In a preschool class of n, exactly n1 children are needed for activity 1, n2 for activity 2, and n3 for activity 3. Luckily n = n1 + n2 + n3. The teachers want to know in how many distinct ways the children can be assigned into these activities. (Two assignments are distinct if at least one student is in a different activity in each.) (a) They figure they could start by lining up the children arbitrarily. How many different line-ups are possible? (b) They then take the first n1 for activity 1, the next n2 for activity 2, and the rest for activity 3. How many different line-ups will create the exact same assignment of children to activities? (c) Use your answers from (a) and (b) to deduce the total number of distinct assignments.

Problem 16-13 (Algorithmic)

The wedding date for a couple is quickly approaching, and the wedding planner must provide the caterer an estimate of how many people will attend the reception so that the appropriate quantity of food is prepared for the buffet. The following table contains information on the number of RSVP guests for the 145 invitations. Unfortunately, the number of guests does not always correspond to the number of RSVPed guests.

Based on her experience, the wedding planner knows it is extremely rare for guests to attend a wedding if they notified that they will not be attending. Therefore, the wedding planner will assume that no one from these 50 invitations will attend. The wedding planner estimates that the each of the 25 guests planning to come solo has a 75% chance of attending alone, a 20% chance of not attending, and a 5% chance of bringing a companion. For each of the 60 RSVPs who plan to bring a companion, there is a 90% chance that she or he will attend with a companion, a 5% chance of attending solo, and a 5% chance of not attending at all. For the 10 people who have not responded, the wedding planner assumes that there is an 80% chance that each will not attend, a 15% chance each will attend alone, and a 5% chance each will attend with a companion.

1. Assist the wedding planner by constructing a spreadsheet simulation model to determine the expected number of guests who will attend the reception. Round your answer to 2 decimal places.
   
   _________ guests
   
2. To be accommodating hosts, the couple has instructed the wedding planner to use the Monte Carlo simulation model to determine X, the minimum number of guests for which the caterer should prepare the meal, so that there is at least a 90% chance that the actual attendance is less than or equal to X. What is the best estimate for the value of X? Round your answer to the nearest whole number.
   
   __________ guests

What is the best measure of center (mean, median, or mode) for each topic? And why?

Country

Infant mortality

Health expenditure

Obesity rate

Average income

Life expectancy

Diabetes rate

Leading cause of death

Problem 16-01

The management of Brinkley Corporation is interested in using simulation to estimate the profit per unit for a new product. The selling price for the product will be $45 per unit. Probability distributions for the purchase cost, the labor cost, and the transportation cost are estimated as follows:

1. Compute profit per unit for the base-case, worst-case, and best-case.
   
   Profit per unit for the base-case: $ __________
   
   Profit per unit for the worst-case: $ __________
   
   Profit per unit for the best-case: $ __________
   
2. Construct a simulation model to estimate the mean profit per unit. If required, round your answer to the nearest cent.
   
   Mean profit per unit = $ _________
   
3. Why is the simulation approach to risk analysis preferable to generating a variety of what-if scenarios?
   
4. Management believes the project may not be sustainable if the profit per unit is less than $5. Use simulation to estimate the probability the profit per unit will be less than $5. If required, round your answer to two decimal places.
   
   __________%

The sample mean is 19.0, the sample standard deviation is 6.6, and n = 41. Establish a 90% confidence interval for the population mean. (by default two tailed test, α is reliability factor)

6) An economist is interested in studying the average income of consumers in a particular country. The population standard deviation of incomes is known to be $1,000.

What type of confidence interval should the economist build?

Group of answer choices

A t-based interval for the population mean.

No answer text provided.

A Z-based interval for the population proportion.

A Z-based interval for the population mean.

7) A race car driver tested his car for the time it takes to go from 0 to 60 mph, and for 20 tests obtained a mean of 4.85 seconds with a standard deviation of 1.47 seconds. These times are known to be normally distributed.

What is the value of t used to construct a 95% confidence interval for the mean time it takes the car to go from 0 to 60? Round your answer to 2 decimal places.

8) A quality control engineer is interested in the mean length of sheet insulation being cut automatically by machine. The desired mean length of the insulation is 12 feet.A sample of 70 cut sheets yields a mean length of 12.14 feet and a sample standard deviation of 0.15 feet.

What type of confidence interval should the engineer build?

Group of answer choices

A Z-based interval for the population proportion.

A Z-based interval for the population mean.

A t-based interval for the population mean.

Use the following information for Questions 8, 9, and 10

Soda bottles are filled so that they contain an average of 330 ml of soda in each bottle. Suppose that the amount of soda in a bottle is normally distributed with a standard deviation of 4 ml.

8) What is the probability that a randomly selected bottle will have less than 325 ml of soda?  Round your answer to 4 decimal places.

