Dylan Jones kept careful records of the fuel efficiency of his new car. After the first twelve times he filled up the tank, he found the mean was 22.9 miles per gallon (mpg) with a sample standard deviation of 1.2 mpg.

1. Compute the 98% confidence interval for his mpg. (Use t Distribution Table.) (Round your answers to 3 decimal places.)

2. How many times should he fill his gas tank to obtain a margin of error below 0.15 mpg? (Use z Distribution Table.) (Round your answer to the next whole number.)

question about R:

The vectors state.name, state.area, and state.region are pre-loaded in R and contain US state names, area (in square miles), and region respectively.

(a) Identify the data type for state.name, state.area, and state.region.

(b) What is the longest state name (including spaces)? How long is it?

(c) Compute the average area of the states which contain the word “New” at the start of the state name. Use the function substr().

(d) Use the function table() to determine how many states are in each region. Use the function kable() to include the table in your solutions. (Notes: you will need the R package knitr to be able to use kable().

Suppose you are trying to determine the capacity (in gallons) of the gas tank needed on an airplane you are constructing. You want to be able to travel 3200 nautical miles without stopping, and have gathered data on the amount of fuel similar planes used during flights of comparable length. Show complete calculation and your steps, also interpetation and explanation as asked.

Consider a sample with the following properties: x̅ = 261.5, s = 18.73, n = 26

A) Calculate a confidence interval with α = 0.10

B) Calculate a confidence interval with α = 0.01

C) How would you interpret the results for the confidence interval from part B?

You are given the sample mean and the population standard deviation. Use this information to construct the​ 90% and​ 95% confidence intervals for the population mean. Interpret the results and compare the widths of the confidence intervals. If​ convenient, use technology to construct the confidence intervals. A random sample of 60 home theater systems has a mean price of ​$118.00. Assume the population standard deviation is ​$19.60. Construct a​ 90% confidence interval for the population mean.

The​ 90% confidence interval is ​( nothing​, nothing​). ​(Round to two decimal places as​ needed.)

Construct a​ 95% confidence interval for the population mean.

The​ 95% confidence interval is ​( nothing​, nothing​). ​(Round to two decimal places as​ needed.)

Interpret the results. Choose the correct answer below

A. With​ 90% confidence, it can be said that the population mean price lies in the first interval. With​ 95% confidence, it can be said that the population mean price lies in the second interval. The​ 95% confidence interval is wider than the​ 90%.

B. With​ 90% confidence, it can be said that the sample mean price lies in the first interval. With​ 95% confidence, it can be said that the sample mean price lies in the second interval. The​ 95% confidence interval is wider than the​ 90%.

C. With​ 90% confidence, it can be said that the population mean price lies in the first interval. With​ 95% confidence, it can be said that the population mean price lies in the second interval. The​ 95% confidence interval is narrower than the​ 90%.

Do people eat more of a snack food when the food is labeled as low-fat? Do people pay attention to serving size? The answer may depend on whether the snack food is labelled low-fat and whether the label includes serving-size information. A study investigated these two questions using staff, grad students, and undergrad students at a large university as subjects.   
Subjects were asked to evaluate a pilot episode for an upcoming TV show at a theater on campus and were given a bag of granola from a respected campus restaurant. They were told to enjoy as much or as little of the granola as they wanted. Each granola bag had two labels: Twenty subjects were assigned to each treatment, and their granola bags were weighed at the end of the session to determine how much granola was eaten.

a) Is the study an observational study or an experiment? Specifically in this study (do not give general definitions),

what are the b) experimental units (abbreviated EU, also called individuals or subjects)

c) response variable and whether it is quantitative or categorical

d) How many factors were there and what were they?

e) How many treatments were there and what were they?

f) How many experimental units were in the study?

Microeconomics

Marginal Productivity and the Law of Diminishing Marginal Returns

You have recently been hired to manage a movie theater. You observe that there are many customers waiting around the concession area to buy snacks. You also observe that there is only one clerk working the counter. This employee has to do everything from get the popcorn going, stocking condiments and supplies, changing the soda canister when the syrup runs out for fountain drinks, helping customers, fill orders, collect cash, and of course, smile at the customers who have waited lengthy periods of time.

You obtain a report that shows the average sales per weekend night are $500 with one clerk. You decide to hire another clerk for the shift and sales increase to $1,000. You add one more clerk, and sales increase to $1,700. Again, you add another clerk, and sales increase to $1,900. Finally you add one more clerk, and sales increase to $2,000.

