I M the Quality Control Inspector for a Motel Chain

COB 191 Name______Statistical Methods Dr. Scott Stevens Exam I Spring 2003

DO NOT TURN TO THE NEXT PAGE UNTIL YOU ARE INSTRUCTED TO DO SO!

The following exam consists of 40 questions, each worth 2.5 points. You will have 75 minutes to complete the test. This means that you have, on average, about 1 minute, 50 seconds per question.

1. Record your answer to each question on the scantron sheet provided. You are welcome to write on this exam, but your scantron will record your graded answer.

2. Read carefully, and check your answers. Don’t let yourself write nonsense.

3. Keep your eyes on your own paper. If you believe that someone sitting near you is cheating, raise your hand and quietly inform me of this. I'll keep an eye peeled, and your anonymity will be respected.

4. If any question seems unclear or ambiguous to you, raise your hand, and I will attempt to clarify it.

5. Be sure your correctly record your student number on your scantron, and blacken in the corresponding digits. Failure to do so will cost you 10 points on this exam!

Pledge: On my honor as a JMU student, I pledge that I have neither given nor received unauthorized assistance on this examination.

Signature ______Questions 1-21 deal with the scenario below.

Aaron: I’m the quality control inspector for a motel chain. Part of my job is issuing one of three room cleanliness ratings to our franchises: excellent, satisfactory, or unsatisfactory. Our company guidelines state that a motel should receive an excellent rating if at least 95% of all rooms meet our cleanliness standards. If less than 80% of the rooms in the motel meet our cleanliness standards, the motel is to receive an “unsatisfactory” rating. The guidelines state that the remaining motels (those motels with between 80 and 95-% of all rooms in compliance with our standards) get a “satisfactory” rating. Let me tell you about my recent experiences at our Harrisonburg franchise.

I go to the Harrisonburg branch of our motel, and I inspect a simple random sample of 10 rooms from the motel’s 100 rooms. I rate each room either “compliant” or “not compliant” with our cleanliness standards. Based on what I find in these 10 rooms, I’ll draw my conclusions about the quality of this Harrisonburg motel. My inspection reveals that 5 of the 10 rooms inspected were noncompliant, so I issue an “unsatisfactory” rating to the motel.

1. What is the population for this scenario? 6. Aaron based his “unsatisfactory” rating a) the 5 rooms that failed the inspection on a statistic (the 50% mentioned in b) the 10 motel rooms inspected question 3), not the corresponding c) the 90 motel rooms not inspected parameter. This is probably because d) the 100 motel rooms in the motel a) conclusions based on statistics are more e) the motel rooms in all motels in the chain reliable than those based on parameters. b) finding the parameter would have taken 2. According to the company guidelines, the more time. cleanliness rating should be based upon c) it would be impossible to compute the a) a sample statistic corresponding parameter in this problem. b) a standard deviation d) in this case, Aaron can be sure that basing c) a population parameter his decision on the parameter’s value would d) a histogram have led to the same motel rating. e) a sample size of 10 e) All of the above (a-d) are true.

3. 50% of the rooms examined failed the inspection, 7. Suppose that Aaron had decided conduct his so Aaron gave the motel an “unsatisfactory” rating, inspection by inspecting one room chosen at even though 90 rooms remained uninspected. In random from each floor of the motel. This so doing, Aaron was applying would have been an example of a) a census analysis. a) cluster sampling b) descriptive statistics. b) progressive sampling c) deductive reasoning. c) simple random sampling d) inferential statistics. d) stratified sampling e) the standard given in the guidelines. e) systematic sampling

4. Each motel visited receives a rating of 7. Suppose that two of the ten rooms that Aaron “excellent”, “satisfactory”, or “unsatisfactory”. had randomly chosen to inspect were occupied The level of this measurement is best described as by guests, and hence unavailable. This a) continuous complication might introduce which type of error b) nominal into his inspection results? c) ratio a) coverage (frame) d) nonresponse d) ordinal b) sampling e) measurement e) interval c) adjustment 5. Questions 8-13 continue with the motel scenario, and focus on some particular measures that are used to determine if a room is or is not compliant with the company standards. The specifics of these measures are detailed in the table below.

Measure How measure is defined Fire extinguisher The installation year of the extinguisher, recorded from the arbitrary reference year of 1992. Thus, an extinguisher installed in 1995 receives an extinguisher rating of 3. Burns The number of cigarette burns on tables in the room Décor American, European, or Other Air quality 1 = foul, 2 = unpleasant, 3 = good, 4 = exceptional Windows 1 = windows dirty but intact, 2 = windows broken, 3 = acceptable, 4 = no windows Carpet Percentage of the carpet that is threadbare or stained

8. The level of measurement for the fire extinguisher variable is best characterized as

a) interval b) nominal c) ordinal d) ratio e) constant

9. The burns measure would be an example of a numeric variable that is

a) correlative b) continuous c) disjoint d) discrete e) randomized

10. After Aaron’s inspection, the value of the sample mean would be computable and meaningful for which of these measures?

a) décor b) air quality c) windows d) carpet e) all of these

11. Which of the measures above would most sensibly be considered a continuous variable?

a) burns b) décor c) air quality d) windows e) carpet

12. Which of the following explains why the window measure would generally be treated as a categorical (qualitative), rather than a numeric (quantitative) variable?

a) The windows measure can only take on four different values. b) The windows measure is only at the nominal level of measurement. c) Each room can be assigned only one windows measure. d) The windows measure deals with objective information. e) The windows measure deals with subjective information.

