Because of safety considerations, in May 2003 the Federal Aviation Administration (FAA) changed its guidelines for how small commuter airlines must estimate passenger weights. Under the old rule, airlines used 180 pounds as a typical passenger weight (including carry-on luggage) in warm months and 185 pounds as a typical weight in cold months. The Alaska Journal of Commerce (May 25, 2003) reported that Frontier Airlines conducted a study to estimate average passenger plus carry-on weights. They found an average summer weight of 183 pounds and a winter average of 190 pounds. Suppose that each of these estimates was based on a random sample of 100 passengers and that the sample standard deviations were 20 pounds for the summer weights and 23 pounds for the winter weights. a. Construct and interpret a $95\%$ confidence interval for the mean summer weight (including carry-on luggage) of Frontier Airlines passengers. b. Construct and interpret a $95\%$ confidence interval for the mean winter weight (including carry-on luggage) of Frontier Airlines passengers. c. The new FAA recommendations are 190 pounds for summer and 195 pounds for winter. Comment on these recommendations in light of the confidence interval estimates from Parts (a) and (b).

Suppose that the distribution of the number of items x produced by an assembly line during an 8-hour shift can be approximated by a normal distribution with mean value 150 and standard deviation 10. a. What is the probability that the number of items produced is at most 120? b. What is the probability that at least 125 items are produced? c. What is the probability that between 135 and 160 (inclusive) items are produced?

A population has a proportion equal to 0.30. Calculate the following probabilities with n=100 :

a. $P(\bar{p} \leq 0.35)$

b. $P(\bar{p}>0.40)$

c. $P(0.25<\bar{p} \leq 0.40)$

d. $P(\bar{p} \geq 0.27)$.

A marketing researcher wants to estimate the mean amount spent ($) on Amazon.com by Amazon Prime member shoppers. Suppose a random sample of 100 Amazon Prime member shoppers who recently made a purchase on Amazon.com yielded a mean of$1,500 and a standard deviation of $200. Interpret the interval constructed in (a).

Use the Integral Test to determine whether each series converges or diverges. $\sum_{k=1}^{\infty} \frac{1}{k^{2}+1}$

The town of Paloma has a total of 90 businesses. Of these businesses, 38 are registered with the Chamber of Commerce, 26 are restaurants, and 50 are incorporated. Furthermore, 12 are restaurants and are incorporated, 11 are registered with the Chamber of Commerce and are restaurants, 10 are registered with the Chamber of Commerce and are incorporated, and 4 are in all 3 categories. Let C = the event that the business is registered with the Chamber of Commerce, R = the event that the business is a restaurant, and I = the event that the business is incorporated. A business is randomly chosen.

Find $P(R \cup I)$.

Princess Cruises offered a 9-day voyage from Beijing to Bangkok during the time period from May to August. The announced price, excluding airfare, for a room with an ocean view or a balcony was listed as $3,475. Cruise fares usually are quite variable due to discounting by the cruise line and travel agents. A sample of 20 passengers who purchased this cruise paid the following amounts (in dollars):

a. Calculate the sample mean cruise fare.

b. Determine the sampling error for this sample.

c. Would the results obtained in part b indicate that the average cruise fare during this period for this cruise is different from the listed price? Explain your answer from a statistical point of view.

Discuss each potential problem in obtaining a random survey of a population. Include suggestions for overcoming the problem.

a. The people selected might not be a random sample of the population.

b. The people selected might not be willing to participate in the survey.

c. The people selected might not be truthful when answering the question.

d. The peopled selected might not understand the survey question.

According to a study conducted by the Gallup organization, the proportion of Americans who are afraid to fly is 0.10. A random sample of 1100 Americans results in 121 indicating that they are afraid to fly. Explain why this is not necessarily evidence that the proportion of Americans who are afraid to fly has increased since the time of the Gallup study.

Consider a random variable, z, that has a standardized normal distribution. Determine the following probabilities:

a. P(0<z<1.96)

b. P(z>1.645)

c. $P(1.28<z \leq 2.33)$

d. $P(-2 \leq z \leq 3)$

e. P(z>-1)

The time that a randomly selected individual waits for an elevator in an office building has a uniform distribution over the interval from 0 to 1 minute. For this distribution $\mu=0.5 \$ and$\sigma=0.289$. a. Let be the sample mean waiting time for a random sample of 16 individuals. What are the mean and standard deviation of the sampling distribution of ? b. Answer Part (a) for a random sample of 50 individuals. Draw a picture of a good approximation to the sampling distribution of when n = 50.$$$\overline{x}$.

A sample of 548 ethnically diverse students from Massachusetts were followed over a 19-month period from 1995 and 1997 in a study of the relationship between TV viewing and eating habits (Pediatrics [2003]: 1321–1326). For each additional hour of television viewed per day, the number of fruit and vegetable servings per day was found to decrease on average by 0.14 serving. a. For this study, what is the dependent variable? What is the predictor variable? b. Would the least-squares line for predicting number of servings of fruits and vegetables using number of hours spent watching TV as a predictor have a positive or negative slope? Explain.

Show all your work. Indicate clearly the methods you use. "I can't get through my day without coffee" is a common statement from many students. Assumed benefits include keeping students awake during lectures and making them more alert for exams and tests. Students in a statistics class designed an experiment to measure memory retention with and without drinking a cup of coffee one hour before a test. This experiment took place on two different days in the same week (Monday and Wednesday). Ten students were used. Each student received no coffee or one cup of coffee one hour before the test on a particular day. The test consisted of a series of words flashed on a screen, after which the student had to write down as many of the words as possible. On the other day, each student received a different amount of coffee (none or one cup). One of the researchers suggested that all the subjects in the experiment drink no coffee before Monday's test and one cup of coffee before Wednesday's test. Explain to the researcher why this is a bad idea and suggest a better method of deciding when each subject receives the two treatments.

The times between failures on a personal computer follow an exponential distribution with a mean of 300,000 hours. What is the probability of: a. A failure in less than 100,000 hours? b. No failure in the next 500,000 hours? c.The next failure occurring between 200,000 and 350,000 hours? d.What are the mean and standard deviation of the time between failures?