1.         Calculate the marginal product associated with each clerk. Draw a table to do this.

2.         At what point did the law of diminishing marginal return become evident?

3.         Why did the marginal product increase as more clerks were added initially?

4.         Why did the marginal product start to diminish?

Your company has sent you on business to the Los Angeles (LA) metropolitan area. Upon your arrival at LAX, you make your way to the Klunker Car Rental counter. As usual, the line at the counter is long, so you enter and begin your wait. While waiting you notice that Klunker is offering a special deal on gas. They are selling gas for $1.579 per gallon. However, you must purchase a full tank when you rent the car. Klunker also says that the average price per gallon of gas in the LA area is $1.60. You are uncertain of several necessary pieces of information to determine whether you should take advantage of this deal. These are: · the total miles you will drive on the trip; · rental car gas mileage; · how much gas the rental car's tank holds; · the true cost of gas in the LA area. First, you expect to drive between 150 and 250 miles on this trip. You believe there is an equal chance that you will drive either of these extreme amounts. However, you may have to make a side trip to Edwards AFB that will increase the total miles to 500. You believe there is a 1 in 5 chance that this will happen. Normally, Klunker rents you a mid-size car. You believe most cars in this class have either a 15 gallon gas tank with 60% confidence or a 18 gallon gas tank with 40% confidence. You've heard that cars in this class get as much as 25 mpg on the highway but may get as little as 18 mpg city driving. You decide there is an 70% chance most of your driving will be on the freeways and the rest in the city. Finally, you don't believe Klunker's posted average price of $1.60 per gallon in the LA area. You guess that there is 40% chance that the gas will be $1.259, 20% chance it will be $1.479 and a 40% chance it will be $1.659. Assume you must decide whether to pre-purchase the tank of gas prior to talking to a Klunker clerk.

a. Draw the decision tree for this problem using Decision Tree in Excel.

b. What is the optimal decision?

c. Now suppose you can delay your decision until you speak to a clerk and find out exactly how much gas your rental car holds. The clerk says the car you will rent holds 18 gallons of gas. What is your optimal decision now? What is the value of this additional information?

*Please show steps in excel *

CardioGood Fitness is a developer of high-quality cardiovascular exercise equipment. The company looks to increase the sales of its treadmill products and has hired The AdRight Agency, a small advertising firm, to create and implement an advertising program. CardioGood Fitness sells three different lines of treadmills: a. The TM195 is an entry-level treadmill with fewer programs and features. It is suitable for individuals who thrive on minimal programming and the desire for simplicity. The TM195 sells for $1,500. b. The middle-line TM498 adds to the features of the entry-level model two user programs and up to 15% elevation upgrade. The TM498 is suitable for individuals who are walkers at a transitional stage from walking to running or midlevel runners. The TM498 sells for $1,750. c. The top-of-the-line TM798 has more features than the other models. This model is designed to handle rigorous, frequent running; the TM798 is therefore appealing to someone who is a power walker or a runner. The selling price is $2,500. As a first step, the market research team at AdRight is assigned the task of identifying the profile of the typical customer for each treadmill product offered by CardioGood Fitness. The team decides to collect data on individuals who purchased a treadmill at a CardioGood Fitness retail store during the prior three months. The team identifies the following customer variables to study: product purchased—TM195, TM498, or TM798; gender; age, in years; education, in years; marital status, single or partnered; annual household income ($); usage, number of times the customer plans to use the treadmill each week; miles, mean number of miles the customer expects to walk/run each week; and Fitness, self-rated fitness on a 1-to-5 scale (where 1 is poor shape and 5 is excellent shape). a. For each of the following variables in the dataset, determine whether the variable is categorical or numerical. If the variable is categorical, determine whether the variable is nominal or ordinal. If the variable is numerical, determine whether the variable is discrete or continuous. i. Education ii. Usage iii. Miles iv. Fitness b. Build three (3) contingency tables based on total percentages for gender, marital status and fitness level: i. One for the model TM195 ii. One for the model TM498 iii. One for the model TM798 c. Build three (3) contingency tables showing the average of annual household income by gender, marital status and fitness level i. One for the model TM195 ii. One for the model TM498 iii. One for the model TM798 d. Based on parts b) and c), describe the features of the typical customer for each treadmill product i. Features typical customer TM195 ii. Features typical customer TM498 iii. Features typical customer TM798.

Some car tires can develop what is known as "heel and toe" wear if not rotated after a certain mileage. To assess this issue, a consumer group investigated the tire wear on two brands of tire, A and B, say. Fifteen cars were fitted with new brand A tires and thirteen with brand B tires, the cars assigned to brand at random. (Two cars initially assigned to brand B suffered serious tire faults other than heel and toe wear, and were excluded from the study.) The cars were driven in regular driving conditions, and the mileage at which heal and toe wear could be observed was recorded on each car. For the cars with brand A tires, the mean mileage observed was 24.99 24.99 (in 103 10 3 miles ) and the variance was 7.75 7.75 (in 106 10 6 miles2 2 ). For the cars with brand B, the corresponding statistics were 32.92 32.92 (in 103 10 3 miles) and 6.47 6.47 (in 106 10 6 miles2 2 ) respectively. The mileage before heal and toe wear is detectable is assumed to be Normally distributed for both brands. Part a) Calculate the pooled variance s2 s 2 to 3 decimal places. During intermediate steps to arrive at the answer, make sure you keep as many decimal places as possible so that you can achieve the precision required in this question. ×106 × 10 6 miles 2 2 Part b) Determine a 95% confidence interval for μA−μB μ A − μ B , the difference in the mean 103 10 3 mileages before heal and toe wear for the two brands of tire. Leave your answer to 2 decimal places. ( ,) Part c) Based on the 95% confidence interval constructed in the previous part, which of the following conclusions can be drawn when we test H0:μA=μB H 0 : μ A = μ B vs. Ha:μA≠μB H a : μ A ≠ μ B with α=0.05 α = 0.05 . A. Do not reject H0 H 0 since 0 is not in the interval found in part (b). B. Reject H0 H 0 since 0 is not within the interval found in part (b). C. Do not reject H0 H 0 since −7.93 − 7.93 is within the interval found in part (b). D. Do not reject H0 H 0 since 0 is within the interval found in part (b). E. Reject H0 H 0 since 0 is in the interval found in part (b)