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Psychology 281 Christmas exam review questions 1. Eight male and eight female rats are going to be assigned to samples of size 2, 6, and 8 to take part in different research experiments. a) In how many ways can this be accomplished if assignment to the samples is done at random? b) In how many ways can the assignment to the samples be accomplished if the sample of size 6 can only include female rats? c) If assignment to the samples is random, what is the probability that each sample will end up being half male and half female? 2. A large grocery chain randomly samples 150 customers and finds that 60 of these have coupons that they use when purchasing specific items. a) Form a 95% confidence interval to estimate the proportion of people in the general population of which these customers are representative who would be expected to use coupons. b) In an earlier study the grocery chain found that soft-sell advertising led to 65% of all of its customers using coupons the week after the advertising campaign was conducted. Is the proportion of coupon-using customers in the present sample of 150 significantly less than this? (α .01). 3. A company that manufactures flashlight batteries is concerned that the mean length of life of its batteries may be significantly less than 100 hours. To see if this is the case, the company takes a random sample of 25 batteries and finds that their mean length of life is 97 hours, with a standard deviation of 8.8 hours. Test the research (alternative) hypothesis that the population mean is less than the desired value of 100 hours (α .05). 4. Calculate the mean and standard deviation of X: the number of spades chosen if two cards are selected without replacement from a standard deck of 52 cards. 5. A growing manufacturing company has kept careful records of applicants who have applied for jobs on their assembly lines. The probability that a new applicant will be called in for a first interview is .30 while the probability that he or she will be rejected outright is .45. Otherwise, the applicant is kept "on file". Of those applicants interviewed, some are rejected immediately but the probability is .50 of being hired immediately and the probability is one in three of being called in for a second interview. Of the people interviewed a second time, half are hired and half are rejected. Of the people "on file", 80% are ultimately interviewed but the probability of one of those interviewees being hired is only .25. a) What is the probability that an applicant for a job at the company's assembly plant will be hired? b) What is the probability that someone who is put "on file" is rejected? c) What is the probability that someone who is interviewed is hired? 6. The technical support group for the 1992 Winter games has ten electronic timers that can be used for five events. The events are run concurrently at different sites. Three of the events each require only one timer, but the other two each require two timers. One of the ten timers has a malfunction that makes it inaccurate. a) The timers are labeled with the numbers one through ten. How many possible numbering arrangements are there in which the malfunctioning timer is "lucky number seven"? b) How many ways can the available timers be distributed to satisfy the requirements of the five events? c) What is the probability that timing is inaccurate in one of the events requiring two timers? 7. A key ring contains 4 keys that appear to be identical, although only one will open a particular door. Describe the probability distribution of X: the number of keys that must be tried before the door is unlocked, assuming that once a key has been tried it is not used again. 8. Historically, the mean grade (μ) for Psychology 281 has been 71.0 with a variance of 196.0. This year, one section of Psychology 281 is employing a new instructional method that is hypothesized to increase understanding and test performance. a) In order to be convinced that the new method is actually effective, you arbitrarily decide that the mean grade achieved by a random sample of 49 students taught by the new method must be 74.0 or higher. If, in fact, the new method increases μ by 6.0 points, what is the probability that you remain unconvinced? b) Because of time limitations you are only able to take the mean grade of 36 students taught by the new method. Assume that you continue to require a mean grade of 74.0 or higher to be convinced of the method's effectiveness. What is the probability that you falsely conclude the method worked if, in fact, it is no different from any other method? 