9) What is the probability that a randomly selected six-pack of soda will have a mean less than 325 ml of soda?  Round your answer to 4 decimal places.

10) What is the probability that a randomly selected twelve-pack of soda will have a mean less than 325 ml of soda?  Round your answer to 4 decimal places.

In triathlons, it is common for racers to be placed into age and gender groups. Friends Leo and Mary both completed the Hermosa Beach Triathlon, where Leo competed in the Men, Ages 30-34 group while Mary competed in the Women, Ages 25-29 group. Leo completed the race in 1:22:28 (4948 seconds) , while Mary completed the race in 1:31:53 (5513 seconds). Obviously Leo finished faster, but they are curious about how they did within their respective groups. Can you help them? Here is some information on the performance of their groups:

• The finishing times of the Men, Ages 30-34 group have a mean of 4313 seconds with a standard deviation of 583 seconds.
• The finishing times of the Women, Ages 25-29 group have a mean of 5261 seconds with a standard deviation of 807 seconds.
• The distributions of finishing times for both groups are approximately Normal.

Remember, a better performance corresponds to a faster finish.

1. Find the Z-scores for Leo’s and Mary’s finishing times. Did Leo or Mary rank better in their respective groups? Explain your reasoning?
   
   

2. What percent of the triathletes did Leo finish faster than in his group?
   
   
   
3. What percent of the triathletes did Mary finish faster than in her age group?
   

4. What is the cutoff time for the fastest 5% of athletes in the men’s group?

5. What is the cutoff time for the slowest 10% of athletes in the women’s group?
   
   

A company has sales of automobiles in the past three years as given in the table below. Using trend and seasonal components, predict the sales for each quarter of year 4.

identify (but don’t collect) a type of dataset that might vary significantly from its mean. (Examples may be adult’s weights or BMIs, a company’s sales, or the number of pieces of mail you receive in a week. Using your imaginary dataset, answer the following questions:

• What is a brief description of the data?
• How much would you expect the data to vary?
• What might the causes of the variation be?

you are an analyst working for the new Joint High Speed Vessel (JHSV) program office. Several tests have been conducted on two different types of experimental test vessels (denoted simply “1” and “2”) to determine their performance characteristics under various loading and sea state conditions.

With detailed data on the fuel consumption of vessel "1," and with the Navy's new focus on energy efficiency, the program manager wants to test whether the vessel beats the design specs in terms of mean hourly fuel consumption. You decide to conduct a large sample hypothesis test of the data with the goal of conclusively demonstrating, if possible, that the data support the claim that the mean hourly fuel consumption is less than 50 gph (gallons per hour) at 35 knots. Given that for 36 (independent) hours of operation at 35 kts you observe y-bar 49.5 gph with s=2 gph, and using a significance level of a=0.05:

a. Write out the null and alternative hypotheses.

b. Calculate the test statistic.

c. Calculate and state the rejection region or p-value.

d. Conduct the test and state the outcome. State the outcome both in terms of accepting or rejecting the null hypothesis and then in terms of what the result means in the context of this particular problem.

A friend who lives in Los Angeles makes frequent consulting trips to Washington, D.C.; 40% of the time she travels on airline #1, 20% of the time on airline #2, and the remaining 40% of the time on airline #3. For airline #1, flights are late into D.C. 40% of the time and late into L.A. 15% of the time. For airline #2, these percentages are 30% and 15%, whereas for airline #3 the percentages are 35% and 20%. If we learn that on a particular trip she arrived late at exactly one of the two destinations, what are the posterior probabilities of having flown on airlines #1, #2, and #3? Assume that the chance of a late arrival in L.A. is unaffected by what happens on the flight to D.C. [Hint: From the tip of each first-generation branch on a tree diagram, draw three second-generation branches labeled, respectively, 0 late, 1 late, and 2 late.] (Round your answers to four decimal places.)

Appropriate sampling is a critical component in developing a good research project. Using your approved research questions and research topic, explain your anticipated sampling method and why this is appropriate for your research proposal. What is your sample size? Next, read and review two of your classmates’ posts and analyze their sampling approach. Are their sampling approaches appropriate? Why or why not?

The number of floods that occur in a certain region over a given year is a random variable having a Poisson distribution with mean 2, independently from one year to the other. Moreover, the time period (in days) during which the ground is flooded, at the time of an arbitrary flood, is an exponential random variable with mean 5. We assume that the durations of the floods are independent. Using the central limit theorem, calculate (approximately)
(a) the probability that over the course of the next 50 years, there will be at least 80 floods in this region. Assume that we do not need to apply half-unit correction for this question.

(b) the probability that the total time during which the ground will be flooded over the course of the next 50 floods will be smaller than 200 days.