Data are gathered on each car in the motor pool, regarding number of miles (in thousand miles) driven in a given year, and maintenance costs (in thousand dollars) for that year:

Part of the linear regression analysis output are shown in below:

3. Construct a 95% confidence interval for the conditional mean of y given x0=50.

   (A) [2.476, 2.879]   (B) [1.538, 2.409] (C) [1.651, 2.30] (D) [2.410, 2.908]

3. Construct a 95% prediction interval at x0=50.

(A) [2.360, 2.996]

(B) [2.064, 3.254]

(C) [1.612, 2.335]

(D) [2.223, 3.132]

The Apex corporation produces corrugated paper. It has collected monthly data from January 2001 through March 2003 on the following two variables:

y= total manufacturing cost per month (In thousands of dollars) (COST)

x= total machine hours used per month (Machine)

The data are shown below.

1102 218
1008 199
1227 249
1395 277
1710 363
1881 399
1924 411
1246 248
1255 259
1314 266
1557 334
1887 401
1204 238
1211 246
1287 259
1451 286
1828 389
1903 404
1997 430
1363 271
1421 286
1543 317
1774 376
1929 415
1317 260
1302 255
1388 281

answer the following questions

a. State the least squares regression line.

b. What percentage of variation in ? has been explained by the regression?

c. Are ? and ? linearly related? Conduct a hypothesis test at the 5% significance level by completing the following steps:

i. State the null and alternative hypotheses

ii. State the value of the test statistic

iii. Provide the p-value

iv. Do you reject the null hypothesis or not? Explain your answer.

v. State you conclusion within context of the problem.

d. Fill in the blanks for the following statement: “I am 95% confident that the average manufacturing cost at the Apex corporation for all months with 350 total machine hours is between ____ and ____.”

Please show me the steps. Thank you

Thirty-two small communities in Connecticut (population near 10,000 each) gave an average of x = 138.5 reported cases of larceny per year. Assume that σ is known to be 41.3 cases per year. (a) Find a 90% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.) lower limit upper limit margin of error (b) Find a 95% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.) lower limit upper limit margin of error (c) Find a 99% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.) lower limit upper limit margin of error (d) Compare the margins of error for parts (a) through (c). As the confidence levels increase, do the margins of error increase? As the confidence level increases, the margin of error decreases. As the confidence level increases, the margin of error increases. As the confidence level increases, the margin of error remains the same. (e) Compare the lengths of the confidence intervals for parts (a) through (c). As the confidence levels increase, do the confidence intervals increase in length? As the confidence level increases, the confidence interval decreases in length. As the confidence level increases, the confidence interval increases in length. As the confidence level increases, the confidence interval remains the same length.

What price do farmers get for their watermelon crops? In the third week of July, a random sample of 41 farming regions gave a sample mean of x = $6.88 per 100 pounds of watermelon. Assume that σ is known to be $1.94 per 100 pounds.

(a) Find a 90% confidence interval for the population mean price (per 100 pounds) that farmers in this region get for their watermelon crop. What is the margin of error? (Round your answers to two decimal places.)

(b) Find the sample size necessary for a 90% confidence level with maximal error of estimate E = 0.41 for the mean price per 100 pounds of watermelon. (Round up to the nearest whole number.)
farming regions

(c) A farm brings 15 tons of watermelon to market. Find a 90% confidence interval for the population mean cash value of this crop. What is the margin of error? Hint: 1 ton is 2000 pounds. (Round your answers to two decimal places.)

Thirty-four small communities in Connecticut (population near 10,000 each) gave an average of x = 138.5 reported cases of larceny per year. Assume that σ is known to be 44.3 cases per year.

(a) Find a 90% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)

(b) Find a 95% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)

(c) Find a 99% confidence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (Round your answers to one decimal place.)

(d) Compare the margins of error for parts (a) through (c). As the confidence levels increase, do the margins of error increase?

As the confidence level increases, the margin of error increases.As the confidence level increases, the margin of error decreases.    As the confidence level increases, the margin of error remains the same.

(e) Compare the lengths of the confidence intervals for parts (a) through (c). As the confidence levels increase, do the confidence intervals increase in length?

As the confidence level increases, the confidence interval increases in length.As the confidence level increases, the confidence interval decreases in length.    As the confidence level increases, the confidence interval remains the same length.

How would we determine the right comparative method to use when analyzing the different studies that we are interested in using in our research?

Numerous studies have shown that IQ scores have been increasing, generation by generation, for years (Flynn, 1984, 1999). The increase is called the Flynn Effect, and the data indicate that the increase appears to be about 7 points per decade. To demonstrate this phenomenon, a researcher obtains an IQ test that was written in 1980. At the time the test was prepared, it was standardized to produce a population mean of 100. The researcher administers the test to a random sample of 16 of today's high school students and obtains a sample mean IQ of 110 with standard deviation of 20. Is this result sufficient to conclude that today's sample scored significantly higher than would be expected from a population with 100? Test this claim at the 5% significance level.