9. Students who read their statistics text have a .98 probability of passing the course. Those who don't read their textbook have a .85 probability of failing. If 70% of the students do the readings, then: a) What is the probability that a student will pass the course? b) If a student were to pass, what would the probability have been that he or she did the readings? 10. The makers of HOBNOB coffee have purchased a new machine to fill their cans with coffee, and are concerned that it may deposit a different volume of coffee per can than the old machine. The old machine deposited an average of 100 ml of coffee per can. The standard deviation of the volume deposited per can is the same for all machines of this kind. Sixty-four cans filled by the new machine are drawn at random from its production. The volume per can is found to have a mean of 99.55 ml and a variance of 4 ml2. a) Do the makers of HOBNOB coffee have cause to be concerned? (Use alpha .05) b) Find the 95% confidence interval for the true volume of coffee deposited per can by the new machine. 11. In a certain sampling distribution of the mean based on sample sizes of 44, a sample mean of 438 is known to correspond to a Z score of -1.80. If the standard error of the mean is 8, what are the population mean and variance? 12. A student wants to calculate the variance of a set of 10 scores. Unfortunately, he doesn't have the raw scores but only has the deviation of each raw score from the mean. Worse yet, he only has 9 of these deviation scores, as listed below. Compute the variance for him. -5, +11, -4, -2, +7, -8, -6, +1, -3 13. What is the probability that in 120 tosses of a fair coin between 40% and 60% (inclusively) will be heads? 14. The actual amount of coffee which an automatic filling machine puts into "4 oz" jars of coffee varies slightly from jar to jar, and may be looked upon as a continuous random variable having a normal distribution with σ = .04. If only 2% of the jars contain less than 4 oz. of coffee, what must the mean fill of the jars be? 15. Assume that the heights of male students at UWO are normally distributed with a mean of 68.0 inches and a variance of 9 inches. If 80 samples of 25 students each are randomly taken, in how many would you expect to find the mean to be between 66.8 and 68.3 inches? 16. Santa Claus believes that some of the elves are trying to sabotage Christmas by deliberately putting fewer candies in the bags he leaves in children's stockings (the elves are angry because they haven't been allowed to take a vacation on Xmas Eve for 500 years). Typically, the number of candies per bag has been normally distributed with a mean of 15 and a standard deviation of 3.5. Santa decides that he can easily test his belief by taking a random sample of bags and seeing how many candies they contain. a) Suppose that Santa takes a sample of 20 bags of candy and decides that he will conclude that his belief is correct if no more than 25% of these bags contain 15 or more candies. What is the probability that he will draw this conclusion even if his belief is in fact incorrect and the distribution of the number of candies in the bags he sampled from is the same as it has always been? b) Suppose that, in fact, the mean number of candies in the bags Santa samples from is 13. What is the probability that Santa will correctly conclude that his belief is correct if he takes a sample of 49 bags and sets α .05? 17. Symptom X occurs 80%, 85%, and 70% of the time with diseases A, B, and C, respectively, which themselves occur with probability .02, .008, and .03. a) What is the probability of symptom X occurring? b) If symptom X occurred, what would the probability be that the patient has disease A? 18. A census shows that 75% of homes in London have color television sets. If an advertising agency takes a random sample of homes: a) What is the probability that exactly 4 of 7 homes surveyed have color televisions? b) What is the probability that between 330 and 350 (inclusively) out of 450 homes surveyed have color televisions? 19. The probability that a lie detector wrongly judges a person to be lying is .10, and the probability that it wrongly judges a person to be telling the truth is .30. a) What is the probability that exactly 2 of 12 people telling the truth are judged to be lying? b) What is the probability that 10 or more of 30 people lying are judged to be telling the truth? 20. The Night Owl Disco notes that 90% of the people making dinner reservations show up for dinner. (Assume one person per reservation.) a) If 4 people make reservations for dinner, find the probability that at least 3 people will fail to show. b) If 25 people make reservations for dinner, find the probability that more than 20 people show. c) If the Night Owl receives 100 reservations for dinner, at most how many people can it count on showing for dinner with 95% certainty? 