M112, Practice Exam 2 Ch 4-6 Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) Compare the relative frequency formula for finding probabilities to the classical formula for finding probabilities. How are the two formulas similar and how are they different? What special requirements does the classical approach have? 2) On an exam on probability concepts, Sue had an answer of 13 8 result was incorrect. for one problem. Explain how she knew that this 3) Describe an event whose probability of occurring is 1 and explain what that probability means. Describe an event whose probability of occurring is 0 and explain what that probability means. 4) Define mutually exclusive events and independent events. Give an example of each. 5) Give an example of events which are independent but not mutually exclusive. 1
6) Cause of Death Cancer Heart Disease Other Total Smoker 135 310 205 650 Nonsmoker 55 155 140 350 Total 190 465 345 1,000 Discuss the methods for finding the following two probabilities and explain the important differences in the computations. 1) If one person is randomly selected, find the probability that he or she died of heart disease. 2) If one person is randomly selected, find the probability that he or she died of heart disease given that he or she was a nonsmoker. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Express the indicated degree of likelihood as a probability value. 7) "Your mother could not have died two years before you were born." A) 0 B) 0.5 C) 0.25 D) 1 8) "It will definitely turn dark tonight." F) 1 G) 0.5 H) 0.30 J) 0.67 Answer the question. 9) Which of the following cannot be a probability? A) 5 3 B) 2 3 C) 1 2 D) 3 5 Find the indicated probability. 10) A sample space consists of 20 separate events that are equally likely. What is the probability of each? 1 F) G) 20 H) 0 J) 1 20 2
11) A bag contains 6 red marbles, 3 blue marbles, and 7 green marbles. If a marble is randomly selected from the bag, what is the probability that it is blue? 3 A) B) 1 C) 1 1 D) 16 3 7 13 12) A class consists of 13 women and 49 men. If a student is randomly selected, what is the probability that the student is a woman? F) 13 1 G) H) 13 J) 49 62 62 49 62 Answer the question, considering an event to be "unusual" if its probability is less than or equal to 0.05. 13) Assume that one student in your class of 28 students is randomly selected to win a prize. Would it be "unusual" for you to win? A) Yes B) No From the information provided, create the sample space of possible outcomes. 14) Flip a coin three times. F) HHH HHT HTH HTT THH THT TTH TTT G) HTT THT HTH HHH TTH TTT H) HHH HTT HTH TTT HTT THH HHT THT J) HHH TTT THT HTH HHT TTH HTH Determine whether the events are mutually exclusive. 15) Meet a man with an umbrella. Meet a man with a raincoat. A) No B) Yes 3
16) Get a full time day job as a teller with a bank. Get a full time day job as a cashier at a store. F) Yes G) No 17) Go to a formal dinner affair. Wear blue jeans. A) Yes B) No Find the indicated probability. 18) The probability that Luis will pass his statistics test is 0.65. Find the probability that he will fail his statistics test. F) 0.35 G) 1.86 H) 0.33 J) 1.54 19) The table below describes the smoking habits of a group of asthma sufferers. Nonsmoker Occasional smoker Regular smoker Heavy smoker Total Men 384 33 64 49 530 Women 349 44 72 38 503 Total 733 77 136 87 1033 If one of the 1033 people is randomly selected, find the probability that the person is a man or a heavy smoker. A) 0.550 B) 0.597 C) 0.502 D) 0.563 4
20) 100 employees of a company are asked how they get to work and whether they work full time or part time. The figure below shows the results. If one of the 100 employees is randomly selected, find the probability of getting someone who carpools or someone who works full time. 1. Public transportation: 7 full time, 10 part time 2. Bicycle: 4 full time, 4 part time 3. Drive alone: 26 full time, 31 part time 4. Carpool: 10 full time, 8 part time F) 0.55 G) 0.49 H) 0.17 J) 0.28 5
21) 100 employees of a company are asked how they get to work and whether they work full time or part time. The figure below shows the results. If one of the 100 employees is randomly selected, find the probability that the person drives alone or cycles to work. 1. Public transportation: 8 full time, 9 part time 2. Bicycle: 3 full time, 3 part time 3. Drive alone: 28 full time, 31 part time 4. Carpool: 9 full time, 9 part time A) 0.65 B) 0.59 C) 0.36 D) 0.31 Is Event B dependent or independent of Event A? 22) A: A mosquito lands on your arm. B: You get a mosquito bite. F) Dependent G) Independent 23) A: A bird lands on your head. B: The bird lays an egg. A) Independent B) Dependent Find the indicated probability. 24) A bin contains 71 light bulbs of which 6 are defective. If 4 light bulbs are randomly selected from the bin with replacement, find the probability that all the bulbs selected are good ones. F) 0.702 G) 0 H) 0.915 J) 0.723 6