Fill in the blanks with the appropriate responses:

Hypotheses
H₀: The mean IQ score is 100
H₁: The mean IQ score is Blank 1 100
(type in “less than”, “greater than”, or “not equal to”)

Results
t = Blank 2 (enter the test statistic, use 2 decimal places)
p-value = Blank 3 (round answer to nearest thousandth of a percent – i.e. 0.012%)

Conclusion
We Blank 4 sufficient evidence to support the claim that the mean IQ is Blank 5 100 (p Blank 6 0.05).
(Use “have” or “lack” for the first blank, “less than”, “greater than” or “not equal to” for the second blank and “<” or “>” for the final blank)

1) The daily demand, D, of sodas in the break room is:

i) Find the probability that the demand is at most 2.
ii) Compute the average demand of sodas.
iii) Compute SD of daily demand of sodas.

2) From experience you know that 83% of the desks in the schools have gum stuck
beneath them. In a random sample of 14 desks.
a) Compute the probability that all of them have gum underneath.
b) Compute the probability that 10 or less desks have gum.
c) What is the probability that more than 10 have gum?
d) What is the expected number of desks in the sample have gum?
e) What is the SD of the number of desks with gum?

3) The number of customers, X, arriving in a ATM in the afternoon can be modeled
using a Poisson distribution with mean 6.5.
a) Compute P(X<3).
b) Compute P(X>4).
c) SD of X.

South Shore Construction builds permanent docks and seawalls along the southern shore of long island, new york. Although the firm has been in business for only five years, revenue has increased from $320,000 in the first year of operation to $1,116,000 in the most recent year. The following data show the quarterly sales revenue in thousands of dollars:

a. Use Excel Solver to find the coefficients of a multiple regression model with dummy variables as follows to develop an equation to account for seasonal effects in the data. Qtr1 = 1 if Quarter 1, 0 otherwise; Qtr2 = 1 if Quarter 2, 0 otherwise; Qtr3 = 1 if Quarter 3, 0 otherwise. Round your answers to two decimal places.

F_t = _ + _Qtr1 + _Qtr2 + _Qtr3

b. Let Period = 1 to refer to the observation in Quarter 1 of year 1; Period = 2 to refer to the observation in Quarter 2 of year 1; . . . and Period = 20 to refer to the observation in Quarter 4 of year 5. Using the dummy variables defined in part (b) and Period, develop an equation to account for seasonal effects and any linear trend in the time series using Excel Solver. Round your answers to two decimal places. If your answer is negative value enter minus sign.

F_t = _ + _Qtr1 + _Qtr2 + _Qtr3 + _Period

Based upon the seasonal effects in the data and linear trend, compute estimates of quarterly sales for year 6. Round your answers to one decimal place.

Quarter 1 forecast =

Quarter 2 forecast =

Quarter 3 forecast =

Quarter 4 forecast =

All airplane passengers at the Lake City Regional Airport must pass through a security screening area before proceeding to the boarding area. The airport has two screening stations available, and the facility manager must decide how many to have open at any particular time. The service rate for processing passengers at each screening station is 4 passengers per minute. On Monday morning the arrival rate is 4.8 passengers per minute. Assume that processing times at each screening station follow an exponential distribution and that arrivals follow a Poisson distribution. When the security level is raised to high, the service rate for processing passengers is reduced to 3 passengers per minute at each screening station. Suppose the security level is raised to high on Monday morning.

1. The facility manager's goal is to limit the average number of passengers waiting in line to 8 or fewer. How many screening stations must be open in order to satisfy the manager's goal?
   
   Having 2  station(s) open satisfies the manager's goal to limit the average number of passengers in the waiting line to at most 8.
   
2. What is the average time required for a passenger to pass through security screening? Round your answer to two decimal places.
   
   W =  minutes

QUESTION FIVE

1. The diameter of shafts produced in a machine follows a normal distribution with the variance of 81mm. A random sample of 36 shafts taken from the production has its mean diameter of 30mm. Find a 95% confidence interval for the diameter of shafts.

2. The Marketing manager of a company feels that 42% of retailers will have enhanced weekly sales after introducing an advertisement at the point of sales. A sample of 36 retailers shops of the company, where the point of sales advertisement has been displayed, reveals that only 18 of them are having enhanced sales after displaying the advertisement. Find the 95% confidence interval for the proportion representing the enhanced sales.

c. The Finance manager of a company feels that 55% of branches will have enhanced yearly collection of deposits after introducing a hike in interest rate. Determine the sample size such that the mean proportion is with plus or minus 0.05 confidence level of 90%?     

