α=0.08α=0.08) if the test statistic (x¯−2180s/n√x¯−2180s/n) is  ? greater than less than  

. Use three decimals.

(e) Calculate the test statistic from this data.

Use two decimals.

(f) What is the p-value?

Input numerical answers to four decimal places.

(g) Should you reject the null hypothesis?  ? Yes No

The Excel data for this assignment shows the annual energy usage in kWh for a random sample of 250 detached single-family homes in the Southeast. A contractor building a new housing development in Town A hypothesizes that mean annual energy usage will be below 16,500 kWh per household. Use the data to run a single mean hypothesis test and calculate a test statistic in Excel. In your method summary, state the hypothesis and your assumptions. In your resulting conclusions, state your conclusion regarding the hypothesis in words, using a 99% confidence level.

The Mean of the data is 14651.548

The Standard Deviation is 7980.99357

Of 250 adults who tried a new multi-grain cereal, "Wow!", 187 rated it excellent; of 100 children sampled, 66 rated it excellent. Using the 0.1 significance level and the alternate hypothesis π1 not equal to π2 , what is the null hypothesis? Select one: a. π1 = π2 b. π1 ≠ π2 c. π1 < π2 d. π1 > π2

A typical adult's resting heart rate is 79 BPM with a standard deviation of 12. If the heart rate of 9 randomly selected adults are taken, what average heart rate is at the 75th percentile? Round answer to 1 decimal place (i.e. 10.2)

Portfolio returns. The Capital Asset Pricing Model is a financial model that assumes returns on a portfolio are normally distributed. Suppose a portfolio has an average annual return of 11.1% (i.e. an average gain of 11.1%) with a standard deviation of 40%. A return of 0% means the value of the portfolio doesn't change, a negative return means that the portfolio loses money, and a positive return means that the portfolio gains money. Round all answers to 4 decimal places.

a. What percent of years does this portfolio lose money, i.e. have a return less than 0%?

b. What is the cutoff for the highest 13% of annual returns with this portfolio?

Changes in demographic outcomes can occur because of changes in within-group rates and/or changes in composition. First, explain what that sentence means. Then take the example of falling rates of death due to heart disease. How might changes in within-group rates account for that outcome? What about changes in composition? Can you think of any way to sort out the effects of changes in within-group rates vs. changes in composition?

11.)  

An experiment consists of rolling two fair dice and adding the dots on the two sides facing up. Assuming each simple event is as likely as any​ other, find the probability that the sum of the dots is greater than 2.

The probability that the sum of the dots is greater than 2 is

12.)

An experiment consists of rolling two fair dice and adding the dots on the two sides facing up. Find the probability of the sum of the dots indicated.

Getting a sum of 1

The probability of getting a sum equal to 1 is

13.)

An experiment consists of tossing 4 fair​ (not weighted)​ coins, except one of the 4 coins has a head on both sides. Compute the probability of obtaining exactly 1 headhead.

The probability of obtaining exactly 1 headhead is

15.)

An experiment consists of rolling two fair​ (not weighted) dice and adding the dots on the two sides facing up. Each die has the number 1 on two opposite​ faces, the number 2 on two opposite​ faces, and the number 3 on two opposite faces. Compute the probability of obtaining the indicated sum.

Sum of 8

The probability of getting a sum of 8 is nothing

16.)

An experiment consists of dealing 7 cards from a standard​ 52-card deck. What is the probability of being dealt exactly 1 ace​?

The probability of being dealt exactly1 ace is approximately

Describe the continuity correction (covered in an earlier course unit), and explain how, when, and why it is used in statistics. Don't limit your thinking to the most common example given of this corrective step: approximation of binomials using the Normal distribution. While that is an example of such a correction, it is only one example. (Hint: You could compare the development of this correction logic for discrete distributions to the use of integration against continuous distributions.) Discuss the more general case and how it affects probabilities you might need to calculate as an engineering working on a project.

Your response of 1-3 paragraphs (about 100-200 words).

According to statistics, the distribution of ages for licensed drivers has a mean of 44.5 years and a standard deviation of 17.1 years. Assume the distribution of ages is normally distributed. (Give your answers correct to one decimal place.)

(a) What percentage of the drivers are between the ages of 17 and 22.
_______%

(b) What percentage of the drivers are younger than 25 years of age.
_____%

(c) What percentage of the drivers are older than 21 years of age.
______ %

(d) What percentage of the drivers are between the ages of 45 and 65.
_______%

(e) What percentage of the drivers are older than 75 years of age.
_______%

Bob and Carol Packer operate a successful outdoor wear store in Vermont called Northwoods Backpackers. They stock mostly cold-weather outdoor items such as hiking and backpacking clothes, gear, and accessories. They established an excellent reputation throughout New England for quality products and service. Eventually, Bob and Carol noticed that more and more of their sales were from customers who did not live in the immediate vicinity but were calling in orders on the telephone. As a result, the Packers decided to distribute a catalog and establish a phone-order service. The order department consisted of five operators working eight hours per day from 10:00 A.M. to 6:00 P.M., Monday through Friday. For a few years the mail-order service was only moderately successful; the Packers just about broke even on their investment. However, during the holiday season of the third year of the catalog order service, they were overwhelmed with phone orders. Although they made a substantial profit, they were concerned about the large number of lost sales they estimated they incurred. Based on information provided by the telephone company regarding call volume and complaints from customers, the Packers estimated they lost sales of approximately $100,000. Also they felt they had lost a substantial number of old and potentially new customers because of the poor service of the catalog order department.

