To investigate an alleged unfair trade practice, the Federal Trade Commission (FTC) takes a random sample of sixteen “5- ounce” candy bars from a large shipment. The mean of the sample weights is 4.85 ounces and the sample standard deviation is 0.1 ounce. It is reasonable to assume the population of candy bar weights is approximately Normally distributed. Based on this sample, does the FTC have grounds to proceed against the manufacturer for the unfair practice of short-weight selling, on average? Answer this question by completing the following steps of a hypothesis test at the 5% significance level.

d. Do you reject or fail to reject the null hypothesis? Why?

e. Does the FTC have grounds to proceed against the manufacturer for the unfair practice of short-weight selling, on average? Explain your answer.

Suppose we have a binomial experiment in which success is defined to be a particular quality or attribute that interests us.

(a)

• Suppose n = 45 and
• p = 0.13.

(For each answer, enter a number. Use 2 decimal places.)
n·p =  
n·q =  

Can we approximate p̂ by a normal distribution? Why? (Fill in the blank. There are four answer blanks. A blank is represented by _____.)

_____, p̂ _____ be approximated by a normal random variable because _____ _____.

first blank

YesNo     

second blank

cancannot     

third blank

both n·p and n·q exceedn·p and n·q do not exceed     n·q exceedsn·q does not exceedn·p exceedsn·p does not exceed

fourth blank (Enter an exact number.)


What are the values of μ_p̂ and σ_p̂? (For each answer, enter a number. Use 3 decimal places.)
μ_p̂ = mu sub p hat =

σ_p̂ = sigma sub p hat =

(b)

Suppose

• n = 25 and
• p = 0.15.

Can we safely approximate p̂ by a normal distribution? Why or why not? (Fill in the blank. There are four answer blanks. A blank is represented by _____.)

_____, p̂ _____ be approximated by a normal random variable because _____ _____.

first blank

YesNo     

second blank

cancannot     

third blank

both n·p and n·q exceedn·p and n·q do not exceed     n·q exceedsn·q does not exceedn·p exceedsn·p does not exceed

fourth blank (Enter an exact number.)

(c)

Suppose

• n = 59 and
• p = 0.40.

(For each answer, enter a number. Use 2 decimal places.)
n·p =  
n·q =  

Can we approximate  p̂ by a normal distribution? Why? (Fill in the blank. There are four answer blanks. A blank is represented by _____.)

_____, p̂ _____ be approximated by a normal random variable because _____ _____.

first blank

YesNo     

second blank

cancannot     

third blank

both n·p and n·q exceedn·p and n·q do not exceed     n·q exceedsn·q does not exceedn·p exceedsn·p does not exceed

fourth blank (Enter an exact number.)


What are the values of μ_p̂ and σ_p̂? (For each answer, enter a number. Use 3 decimal places.)
μ_p̂ = mu sub p hat =

σ_p̂ = sigma sub p hat =

All work must be done in R programing. Consider this dataset provided to you as prob10.txt c1 t1 c2 t2 c3 t3 c4 t4 2650 3115 2619 2933 2331 2799 2750 3200 1200 1101 1200 1309 1888 1901 1315 980 1541 1358 1401 1499 1256 1238 1625 1421 1545 1910 1652 2028 1449 1901 1399 2002 1956 2999 2066 2880 1777 2898 1999 2798 1599 2710 1754 2765 1434 2689 1702 2402 2430 2589 2789 2899 2332 2300 2250 2741 1902 1910 2028 2100 1888 1901 2000 1899 1530 2329 1660 2332 1501 2298 1478 2287 2008 2485 2104 2871 1987 2650 2100 2520 (2) Read it in and set the row names to “Gene 1” through “Gene 10” It should look like this in R > prob10 c1 t1 c2 t2 c3 t3 c4 t4 Gene 1 2650 3115 2619 2933 2331 2799 2750 3200 Gene 2 1200 1101 1200 1309 1888 1901 1315 980 Gene 3 1541 1358 1401 1499 1256 1238 1625 1421 Gene 4 1545 1910 1652 2028 1449 1901 1399 2002 Gene 5 1956 2999 2066 2880 1777 2898 1999 2798 Gene 6 1599 2710 1754 2765 1434 2689 1702 2402 Gene 7 2430 2589 2789 2899 2332 2300 2250 2741 Gene 8 1902 1910 2028 2100 1888 1901 2000 1899 Gene 9 1530 2329 1660 2332 1501 2298 1478 2287 Gene 10 2008 2485 2104 2871 1987 2650 2100 2520 (3 )Perform a one-sample t-test to compare the hypothesis that the mean of the control expression values is 2000.

Determine the margin of error for a confidence interval to estimate the population mean with n=18 and s=12.1 for the confidence levels below.

A) The margin of error for an 80% confidence interval is:

B) The margin of error for an 90% confidence interval is:

C) The margin of error for an 99% confidence interval is:

The board of directors for a particular company consists of 10 members, 6 of whom are loyal to the current company president and 4 of whom want to fire the president. Suppose the chair of the board (who is a loyal supporter of the current president) suggests to randomly select 4 other board members to serve on a committee to decide the president’s fate. Find the probability for the first 3 questions and explain your answer for the fourth question. (75–150 words, or 1–2 paragraphs)

-What is the probability that all 5 committee members will vote to keep the president in place, if no one changes their minds?

-What is the probability that a majority of the committee will vote to keep the president in place, if no one changes their minds?

-What is the probability that the vote is 4 to 1 to replace the president, if no one changes their minds?