An auto insurance company concludes that 30% of policyholders with only collision coverage will have a claim next year, 40% of policyholders with only comprehensive coverage will have a claim next year and 50% of policyholders with both collision and comprehensive coverage will have a claim next year. Records show 60% of policyholders have collision coverage 70% have comprehensive coverage and all policyholders have at least one of these coverages.

Calculate the percentage of policyholders expected to have an accident next year.

1. 10%

2. 20%

3. 31%

4. 36%

5. 40%

**Must be a clear and logical response in 150 to 200 words to the following questions/prompts, providing specific examples to support your answers. Type answers.**

• Which of the statistical techniques have the most usefulness for a business of interest or the job that you are currently in? What is the technique and why is it useful to the business or job?

(St Petersburg Paradox). Suppose you have the opportunity to play the following game. You flip a fair coin, and if it comes up heads on the first flip, then you win $1. If not, then you flip again. If it comes up heads on the second flip, then you win $2, and if not you flip again. On the third flip, a heads pays $4, on the fourth $8, and so on. That is, each time you get tails, you flip again and your prize doubles, and you get paid the first time you flip heads.

a) How much should you be willing to pay to play this amazing game? In other words, compute the expected payout from playing this game.

b) Now suppose the casino (or wherever you’re playing this game) has a limited bankroll of $2^n. So, if you get tails n times in a row, then the game is over automatically and you are paid $2^n. Now what is the expected payout? How much should you be willing to pay to play the game if n = 10?

1. The average production cost for major movies is 57 million dollars and the standard deviation is 22 million dollars. Assume the production cost distribution is normal. Suppose that 46 randomly selected major movies are researched. Answer the following questions. Round all answers to 4 decimal places where possible.

1. What is the distribution of X? X~ N( , )
2. What is the distribution of x¯? x¯ ~ N( , )
3. For a single randomly selected movie, find the probability that this movie's production cost is between 51 and 56 million dollars.
4. For the group of 46 movies, find the probability that the average production cost is between 51 and 56 million dollars.

2. Suppose the age that children learn to walk is normally distributed with mean 11 months and standard deviation 1.1 month. 18 randomly selected people were asked what age they learned to walk. Round all answers to 4 decimal places where possible.

1. What is the distribution of X? X ~ N( , )  
2. What is the distribution of x¯? x¯ ~ N( , )
3. What is the probability that one randomly selected person learned to walk when the person was between 10 and 12.5 months old?
4. For the 18 people, find the probability that the average age that they learned to walk is between 10 and 12.5 months old.
5. For part d), is the assumption that the distribution is normal necessary? Yes or No
6. Find the IQR for the average first time walking age for groups of 18 people.
   Q1 = ______ months
   Q3 = ______ months
   IQR: ______ months

3. The average number of miles (in thousands) that a car's tire will function before needing replacement is 72 and the standard deviation is 12. Suppose that 8 randomly selected tires are tested. Round all answers to 4 decimal places where possible and assume a normal distribution.

1. What is the distribution of X? X ~ N( , )
2. What is the distribution of x¯? x¯ ~ N( , )
3. If a randomly selected individual tire is tested, find the probability that the number of miles (in thousands) before it will need replacement is between 78.2 and 84.2.
4. For the 8 tires tested, find the probability that the average miles (in thousands) before need of replacement is between 78.2 and 84.2.

4. The lengths of adult males' hands are normally distributed with mean 188 mm and standard deviation is 7.2 mm. Suppose that 17 individuals are randomly chosen. Round all answers to 4 decimal places where possible.

1. What is the distribution of x¯? x¯ ~ N( , )
2. For the group of 17, find the probability that the average hand length is more than 187.
3. Find the third quartile for the average adult male hand length for this sample size.

5. Suppose that the average number of Facebook friends users have is normally distributed with a mean of 125 and a standard deviation of about 55. Assume fourteen individuals are randomly chosen. Answer the following questions. Round all answers to 4 decimal places where possible.

1. What is the distribution of x¯? x¯ ~ N( , )
2. For the group of 14, find the probability that the average number of friends is less than 107.
3. Find the first quartile for the average number of Facebook friends

6. The amount of syrup that people put on their pancakes is normally distributed with mean 57 mL and standard deviation 9 mL. Suppose that 41 randomly selected people are observed pouring syrup on their pancakes. Round all answers to 4 decimal places where possible.

1. What is the distribution of X? X ~ N( , )
2. What is the distribution of x¯? x¯ ~ N( , )
3. If a single randomly selected individual is observed, find the probability that this person consumes is between 57.7 mL and 59.2 mL.
4. For the group of 41 pancake eaters, find the probability that the average amount of syrup is between 57.7 mL and 59.2 mL