1. (The Excel file BASE.XLS, found on the CD that came with your Berk and Carey text, has average career batting averages for 263 major league baseball players at the start of the 1988 season.
a. Calculate descriptive data statistics (e.g., mean, median, and standard deviation) for the batting averages.
b. Obtain a histogram for the batting averages of the players and print out with appropriate title, legend and axis scaling. Interpret your results.
c. Divide the data into three sub-populations: rookies (those players with only 1 year of experience); young players (those players with less than 6 years of experience); and veterans (those players with 6 or more years of experience). Use the STATS add-in provided by the Berk and Carey text to create a single graph which contains four boxplots corresponding to the batting averages of: 1) all 263 players; 2) rookies; 3) young players, and, 4) veterans. Please make sure the graph is properly labeled and that the y-scale is modified to go from 0.180 to 0.360.
d. Interpret fully the results found in part c paying particular attention to the presence of outliers, the difference in variation, the relative values of percentiles and any other contrasts that you might find interesting between the batting averages in these populations of baseball players.
2. (You are examining a brick manufacturing facility. The process has historically produced 5% nonconforming bricks. Under normal circumstances, this facility makes 25 bricks per hour.
a. Find the probability that there will be exactly 2 nonconforming bricks during the next hour of production.
b. Management is really more concerned about the probability that at least 1 brick in the next hour's production fails to conform to the standards. Find this probability.
c. Calculate the expected value and standard deviation of the number of nonconforming bricks during a typical hour of production.
3. A cable manufacturer has historically averaged 2.6 flaws per 1000 meters length of wire production.
a. Find the probability that a 1000 meter length of wire has one or fewer flaws.
b. Find the probability that a 1000 meter length of wire has more than one flaw.
c. Find the mean number of flaws per 1000 meter length of wire, the variance and the standard deviation
d. Generate the graph of the probability mass function for the number of flaws in 1000 meters of wire.
e. Suppose you examine a 1000 meter length of wire. What is the probability that there is exactly 1 defect in the first 500 meters and exactly 2 defects in the second 500 meters of wire?
What is the probability that there are exactly 3 defects anywhere in the 1000 meters of wire?
Do your two answers "make sense"? Please explain.
4. Consider a process for making nickel battery plates that has an operator who successfully meets the weight specification 40% of the time. Let Y be the number of plates she makes until she is successful.
a. Find the probability that she requires 4 attempts to make her first successful plate.
b. Generate the graph of the probability mass function for the number of attempts until her first successful attempt.
c. Find the probability that she requires more than 1 attempt to make her first successful plate.
d. Find the probability that she requires 10 attempts to make her third successful plate.
5. Suppose you are purchasing small lots of cathode ray tubes (CRTs) for computer terminals. Since it is very costly to test a single CRT, it may be desirable to test a sample of CRTs from the lot rather than every CRT in the lot. Such a sampling plan would be based on a hypergeometric probability distribution. For example, assume that each lot contains ten CRTs. You decide to sample three CRTs per lot and to reject the lot if you observe one or more defectives in the sample.
a. If the lot contains one defective CRT, what is the probability that you will accept the lot?
b. What is the probability that you will accept the lot if it contains three defective CRTs?
6. (An electrical current traveling through a resistor may take one of three different paths, with probabilities 25. , , and p1 = 30. p2 = p3 = .45, respectively. Suppose we monitor the path taken in n =10 consecutive trials.
a. Find the probability that the electrical current will travel the first path times, the second path times, and the third path 2 y1 = 4 y2 = y3 = 4 times.
b. Find E(y2) and VAR(y2). Interpret the results.