To receive full credit you must show your work for all problems listed below. The point values associated with
each part are clearly marked. Don't spend too much time on one particular problem. The following formulas may be
helpful on this exam.
| Random Variable | Expected Value (Mean) | Variance |
| Number of Successes | n*p | n*p*(1-p) |
| Proportion of Successes | p | p*(1-p)/n |
| Sample Average | mu | (sigma^2)/n |
a. What is the expected difference in the times?
b. What is the standard deviation of the difference in the two times?
c. What is the probability that the times will differ by 10 seconds or more in either direction?
2. Suppose that 65% of the American people agree with the way President Clinton is dealing with the crisis in
Iraq. In a national poll of 5,000 Americans, what is the probability that the sample percentage of people who disagree
with the President's actions in the confrontation with Iraq is greater than 34%?
3. The chance of experiencing side effects with a new medication for children is .20. A pediatrician prescribes
this medication for 12 of her patients.
a. What is the probability that exactly 2 of her patients will experience side effects with this new medication?
b. What is the probability that at least 2 of her patients will experience side effects with this new medication?
c. What is the probability that at most 2 of her patients will experience side effects with this new medication?
d. What is the probability that more than 2 of her patients will experience side effects with this new medication?
e. What is the probability that less than 2 of her patients will experience side effects with this new medication?
4. Past experience indicates that about 40% of all alumni who receive a survey will take the time to complete and return the survey. A self-study committee at a small liberal arts college mails a survey to 4000 randomly selected alumni.
a. What are the mean and standard deviation of the number of alumni who will complete and return the survey?
b. Approximate the probability that at least 1650 alumni will complete and return the survey.
5. Suppose that a golfer's score in a round (18 holes) of golf varies with a standard deviation of 5 shots a
round. How many rounds of golf would the golfer need to play to estimate his average score within ±3 shots
with 96% confidence?
6. A saleswoman earns a 10% commission on all sales of a particular product. The weekly sales of this product varies
according to the normal distribution with a mean of $3700 and a standard deviation of $100.
a. What is the probability that the saleswoman will earn more than $350 in any particular week?
b. What is the probability that the saleswoman will average more than $350 per week in a month which contains 4 weeks?
7. An agronomist examines the cellulose content of a variety of alfalfa hay. Suppose that the cellulose content in the population has standard deviation = 8 milligrams per gram (mg/g). A sample of 15 cuttings has an average cellulose content of 145 mg/g.
a. Give a 90% confidence interval for the mean cellulose content in the population.
b. A previous study claimed that the mean cellulose content was µ = 140 mg/g, but the agronomist believes that the mean is higher than that figure. State H0 and Ha and carry out a significance test to see if the new data support this belief.
c. The statistical procedures used in (a) and (b) are valid when several assumptions are met. What are these assumptions?
Please show your work for all six problems listed below. If you do not show your work, you will not receive credit.
The point values associated with each part are clearly marked. Don't spend too much time on any one of the problems.
The following formulas may be helpful on this exam.
Good luck and enjoy the vacation!
1. A particular airline has 10:00 AM flights from Port Columbus to New York, Atlanta, and Los Angeles. Let A denote
the event that the New York flight is full, and define events B and C analogously for the other two flights. Suppose
that P(A) = .6, P(B) = .5, and P(C) = .4, and that the three events are independent.
a. What is the probability that all three flights are full?
b. What is the probability that at least one flight is not full?
c. What is the probability that only the New York flight is full?
d. What is the probability that exactly one of the three flights is full?
2. A local television station sells 15-second, 30-second, and 60-second advertising spots. Let X denote the length of a randomly selected commercial appearing on this station, and suppose the probability distribution of X is given in the table below.
| Length of Commercial | 15 | 30 | 60 |
| Probability | 0.1 | 0.3 | 0.6 |
a. Find the average length of commercials appearing on this station.
b. If a 15-second spot sells for $750, a 30-second spot for $1100, and a 60-second spot for $1700, find the average amount paid for commercials appearing on this station.
3. Industrial quality control programs often include inspection of incoming materials from suppliers. If parts are purchased in large lots, a typical plan might be to select 20 parts at random from the lot and inspect them. The lot might be judged as being of acceptable quality if 1 or fewer defective parts are found among those inspected. Otherwise the lot is rejected and returned to the supplier. Find the probability of accepting lots that have
a. 5% defective parts;
b. 10% defective parts.