-Imagine that you were the president of the company and you hoped to keep your position. Considering the various probabilities, would you consider the chair of the board’s suggestion to be in your favor or not? If the choice was yours, would you allow the suggestion to proceed?

Summarize the different approach in school safety such as metal detectors arm guard and 0 tolerance Policies

Richard has just been given a 4-question multiple-choice quiz in his history class. Each question has five answers, of which only one is correct. Since Richard has not attended class recently, he doesn't know any of the answers. Assuming that Richard guesses on all four questions, find the indicated probabilities. (Round your answers to three decimal places.)

(a) What is the probability that he will answer all questions correctly?


(b) What is the probability that he will answer all questions incorrectly?


(c) What is the probability that he will answer at least one of the questions correctly? Compute this probability two ways. First, use the rule for mutually exclusive events and the probabilities shown in the binomial probability distribution table.


Then use the fact that P(r ≥ 1) = 1 − P(r = 0).


Compare the two results. Should they be equal? Are they equal? If not, how do you account for the difference?

(d) What is the probability that Richard will answer at least half the questions correctly?

What is the age distribution of patients who make office visits to a doctor or nurse? The following table is based on information taken from a medical journal.

Suppose you are a district manager of a health management organization (HMO) that is monitoring the office of a local doctor or nurse in general family practice. This morning the office you are monitoring has eight office visits on the schedule. What is the probability of the following?

(a) At least half the patients are under 15 years old. (Round your answer to three decimal places.)


Explain how this can be modeled as a binomial distribution with 8 trials, where success is visitor age is under 15 years old and the probability of success is 10%?

Let n = 15, p = 0.10 and compute the probabilities using the binomial distribution.Let n = 8, p = 0.10 and compute the probabilities using the binomial distribution.    Let n = 8, p = 0.90 and compute the probabilities using the binomial distribution.Let n = 8, p = 0.15 and compute the probabilities using the binomial distribution.

(b) From 2 to 5 patients are 65 years old or older (include 2 and 5). (Round your answer to three decimal places.)


(c) From 2 to 5 patients are 45 years old or older (include 2 and 5). (Hint: Success if 45 or older. Use the table to compute the probability of success on a single trial. Round your answer to three decimal places.)


(d) All the patients are under 25 years of age. (Round your answer to three decimal places.)


(e) All the patients are 15 years old or older. (Round your answer to three decimal places.)

1. Generate 1000 random numbers from ??3? starting with standard normal random numbers in R.

1. Generate 1000 random numbers from ??2, 5? starting with standard normal random numbers in R.

In an article in the Journal of Marketing, Bayus studied the differences between "early replacement buyers” and "late replacement buyers” in making consumer durable good replacement purchases. Early replacement buyers are consumers who replace a product during the early part of its lifetime, while late replacement buyers make replacement purchases late in the product’s lifetime. In particular, Bayus studied automobile replacement purchases. Consumers who traded in cars with ages of zero to three years and mileages of no more than 35,000 miles were classified as early replacement buyers. Consumers who traded in cars with ages of seven or more years and mileages of more than 73,000 miles were classified as late replacement buyers. Bayus compared the two groups of buyers with respect to demographic variables such as income, education, age, and so forth. He also compared the two groups with respect to the amount of search activity in the replacement purchase process. Variables compared included the number of dealers visited, the time spent gathering information, and the time spent visiting dealers.

(a) Suppose that a random sample of 796 early replacement buyers yields a mean number of dealers visited of x¯x¯ = 3.1, and assume that σ equals .77. Calculate a 99 percent confidence interval for the population mean number of dealers visited by early replacement buyers. (Round your answers to 3 decimal places.)

The 99 percent confidence interval is            [, ].

(b) Suppose that a random sample of 496 late replacement buyers yields a mean number of dealers visited of x¯x¯ = 4.8, and assume that σ equals .64. Calculate a 99 percent confidence interval for the population mean number of dealers visited by late replacement buyers. (Round your answers to 3 decimal places.)

The 99 percent confidence interval is            [, ].

(c) Use the confidence intervals you computed in parts a and b to compare the mean number of dealers visited by early replacement buyers with the mean number of dealers visited by late replacement buyers. How do the means compare?

1. Generate 1000 random numbers from a normal distribution with mean 1 and variance 2 using Box‐Muller transformation in R.

Trevor is interested in purchasing the local hardware/sporting goods store in the small town of Dove Creek, Montana. After examining accounting records for the past several years, he found that the store has been grossing over $850 per day about 70% of the business days it is open. Estimate the probability that the store will gross over $850 for the following. (Round your answers to three decimal places.)

(a) at least 3 out of 5 business days


(b) at least 6 out of 10 business days


(c) fewer than 5 out of 10 business days


(d) fewer than 6 out of the next 20 business days

(e) more than 17 out of the next 20 business days

1. Urban planners in Melbourne average 10.2 years’ experience. We know this from a Census of urban planners, but the raw data was lost, so we cannot use the data to compute σ. What an inconvenience!    (a) In a random sample of 65 urban planners in Melbourne’s growth areas, the average was 8.7 years with a standard deviation of 0.52. Are growth-areas planners significantly less experienced than the metropolitan average? Presume alpha = 0.05 (b) The same sample of urban planners reports an annual salary of $25,665 with s=$722. The statewide average salary is $23,899. Is there statistical evidence to suggest that growth area planners salaries are significantly higher than Melbourne overall? Presume alpha = 0.05

What is the relationship between transportation models and more general logistics models? Explain how these two types of linear optimization models are similar and how they are different.