25) The table below describes the smoking habits of a group of asthma sufferers. Light Heavy Nonsmoker smoker smoker Total Men 314 42 33 389 Women 329 47 35 411 Total 643 89 68 800 If two different people are randomly selected from the 800 subjects, find the probability that they are both heavy smokers. A) 0.007128 B) 0.007225 C) 0.0002163 D) 0.001702 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 26) List the four requirements for a binomial distribution. Describe an experiment which is binomial and discuss how the experiment fits each of the four requirements. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Identify the given random variable as being discrete or continuous. 27) The number of oil spills occurring off the Alaskan coast A) Discrete B) Continuous 28) The braking time of a car F) Continuous G) Discrete 7
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Determine whether the following is a probability distribution. If not, identify the requirement that is not satisfied. 29) In a certain town, 20% of adults have a college degree. The accompanying table describes the probability distribution for the number of adults (among 4 randomly selected adults) who have a college degree. x P(x) 0 0.4096 1 0.4096 2 0.1536 3 0.0256 4 0.0016 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 30) A police department reports that the probabilities that 0, 1, 2, and 3 burglaries will be reported in a given day are 0.46, 0.43, 0.09, and 0.02, respectively. Find the standard deviation for the probability distribution. Round answer to the nearest hundredth. F) 0.72 G) 0.99 H) 0.98 J) 0.52 31) A contractor is considering a sale that promises a profit of $31,000 with a probability of 0.7 or a loss (due to bad weather, strikes, and such) of $13,000 with a probability of 0.3. What is the expected profit? A) $17,800 B) $21,700 C) $18,000 D) $30,800 8
Determine whether the given procedure results in a binomial distribution. If not, state the reason why. 32) Rolling a single die 34 times, keeping track of the numbers that are rolled. F) Not binomial: there are more than two outcomes for each trial. G) Not binomial: the trials are not independent. H) Procedure results in a binomial distribution. J) Not binomial: there are too many trials. 33) Rolling a single "loaded" die 39 times, keeping track of the "fives" rolled. A) Procedure results in a binomial distribution. B) Not binomial: the trials are not independent. C) Not binomial: there are more than two outcomes for each trial. D) Not binomial: there are too many trials. Find the indicated probability. 34) A machine has 7 identical components which function independently. The probability that a component will fail is 0.2. The machine will stop working if more than three components fail. Find the probability that the machine will be working. F) 0.967 G) 0.033 H) 0.029 J) 0.996 35) In a study, 45% of adults questioned reported that their health was excellent. A researcher wishes to study the health of people living close to a nuclear power plant. Among 11 adults randomly selected from this area, only 3 reported that their health was excellent. Find the probability that when 11 adults are randomly selected, 3 or fewer are in excellent health. A) 0.1911 B) 0.1268 C) 0.1259 D) 0.0652 9
Solve the problem. 36) The probability that a radish seed will germinate is 0.7. A gardener plants seeds in batches of 4. Find the standard deviation for the number of seeds germinating in each batch. F) 0.917 G) 0.794 H) 0.906 J) 0.784 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 37) Consider the uniform distribution shown below. Find the probability that x is greater than 6. Discuss the relationship between area under a density curve and probability. 38) Explain how a nonstandard normal distribution differs from the standard normal distribution. Describe the process for finding probabilities for nonstandard normal distributions. 39) Under what conditions can we apply the results of the central limit theorem? 10
40) Complete the following table for a distribution in which µ = 16. It might be helpful to make a diagram to help you determine the continuity factor for each entry. Find the probability that The continuity correction factor is: x is at least 12 x is at most 12 x is more than 12 x is less than 12 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Using the following uniform density curve, answer the question. 41) What is the probability that the random variable has a value less than 6? A) 0.750 B) 0.625 C) 0.500 D) 0.875 Assume that the weight loss for the first month of a diet program varies between 6 pounds and 12 pounds, and is spread evenly over the range of possibilities, so that there is a uniform distribution. Find the probability of the given range of pounds lost. 42) More than 10 pounds F) 1 3 G) 1 7 H) 2 3 J) 5 6 If Z is a standard normal variable, find the probability. 43) The probability that Z lies between 0 and 3.01 A) 0.4987 B) 0.9987 C) 0.1217 D) 0.5013 11