13. Suppose that 20% of all of the motel’s rooms have no windows and that 50% of the motel’s rooms have foul air. Since 0.2  0.5 = 0.1, one might conclude from this information that 10% of all of the motel’s rooms are windowless and have foul air. This conclusion

a) assumes that windows and foul air are dependent, and so is probably right. b) assumes that windows and foul air are independent, and so is probably right. c) assumes that windows and foul air are dependent, and so is probably wrong. d) assumes that windows and foul air are independent, and so is probably wrong. e) is completely inappropriate, since the correct answer is 0.2 + 0.5 = 0.7. Questions 14-21 continue with the motel scenario, and deal with the situation below.

The Harrisonburg motel protests Aaron’s evaluation, claiming that he inspected only 10 on the 100 rooms of the motel, and hence his conclusion that the motel deserved an unsatisfactory rating was premature. The company grudgingly agrees to have Aaron inspect all 100 rooms of the motel. The results of this 100 room inspection are presented below in a variety of formats. The “room rating” is the overall score that a room receives upon inspection; the higher the rating, the more problems with the room. Values shown on the horizontal axis of the percentage polygon and Pareto diagram are for the midpoints of the class intervals. Percentage Polygon for Extinguisher Code Décor Versus Compliance (Harrisonburg) 35% Compliant? 30% Décor No Yes Total 25% American 21 35 56 20% European 20 12 32 Other 1 11 12 15% Total 42 58 100 10% 5% % of Carpet Worn or Stained in Room 0% (Harrisonburg) --- 1 4 7 10 ---

25 100.00%

20 80.00% y c 15 60.00% n e u q 10 40.00% e r F 5 20.00%

0 .00% --- 5 15 25 35 45 55 65 75 85 95 Percentage

Air Quality of Room 0 10 20 30 40 50 Room Ratings in Harrisonburg Motel 9 24 14. 14. What fraction of rooms with American Foul décor were compliant with the company 25 Unpleasant guidelines? Good Exceptional a) 35% b) 56% c) 58% d) 60.3% e) 62.5%

42 15. About what fraction of rooms in the motel received room ratings of 20 or more?

a) 25% b) 40% c) 50% d) 60% e) 75%

16. The numbers on the air quality pie chart represent the number of the rooms given each rating. If we wished instead to create a relative frequency pie chart for air quality, the value of 42 in the chart above would have been replaced with a) 21% b) 23.8% c) 39.6% d) 42% e) 72.8%

17. What fraction of the motel’s rooms are done in European décor? a) 12% b) 20% c) 32% d) 37.5% e) 62.5%

18. What fraction of the motel’s rooms have less than 20% of their carpet stained or worn? a) about 18% b) about 21% c) about 34% d) about 55% e) about 75%

19. The first dot above the x-axis on the frequency polygon tells us that about 20% of all rooms have fire extinguishers whose codes are a) 1 b) 0 or 1 c) 1 or 2 d) 0, 1, or 2 e) 1, 2, or 3

20. If you computed the IQR of the room ratings at the Harrisonburg Motel, your result would lie between a) 0 and 4 b) 5 and 9 c) 10 and 20 d) 21 and 40 e) 41 and 60

21. What fraction of noncompliant rooms have European décor? a) 12/20 b) 20/22 c) 20/32 d) 20/42 e) 32/42

End of Harrisonburg Motel Scenario

22. Which of the following statistical measures could be sensibly be computed on a set of ORDINAL data?

a) mean b) standard deviation c) range d) all of the these (a, b, and c) e) none of these

23. What is the only way to avoid sampling error when conducting a survey? a) Be sure that your sample is completely random. b) Be sure that the group from which you draw your sample is the same group as the one about which you wish to draw conclusions. c) Be sure that each person whom you select for your sample submits a completed survey. d) Be sure that no one gives false information when filling out the survey. e) Survey every single member of the population.

24. Find the mean of these five numbers: 31, 41, 58, 28, 52. The answer is a) 41 b) 42 c) 51.5 d) 52 e) 58

25. Find the median of these five numbers: 31, 41, 58, 28, 52. The answer is a) 41 b) 42 c) 51.5 d) 52 e) 58

26. Find the median of these eight numbers: 1, 7, 8, 20, 9, 8, 0, 4. The answer is a) 7.125 b) 7.5 c) 8 d) 14.5 e) does not exist

27. Find the mode of these eight numbers: 1, 7, 8, 20, 9, 8, 0, 4. The answer is a) 7.125 b) 7.5 c) 8 d) 14.5 e) does not exist 28. Find the standard deviation of this 7 number population: 1, 2, 2, 3, 3, 5, 5. Give your answer to two decimal places (Hint: the mean of this population is 3.) a) 1.07 b) 1.14 c) 1.41 d) 2.00 e) 2.33

29. Suppose that the 7 number population in question #28 were actually a sample from a larger population. Then the variance of this sample would be a) 1.07 b) 1.14 c) 1.41 d) 2.00 e) 2.33

Questions 30-35 deal with the scenario below.

