1. Almost all medical schools in the United States require students to take the Medical College Admission Test (MCAT). To estimate the mean score of those who took the MCAT on your campus, you will obtain the scores of an SRS of students. The scores follow a Normal distribution, and from published information, you know that the standard deviation of scores for all MCAT takers is 10.6.10.6. Suppose that (unknown to you) the mean score of those taking the MCAT on your campus is495. a. If you choose one student at random, what is the probability that the student's score is between 490490 and 500? answer to four decimalplaces.

b. You sample 2525 students. What is the mean of the sampling distribution of their average score x?

Give your answer as a wholenumber.

c. What is the standard deviation of the sampling distribution of their average score x?

Give your answer to two decimalplaces.

d. Using the same sample size of 25 students, what is the probability that the mean score of your sample is between 490490 and 500?500?

Give your answer to four decimalplaces.

2. It appears that people who are mildly obese are less active than leaner people. One study looked at the average number of minutes per day that people spend standing or walking. Among mildly obese people, the mean number of minutes of daily activity (standing or walking) is approximately Normally distributed with 374374 minutes and standard deviation 6464 minutes. The mean number of minutes of daily activity for lean people is approximately Normally distributed with 522522 minutes and standard deviation 104104 minutes. A researcher records the minutes of activity for an SRS of 55 mildly obese people and an SRS of 55 leanpeople.

a. What is the probability that the mean number of minutes of daily activity of the 55 mildly obese people exceeds 400minutes?400minutes? Give your answer to four decimalplaces.

b. What is the probability that the mean number of minutes of daily activity of the 55 lean people exceeds 400minutes?400minutes? Give your answer to four decimalplaces.

3. In 2017, the entire fleet of lightduty vehicles sold in the United States by each manufacturer must emit an average of no more than 8686 milligrams per mile (mg/mi) of nitrogen oxides (NOX) and nonmethane organic gas (NMOG) over the useful life (150,000150,000 miles of driving) of the vehicle. NOX ++ NMOG emissions over the useful life for one car model vary Normally with mean 8080 mg/mi and standard deviation4mg/mi. a. What is the probability that a single car of this model emits more than 86mg/mi86mg/mi of NOX ++ NMOG? Give your answer to four decimalplaces.

b. A company has 25cars25cars of this model in its fleet. What is the probability that the average NOX ++ NMOG level x of these cars is above 86mg/mi?86mg/mi? Give your answer to four decimalplaces.

4. The level of nitrogen oxides (NOX) and nonmethane organic gas (NMOG) in the exhaust over the useful life (150,000miles150,000miles of driving) of cars of a particular model varies Normally with mean 80mg/mi80mg/mi and standard deviation 6mg/mi.6mg/mi. A company has 1616 cars of this model in itsfleet.

a. find the level L such that the probability that the average NOX+NMOGNOX+NMOG level x for the fleet is greater than L is only0.03.only0.03. Give your answer to three decimalplaces.

5. Andrew plans to retire in 40years.40years. He plans to invest part of his retirement funds in stocks, so he seeks out information on past returns. He learns that from 1969 to 2018, the annual returns on S&P 500500 had mean 9.8%9.8% and standard deviation 16.8%.16.8%. The mean return over even a moderate number of years is close toNormal.

a. What is the probability, 1,p1, (assuming that the past pattern of variation continues) that the mean annual return on common stocks over the next 40years will exceed 10%? Give your answer to two decimalplaces.

b. What is the probability,,p2, that the mean return will be less than 5%? Give your answer to two decimalplaces.

6. In response to the increasing weight of airline passengers, the Federal Aviation Administration (FAA) in 2003 told airlines to assume that passengers average 195pounds in the winter, including clothing and carryon baggage. But passengers vary, and the FAA did not specify a standard deviation. A reasonable standard deviation is .35pounds. Weights are not normally distributed, especially when the population includes both men and women, but they are not verynonNormal.

a. A commuter plane carries 22 passengers. What is the approximate probability that the total weight of the passengers exceeds 4500pounds?Use the fourstep process to guide your work. Give your answer as a percentage precise to two decimal places.

7. To estimate the mean score of those who took the Medical College Admission Test on your campus, you will obtain the scores of an SRS of students. From published information you know that the scores are approximately Normal with standard deviation about 10.6. You want your sample mean x to estimate with an error of no more than 1point1point in eitherdirection.

a. What standard deviation must x have so that99.7% of all samples give an x within 1point of ?

Use the 68-95-99.7 rule. Give your answer to four decimalplaces.

b. How large an SRS do you need in order to reduce the standard deviation of x to the value you found?

Give your answer rounded up to the nearest wholenumber.

8. The numbers racket is a wellentrenched illegal gambling operation in most large cities. One version works as follows: you choose one of the 1000 threedigit numbers 000 to 999 and pay your local numbers runner a dollar to enter your bet. Each day, one threedigit number is chosen at random and pays off$600.

The law of large numbers tells us what happens in the long run. Like many games of chance, the numbers racket has outcomes that vary considerablyone threedigit number wins $600 and all others win nothingthat gamblers never reach "the long run." Even after many bets, their average winnings may not be close to the mean. For the numbers racket, the mean payout for single bets is $0.60 (60 cents) and the standard deviation of payouts is about .$18.96. If Joe plays 350 days a year for 40 years, he makes 14,000bets.

Unlike Joe, the operators of the numbers racket can rely on the law of large numbers. It is said that the New York City mobster Casper Holstein took as many as 25,000 bets per day in the Prohibition era. That is 150,000 bets in a week if he takes Sunday off. Casper's mean winnings per bet are $0.40 (he pays out 6060 cents of each dollar bet to people like Joe and keeps the other 40 cents). His standard deviation for single bets is about $18.96, the same asJoe's.

a. What is the mean of Casper's average winnings x on his 150,000 bets? Give your answer to two decimalplaces.

b. What is the standard deviation of Casper's average winnings x on his 150,000 bets? Give your answer to three decimalplaces.

c. According to the central limit theorem, what is the approximate probability that Casper's average winnings per bet are between $0.30 and $0.50? give answer to four decimal places