44) The probability that Z is greater than -1.82 F) 0.9656 G) 0.4656 H) 0.0344 J) -0.0344 The Precision Scientific Instrument Company manufactures thermometers that are supposed to give readings of 0 C at the freezing point of water. Tests on a large sample of these thermometers reveal that at the freezing point of water, some give readings below 0 C (denoted by negative numbers) and some give readings above 0 C (denoted by positive numbers). Assume that the mean reading is 0 C and the standard deviation of the readings is 1.00 C. Also assume that the frequency distribution of errors closely resembles the normal distribution. A thermometer is randomly selected and tested. Find the temperature reading corresponding to the given information. 45) Find P96, the 96th percentile. A) 1.75 B) 1.82 C) 1.03 D) -1.38 46) If 7% of the thermometers are rejected because they have readings that are too high, but all other thermometers are acceptable, find the temperature that separates the rejected thermometers from the others. F) 1.48 G) 1.39 H) 1.26 J) 1.45 Solve the problem. 47) Assume that z scores are normally distributed with a mean of 0 and a standard deviation of 1. If P(0 < z < a) = 0.4608, find a. A) 1.76 B) 0.1772 C) -0.10 D) 0.61 48) Assume that z scores are normally distributed with a mean of 0 and a standard deviation of 1. If P(z > c) = 0.1093, find c. F) 1.23 G) 0.4562 H) -1.23 J) 0.27 12
Assume that X has a normal distribution, and find the indicated probability. 49) The mean is µ = 60.0 and the standard deviation is = 4.0. Find the probability that X is less than 53.0. A) 0.0401 B) 0.9599 C) 0.0802 D) 0.5589 50) The mean is µ = 137.0 and the standard deviation is = 5.3. Find the probability that X is between 134.4 and 140.1. F) 0.4069 G) 0.6242 H) 1.0311 J) 0.8138 Solve the problem. 51) A bank's loan officer rates applicants for credit. The ratings are normally distributed with a mean of 200 and a standard deviation of 50. Find P60, the score which separates the lower 60% from the top 40%. A) 212.5 B) 187.5 C) 207.8 D) 211.3 Find the indicated probability. 52) The weekly salaries of teachers in one state are normally distributed with a mean of $490 and a standard deviation of $45. What is the probability that a randomly selected teacher earns more than $525 a week? F) 0.2177 G) 0.2823 H) 0.7823 J) 0.1003 13
53) The lengths of human pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days. What is the probability that a pregnancy lasts at least 300 days? A) 0.0166 B) 0.4834 C) 0.9834 D) 0.0179 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 54) A poll of 1400 randomly selected students in grades 6 through 8 was conducted and found that 30% enjoy playing sports. Is the 30% result a statistic or a parameter? Explain. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 55) The weights of the fish in a certain lake are normally distributed with a mean of 15 lb and a standard deviation of 6. If 4 fish are randomly selected, what is the probability that the mean weight will be between 12.6 and 18.6 lb? A) 0.6730 B) 0.0968 C) 0.3270 D) 0.4032 14
56) Assume that women's heights are normally distributed with a mean of 63.6 inches and a standard deviation of 2.5 inches. If 90 women are randomly selected, find the probability that they have a mean height between 62.9 inches and 64.0 inches. F) 0.9318 G) 0.0424 H) 0.1739 J) 0.7248 57) A study of the amount of time it takes a mechanic to rebuild the transmission for a 1992 Chevrolet Cavalier shows that the mean is 8.4 hours and the standard deviation is 1.8 hours. If 40 mechanics are randomly selected, find the probability that their mean rebuild time exceeds 8.7 hours. A) 0.1469 B) 0.1346 C) 0.1946 D) 0.1285 Use the continuity correction and describe the region of the normal curve that corresponds to the indicated binomial probability. 58) The probability of more than 56 correct answers F) The area to the right of 56.5 G) The area to the right of 56 H) The area to the left of 56.5 J) The area to the right of 55.5 59) The probability of at least 53 boys A) The area to the right of 52.5 B) The area to the right of 53 C) The area to the left of 52.5 D) The area to the right of 53.5 60) The probability of exactly 37 green marbles F) The area between 36.5 and 37.5 G) The area between 36.5 and 38.5 H) The area between 36.5 and 37 J) The area between 37 and 37.5 15
For the binomial distribution with the given values for n and p, state whether or not it is suitable to use the normal distribution as an approximation. 61) n = 18 and p =.2 A) Normal approximation is not suitable. B) Normal approximation is suitable. Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution. 62) With n = 20 and p = 0.60, estimate P(fewer than 8). F) 0.0202 G) 0.4953 H) 0.0668 J) 0.4332 63) Two percent of hair dryers produced in a certain plant are defective. Estimate the probability that of 10,000 randomly selected hair dryers, exactly 225 are defective. A) 0.0057 B) 0.0034 C) 0.0065 D) 0.0051 Use the normal distribution to approximate the desired probability. 64) Find the probability that in 200 tosses of a fair die, we will obtain at most 30 fives. F).2946 G).3229 H).1871 J).4936 16