4. Suppose your instructor decides over the Thanksgiving break to construct a 250 question multiple choice final exam for a course you are enrolled in this semester. Suppose further that you get so busy after break that you cannot find time to study for this final exam. Thus, during the exam period for this course you must guess (select one of the 5 answers in a completely random fashion) on each question. Let X represent the number of correct answers on your test.
a. What kind of probability distribution does X have?
b. What is your expected score on the exam?
c. Compute the variance and standard deviation of X?
d. Is it likely that you would score over 50 on this exam? Justify your response.
5. Suppose that 60% of all students using the Roth classroom write their names on their diskettes. A random sample of 300 students is selected.
a. What is the mean value of the proportion among the 300 students sampled who have their names on their diskettes?
b. What is the standard deviation of the sample proportion?
c. What is the chance that at most 55% of those sampled have their names on their diskettes?
6. Although our class periods are exactly 50 minutes in length, my actual lecture time on a particular day can be represented by a random variable with mean 52 minutes and standard deviation 2 minutes. Suppose that times of different lectures are independent of one another and I give 36 lectures during the semester.
a. What is the probability that my lecture on Monday, Nov. 27 will be less than 50 minutes in length?
b. What is the probability that my average lecture time over the entire semester will be less than 50 minutes in length?
To receive full credit you must show your work for all problems listed below. The point values associated with
each part are clearly marked. Don't spend too much time on one particular problem. The following formulas may be
helpful on this exam.
1. A used car dealer finds that during 40% of the days, no cars are sold; 20% of the time, one car is sold; 15%
of the time, two cars are sold; 10% of the time, three are sold; 8% of the time, four are sold; 6% of the time,
five are sold; and 1% of the time, six are sold.
a. Find the mean and standard deviation of the number of cars sold.
b. What percentage of the time can the dealer expect the number of cars sold to be within one standard deviation of the mean?
c. The dealer's salesman, Mike, gets a straight commission of $50 on the first car sold and $100 for each additional car sold, no matter who sells it. Mike gets $30 on days when no cars are sold. What are Mike's expected daily earnings?
2. A judge is scheduled to hear 20 appeals for traffic tickets. Each appeal has a probability of .4 of being approved, independent of the other appeals. Find the probability that
a. Exactly 8 of the appeals are approved;
b. At most 8 of the appeals are approved;
c. At least 8 of the appeals are approved;
d. Exactly 8 of the appeals are not approved;
e. Less than half of the appeals are not approved.
3. Previous studies have shown that the standard deviation on an achievement test is 25. How large a sample must be taken so that a 95% confidence interval for the average test score will have a width of 3?
4. A frozen yogurt store offers a 5-ounce cup. The weight of the contents of a cup of yogurt has a normal distribution with µ = 5 oz. and = .2 oz.
a. What is the probability that a randomly selected cup of yogurt will weigh more than 5 oz.?
b. Find the probability that five randomly selected cups will have an average weight greater than 5 oz.
c. Redo (a) and (b) by replacing 5 oz. with 5.3 oz.
d. Explain your results to another student that is skeptical about your calculations. That is, explain why these answers (4 probabilities) intuitively make sense.
5. The caffeine content, in milligrams (mg), was examined for a random sample of 50 cups of black coffee dispensed by a new machine. The mean was 110 mg.
a. Assume, as the manufacturer claims, that = 7 mg to construct a 98% confidence interval for µ, the mean caffeine content for cups dispensed by the machine.
b. How would you modify the confidence interval in part (a) if you found the standard deviation of caffeine content to be s = 5 mg for your sample of 50 cups? That is, you no longer believe the manufacturer's claim.
6. A military analyst wants to know how the success rate of a laser-guided bomb compares with the 90% accuracy that is claimed for it under combat conditions. During one week of conflict, it was reported that the bomb successfully hit 235 of 250 intended targets. The analyst wants to know the probability of hitting at least 235 targets if the probability of success on any one target is .90. Find the appropriate probability for the analyst using:
a. the binomial distribution;
b. the normal approximation to the binomial (with continuity correction).
c. Which answer (a) or (b) would you report to the analyst? Why?
7. Consider the question of whether the home team wins more than half of its games in the National Basketball Association. Suppose that the home team actually wins 50% of all games and that you study a simple random sample of 30 games.
a. According to the Central Limit Theorem, how would the sample proportion of home victories vary from sample to sample?
b. Determine the probability that the home team would win 60% or more of the games in a sample of 30 games (still assuming that the home team wins 50% or all games).
c. Of the 30 NBA games played on January 3-7,1992, 18 were won by the home team. In light of your answer to (b), does this sample information provide strong evidence of a home court advantage?