21. An airline advertises that its flights from Toronto to Calgary are faster than any of its competitors. In fact, the airline claims that it can do this trip in 3.5 hours, on average, and that the distribution of its flight times is normally distributed. As an employee of a rival airline, you have been given the task of assessing whether the first airline's claim is true (you believe that their mean flight time is more than 3.5 hours). As a first test, you plan to record the actual flight times of 15 of their planes and decide that if 9 or more of these take more than 3.5 hours to make the trip, you will conclude that the airline's claim is untrue. a) Suppose the airline's claim is true. What is the probability that you will nonetheless conclude that their claim is untrue? b) As a second test, you record the flight times of a random sample of 30 of the airline's planes and plan to use α .05 to test your hypotheses. Suppose the airline's true mean flight time is 3.8 hours (with a standard deviation of .80 hours). What is the probability that you will correctly conclude that the airline's claim is untrue? 22. The weights of young rats used in Psychology experiments are normally distributed in the population, with a mean of 310 grams and a 95th percentile of 326.45 grams. a) What is the probability that any randomly selected rat from the population will weigh between 285 and 298 grams? b) A researcher obtains 160 young male rats whose weights are normally distributed with a median of 315 grams. 140 of these 160 male rats are heavier than the average rat in the population. Given this, how many of the 160 male rats are heavier than 325 grams? c) The researcher also obtains 150 young female rats. The weights of these female rats are normally distributed with a variance of 42.25, but only 1.5% of the female rats are heavier than the average young male rat in the sample that the researcher obtained. What percentage of the female rats weigh less than 312 grams? 23. The theory that categorically related concepts (e.g., bird, robin) are directly linked in memory predicts that reading one word will produce a speedup in the time required to read a related second word. A psychologist investigating this theory estimates from previous research that the variance for speedup in her experimental sample will be 420 milliseconds. a) Given a sample of 32 observations and α .05, what is the probability of failing to reject the null hypothesis of no speedup if, in fact, the true population mean for speedup is 10 milliseconds? b) 24. In a second experiment she wants the probability of a Type I error to be .05, and the probability of a Type II error to be .10. If speedup has a true mean of exactly 10 milliseconds, how many subjects should she test? The probability distribution for X, the number of eggs in a bald eagle's nest, is given below. These values are based on long-term historical records of the reproductive practices of bald eagles. X 0 1 2 a) b) c) 25. P(X) .25 .62 .13 What is the probability that the mean number of eggs in a random sample of three nests is 1.00? Biologists plan to randomly sample 38 nests to determine whether environmental changes have harmed the reproductive capacities of bald eagles. Given the historical probability distribution for X, what are the mean and standard deviation of the sampling distribution of the mean number of eggs per nest based on samples of 38 nests? What is the probability that two randomly chosen samples of 38 nests both contain more than 40 eggs? Twenty percent of all beer sold in Canada is imported, 30% is a Labatt's product, 25% is a Molson's product, and the remainder is made by "other" Canadian manufacturers. A person's beer preference is independent of that of others. a) What is the probability that of 12 customers who order a single beer each, exactly half order imported beer? b) Of 20 customers who each order a single beer, what is the probability that at least 75% order anything but a Labatt's product? c) Two hundred customers visit Joe's bar on each of two nights and each customer buys a single beer. What is the probability that on both nights fewer Molson's beers are sold than the expected number of imported beers sold on days when 200 customers buy a single beer each? Answers 1. a) 360360 b) 1260 c) .218 2. a) .3216 to .4784 b) Z = -6.41, reject Ho 3. t = -1.705, Do not reject Ho 4. μ = .50, σ = .606 5. a) .25 b) .80 c) .50 6. a) 362880 b) 151200 c) .40 7. μ = 2.5, σ = 1.118 8. a) .0668 b) .0985 9. a) .731 b) .938 10. a) Z = -1.80, Do not reject Ho b) 99.06 to 100.04 11. μ = 452.40, σ2 = 2816 12. S2 = 45.11 13. .9774 14. μ = 4.082 15. 54 16. a) .021 b) .9909 17. a) .0438 b) .365 18. a) .173 b) .73 19. a) .230 b) .4207 20. a) .0037 b) .902 c) 95 21. a) .304 b) .6591 22. a) .1089 b) 1.7 or about 2 c) 95.64% 23. a) .1314 b) 36 24. a) .3592 b) μx = .88, σx = .098 c) .0015 25. a) .0155 b) .416 c) .0018