UNIVERSITY OF MINNESOTA, MORRIS
INTRODUCTION TO STATISTICS
(GENED WEB)
MATH. 1601
SOLUTION TO THE PRACTICE TEST
SUMMER, 1999
FOR INSTRUCTOR'S USE
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b........./7 |
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TOTAL |
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1. A random sample of 25 college students is surveyed about their political views. Five of them have liberal views.Define the parameter of interest and the statistic to estimate this parameter. |
PARAMETER = The proportion of all students who have liberal views
2. The table below gives the results of a study to determine better treatments for cocaine addiction. There are three treatments, despramine, lithium, and placebo.
a. What is the probability that the subject did not receive placebo? b. What is the probability that the subject was on the placebo treatment or had a relapse? c. Find the probability that the subject was in the placebo group given that there was no relapse into drug use. d. Find the probability that the subject had a relapse given that they were on the Lithium treatment. e. Are the events "being on lithium treatment" and "having a relapse" disjoint? Are they independent? Please justify your answer to receive a credit. f. Are the events "being on lithium treatment" and "being on despramine treatment" independent? Are they disjoint? Please justify your answer to receive a credit. |
Let Y="yes", N="no", A="Despramine", B="Lithium", C="Placebo"
3. According to the Arizona Chapter of the Lung Association, 7.0% of the population has a lung disease. Of those having lung disease, 90.0% are smokers; and of those not having lung disease, 25.3% are smokers. Determine the probability that a randomly selected smoker has a lung disease. |
Let L="having lung disease", S="being a smoker"
4. According to Maureen and Jay Neitz of the Medical College of Wisconsin Eye Institute, 9% of men are color blind. For four randomly selected men, find the probability that a. none of them are color blind. b. exactly one of the four is color blind.
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a. 0.914=0.6857
5. An investor plans to invest $50,000 in one of four investments. The return on each investment depends on whether next year's economy is strong or weak. The following table summarizes the possible payoffs, in dollars, for the four investments.
Assume that next year's economy has a 40% chance of being strong and a 60% chance of being weak. Which investment has the best expected payoff/ Which one has the worst? Which investment would you select? Explain your answer. |
Investment |
EXPECTED PAYOFF |
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Certificate of Deposit |
6,000 |
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Office Complex |
9,000 |
Best |
Land Speculation |
3,000 |
Worst |
Technical School |
8,200 |
6.The Arizona state lottery, Lotto,
is played as follows:
The player selects six numbers from the numbers 1-42 and buys a ticket
for $1. There are six winning numbers, which are selected at random from
the numbers 1-42. To win a prize, a Lotto ticket must contain
three or more of the winning numbers. Following is a probability
distribution for the number of winning numbers for a single ticket.
a. Determine the mean and the standard deviation. b. If you buy one Lotto ticket, determine the probability that you win a price. c. If you buy one Lotto ticket per week for a year, determine the probability that you win a prize at least once in the 52 tries. d. If you have more than 1 winning numbers on a ticket, what is the probability you have 3 winning numbers on that ticket? |
a.mean=mu=Sum(xp)=0+0.4311941+0.3368704+0.0816657+0.0072056+0.000206+0.0000012=0.857143
7. a. On a recent trip to Mexico I noticed that there is a randomizated system for inspecting incoming passenger baggage. Suppose a passenger pushes a light that comes up red with probability 0.1 and green with probability 0.9. If your light is red the customs official examines your baggage for illegal items, and if the light is green you pass through without inspection. Let us assume that the probability of customs official being able to find illegal items to be 0.3 if an inspection is conducted. Let us also imagine testing this system with a large number of smugglers to observe the system's effectiveness. What proportion of smugglers will not get caught from this system? Draw a tree diagram to answer the question.
b. A somewhat absent-minded hiker forgets to bring her insect repellent on 30% of his hikes. The probability of being bitten is 90% if he forgets the repellent, and 20% if she uses the repellent. What is the probability that she forgot her repellent given that she was bitten. |
a.Let R="red light", G="green light", F="finding illegal item"
8. An airline knows that 10% of the people holding reservations on a given flight will not appear. The plane holds 18 people. If 20 reservations have been sold, find the probability that the airline can accommodate everyone appearing for the flight. |
Let X= Number of people who will not show up out of 20.
9. According to tables provided by the U.S. National Center for Health Statistics in Vital Statistics of the United States, there is about an 80% chance that a person age 20 will be alive at age 65. Suppose 15 people age 20 are selected at random. a. Find the probability that the number alive at age 65 will be exactly five. b. Find the probability that the number alive at age 65 will be at least five. c. Suppose that 200 people selected at random. What is the probability that the proportion of them alive at age 65 will be more than 50%? |
a. Let X=number of people alive at age 65 Y=number of people not alive at age 65
10. Gestation period of humans are normally distributed with a mean of 266 days and a standard deviation of 16 days. Suppose that we observe the gestation periods for a sample of nine humans. a. Find the probability that the average gestation period for the nine humans selected is more than 270 days. b. Find the probability that the average gestation period for the nine humans is less than 250 days. |