Prior to the next holiday season, the Packers explored several alternatives for improving the catalog order service. The current system includes the five original operators with computer terminals who work eight-hour days, five days per week. The Packers have hired a consultant to study this system, and she reported that the time for an operator to take a customer order is exponentially distributed with a mean of 3.6 minutes. Calls are expected to arrive at the telephone center during the six-week holiday season according to a Poisson distribution with a mean rate of 175 calls per hour. When all operators are busy, callers are put on hold, listening to music until an operator can answer. Waiting calls are answered on a first-in, first-out basis. Based on her experience with other catalog telephone order operations and data from Northwoods Backpackers, the consultant has determined that if Northwoods Backpackers can reduce customer call waiting time to approximately one-half minute or less, the company will save $135,000 in lost sales during the coming holiday season.

Therefore, the Packers have adopted this level of call service as their goal. However, in addition to simply avoiding lost sales, the Packers believe it is important to reduce waiting time to maintain their reputation for good customer service. Thus, they would like about 70 percent of their callers to receive immediate service.

The Packers can maintain the same number of workstations/computer terminals they currently have and increase their service to sixteen hours per day with two operator shifts running from 8:00 A.M. to midnight. The Packers believe when customers become aware of their extended hours the calls will spread out uniformly, resulting in a new call average arrival rate of 87.5 calls per hour (still Poisson distributed). This schedule change would cost Northwoods Backpackers approximately $11,500 for the six-week holiday season.

Another alternative for reducing customer waiting times is to offer weekend service. However, the Packers believe that if they do offer weekend service, it must coincide with whatever service they offer during the week. In other words, if they have phone order service eight hours per day during the week, they must have the same service during the weekend; the same is true with sixteen-hours-per-day service. They feel that if weekend hours differ from weekday hours it will confuse customers. If eight-hour service is offered seven days per week, the new call arrival rate will be reduced to 125 calls per hour at a cost of $3,600. If Northwoods offers sixteen-hour service, the mean call arrival rate will be reduced to 62.5 calls hour, at a cost of $7,300.

Still another possibility is to add more operator stations. Each station includes a desk, an operator, a phone, and a computer terminal. An additional station that is in operation five days per week, eight hours per day, will cost $2,900 for the holiday season. For a sixteen-hour day the cost per new station is $4,700. For seven-day service the cost of an additional station for eight-hour per-day service is $3,800; for sixteen-hour-per-day service the cost is $6,300.

The facility Northwoods Backpackers uses to house its operators can accommodate a maximum of ten stations. Additional operators in excess of ten would require the Packers to lease, remodel, and wire a new facility, which is a capital expenditure they do not want to undertake this holiday season. Alternatively, the Packers do not want to reduce their current number of operator stations.

Determine what order service configuration the Packers should use to achieve their goals, and explain your recommendation.

Suppose a multiple regression model is given by modifying above y with caretequals0.21x 1minus9.52x 2minus28.56. What would an interpretation of the coefficient of x 1 be? Fill in the blank below. An interpretation of the coefficient of x 1 would be, "if x 1 decreases by ?? unit, then the response variable will decrease by nothing units, on average, while holding x 2 constant."

By how many units?

Ex. 2.40 European roulette.

The game of European roulette involves spinning a wheel with 37 slots: 18 red, 18 black, and 1 green. A ball is spun onto the wheel and will eventually land in a slot, where each slot has an equal chance of capturing the ball. Gamblers can place bets on red or black. If the ball lands on their colour, they double their money. If it lands on another colour, they lose their money.

(a) Suppose you play roulette and bet $3 on a single round. What is the expected value and standard deviation of your total winnings?

(b) Suppose you bet $1 in three different rounds. What is the expected value and standard deviation of your total winnings?

(c) How do your answers to parts (a) and (b) compare? What does this say about the riskiness of the two games?

Ex. 2.34 Ace of clubs wins.

Consider the following card game with a well-shuffled deck of cards. If you draw a red card, you win nothing. If you get a spade, you win $5. For any club, you win $10 plus an extra $20 for the ace of clubs.

(a) Create a probability model for the amount you win at this game. Also, find the expected winnings for a single game and the standard deviation of the winnings.

(b) What is the maximum amount you would be willing to pay to play this game? Explain your reasoning.

Plz use both pure NE and MIXED strategy (with Probability)

Consider a firm with two agents – 1 and 2. Both agents have to choose between two options: Client Focus or Cost Focus. If both choose Client the payoffs to 1 are 20 and 10 to agent 2. If both agents choose to play Cost the payoffs are 15 to agent 1 and 25 to agent 2, respectively. Finally, if any other combination of actions is chosen the payoffs to each agent are 0.

a. Assume that the agent choose their actions simultaneously. Draw the normal form of the game and derive all of the Nash equilibria.

b. Now assume that the game is played sequentially: Agent 1 makes her choice of action first, this is observed by Agent 2, who then makes his choice. Draw the extensive form of the game and find the subgame perfect equilibria. Briefly interpret this game in the context of: (i) leadership and corporate culture; and (ii) the Basic Value Maximisation Principle.

newspaper publisher is considering launching a new "national" newspaper in Anytown. It is believed that the newspaper would have to capture over 12% of the market in order to be financially viable. During the planning stages of this newspaper, a market survey was conducted of a sample of 400 readers. After providing a brief description of the proposed newspaper, one question asked if the survey participant would subscribe to the newspaper if the cost did not exceed $20 per month. Suppose that 58 participants said they would subscribe.

a. Can the publisher conclude that the proposed newspaper will be financially viable? Perform the appropriate test at a 1% level of significance.

b. Suppose the actual value of the overall proportion of readers who would subscribe to this newspaper is 0.13. Was the decision made in part (a) correct? If not, what type of error was made?

c. State the meaning of a Type I and Type II error in the context of this scenario. And what would be the repercussions of making these errors to the publisher?