You may not use any extra sheets of paper to answer the following questions, but you may use the back of page 3. Whenever possible, show your work for potential partial credit. NOTE: When performing numerical calculations, keep at least 4 digits after a decimal. (I.e., do NOT round .2265 to .23 or .227) BUDGET YOUR TIME WISELY!
1. The G.PA. for the population of Hanover College students is a continuous random variable that is distributed normally with a mean of 2.94 and a standard deviation of .40.
a. Any student who graduates with a G.P.A. of 3.90 or better, earns the distinction of graduating summa cum laude. How many students in a college of 1100 would you expect to earn such an honor? (5 points)
b. A student earns a spot on the Dean’s List if their GPA is above a 3.50. What is the probability that a student will be on the Dean’s List, but not a candidate for summa cum laude? (5 points)
2. Alcohol consumption for the population of Hanover College students is a continuous random variable that is distributed normally with a mean of 4.3583 beverages per week and a standard deviation of 7.4732. What is the probability of a randomly selected college student having at least 4 drinks in a typical week? (5 points)
3. A particular make of automobile is listed as weighing 4000 pounds. Because of weight differences due to the options ordered with the car, the actual weight for this make varies uniformly between 3600 and 4500 pounds. What is the mathematical expression for the probability density function? What is the probability that the car will weigh less than 3950 pounds? (8 points)
4. A new soft drink is being market tested. It is estimated that 60% of consumers will like the new drink. Suppose a market research firm wishes to sample consumers at a mall. Interviewers contact every 40th person who enters the mall and asks them to sample the soft drink, until a sample of 100 people has been gathered.
a. Would you agree with the research group’s assumption that the population is an infinite one? Explain. (4 points)
b. Does this sampling method appear to provide a simple random sample? Explain. (8 points)
c. Determine the standard error of the proportion. (4 points)
d. What important role does the Central Limit Theorem serve whenever a point estimator is used to estimate a population parameter? (10 points)
e. What is the probability that the sample proportion will be within .02 of the true population proportion? (6 points)
f. How would your answer to the previous question differ if 200 people were randomly sampled? (5 points)
g. Why is sample size so critical to sampling a population? (8 points)
5. A confidence interval consists of two parts: a point estimate and a margin of error. Show and interpret each part of a confidence interval. How does the sample size affect the confidence interval? Explain. (10 points)
6. According to Nielsen Media Research viewership data, the top television broadcast of all time was the last episode of M*A*S*H, which aired on February 28, 1983, and was viewed by an estimated 60.2% of all TV households. Assuming this estimate was based on a simple random sample of 1800 TV households, what is the 95% confidence interval for p, the proportion of all TV households who viewed the last episode of M*A*S*H? (10 points)
7. In a major industry, where well over 100,000 manufacturing employees are represented by a single union, a simple random sampling of n=100 union members finds that 57% of those in the sample intend to vote "yes" for the new labor contract negotiated by union and management representatives. Use a 99% confidence interval to forecast whether or not this contract will be approved with a simple majority vote. Is it a "sure thing"? Why or why not? (12 points)