Anne and Bob have played 100 games of poker. In each game, one of the two players has won some money from the other. The amount won is always a whole number of dollars, and never exceeds $5. Losses are recorded as negative winnings, so if Anne wins $3 from Bob in a game, then Anne’s winnings for the game are 3 and Bob’s winnings are –3. The record of winnings for each player in all 100 games is shown below. As you can see, Anne won $3 (and hence Bob lost $3) in 20 of the 100 games.

Anne's Winnings Bob's Winnings

30 30 y y 25 25 c c n n 20 20 e e u u 15 15 q q e

e 10 10 r r F F 5 5 0 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 Dollars Dollars

30. 30. Which of the following terms would be most appropriate to apply to the distribution of Anne’s winnings?

a) skewed b) symmetric c) positive d) monotonic e) biased

31. Viewing the 100 games as sample, compare the standard deviation of Anne’s winnings with the standard deviation of Bob’s winnings. Which of the following statements is true?

a) Anne’s standard deviation is greater than Bob’s. Both standard deviations are positive. b) Anne’s standard deviation is equal to Bob’s. Both standard deviations are positive. c) Anne’s standard deviation is less than Bob’s. Both standard deviations are positive. d) Anne’s standard deviation is greater than Bob’s. Bob’s standard deviation is negative. e) Anne’s standard deviation is less than Bob’s. Anne’s standard deviation is negative.

32. We have defined the term range in statistics as a measure of dispersion. What is the range of Anne’s winnings?

a) the standard deviation b) -3 to 5 c) 1 to 27 d) 8 e) none of the above 33. Suppose we use the results of the 100 games of poker that Anne and Bob have played to assign probabilities for the outcomes of future games. Thus, we would say that the probability that Anne wins $3 in a game of poker with Bob is 0.2. Anne and Bob are now going to play another game of poker. From the graph, estimate P(Bob wins exactly $1 in the game | Bob wins). The answer is between

a) 0 and 0.05 b) 0.05 and 0.10 c) 0.1 and 0.2 d) 0.2 and 0.5 e) 0.5 and 10.

34. We have said that the probability that Anne wins $3 is a game of poker with Bob is 0.2. The probability that Anne wins $5 in a game of poker with Bob is 0.12. With these facts in mind, and making no other assumptions than those given in the scenario, which of the following statements is NOT necessarily true?

a) The probability that Bob loses $3 in a game of poker with Anne is 0.2. b) If Anne and Bob play two games in a row, there is a 0.04 probability of Anne winning exactly $3 in each of the two games. c) There is a 32% chance that Anne will win either $3 or $5 when playing a game of poker with Bob. d) If Anne and Bob play a single game of poker, P(Anne wins exactly $5 | Anne wins exactly $3) is 0. e) Anne winning the game and Bob winning the game are mutually exclusive events.

Questions 35-38 are based on the information below.

A 1988 Gallup poll indicated that 50% of Americans believe in angels. A 1996 Newsweek poll indicated that 48% of Americans believe that alien UFO visitations have occurred. Take these figures as accurate, and further assume that 40% of Americans believe both in angels and in alien UFO visitations.

35. The probability that a randomly selected American believes in angels or in alien UFO visitations, or in both is

a) 24% b) 48% c) 58% d) 80% e) 98%

36. If an American believes in angels, how likely is it that he or she also believes in alien UFO visitations?

a) 40% b) 60% c) 80% d) 90% e) 98%

37. The information given shows that, for Americans, the events of believing in angels and believing in alien UFO visitations are

a) independent b) dependent c) mutually exclusive d) complementary e) biased

38. Choose two Americans at random. How likely is it that the first person chosen believe in angels, and that the second person believes in UFO visitations?

a) 24% b) 40% c) 48% d) 90% e) 98%

39. The symbol used to represent the standard deviation of a population is

a) s b) s2 c)  d) 2 e) 2

40. For a normal (bell shaped) population distribution, about what fraction of all population values lie within one standard deviation of the mean? a) ½ b) 2/3 c) ¾ d) 4/5 e) 5/6 Answers:

1. d 2. c 3. d 4. d 5. b 6. d 7. d 8. a 9. d 10. d 11. e 12. b 13. d 14. e 15. a 16. d 17. c 18. c 19. d 20. b 21. d 22. e 23. e 24. b 25. a 26. b 27. c 28. c 29. e 30. a 31. b 32. d 33. e 34. b 35. c 36. c 37. b 38. a 39. c 40. b