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Test Information Guide: College-Level Examination ProgramÂŽ 2011-12 College Mathematics

ÂŠ 2011 The College Board. All rights reserved. College Board, College-Level Examination Program, CLEP, and the acorn logo are registered trademarks of the College Board.

CLEP TEST INFORMATION GUIDE FOR COLLEGE MATHEMATICS

worldwide through computer-based testing programs and also — in forward-deployed areas — through paper-based testing. Approximately one-third of all CLEP candidates are military service members.

History of CLEP

2010-11 National CLEP Candidates by Age*

Since 1967, the College-Level Examination Program (CLEP®) has provided over six million people with the opportunity to reach their educational goals. CLEP participants have received college credit for knowledge and expertise they have gained through prior course work, independent study or work and life experience.

Under 18 9% 30 years and older 30%

18-22 years 39%

23-29 years 22%

Over the years, the CLEP examinations have evolved to keep pace with changing curricula and pedagogy. Typically, the examinations represent material taught in introductory college-level courses from all areas of the college curriculum. Students may choose from 33 different subject areas in which to demonstrate their mastery of college-level material.

* These data are based on 100% of CLEP test-takers who responded to this survey question during their examinations.

2010-11 National CLEP Candidates by Gender

Today, more than 2,900 colleges and universities recognize and grant credit for CLEP.

41%

Philosophy of CLEP Promoting access to higher education is CLEP’s foundation. CLEP offers students an opportunity to demonstrate and receive validation of their college-level skills and knowledge. Students who achieve an appropriate score on a CLEP exam can enrich their college experience with higher-level courses in their major field of study, expand their horizons by taking a wider array of electives and avoid repetition of material that they already know.

58%

Computer-Based CLEP Testing The computer-based format of CLEP exams allows for a number of key features. These include: • a variety of question formats that ensure effective assessment • real-time score reporting that gives students and colleges the ability to make immediate creditgranting decisions (except College Composition, which requires faculty scoring of essays twice a month) • a uniform recommended credit-granting score of 50 for all exams • “rights-only” scoring, which awards one point per correct answer • pretest questions that are not scored but provide current candidate population data and allow for rapid expansion of question pools

CLEP Participants CLEP’s test-taking population includes people of all ages and walks of life. Traditional 18- to 22-year-old students, adults just entering or returning to school, homeschoolers and international students who need to quantify their knowledge have all been assisted by CLEP in earning their college degrees. Currently, 58 percent of CLEP’s test-takers are women and 52 percent are 23 years of age or older. For over 30 years, the College Board has worked to provide government-funded credit-by-exam opportunities to the military through CLEP. Military service members are fully funded for their CLEP exam fees. Exams are administered at military installations

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CLEP Exam Development

The Committee

Content development for each of the CLEP exams is directed by a test development committee. Each committee is composed of faculty from a wide variety of institutions who are currently teaching the relevant college undergraduate courses. The committee members establish the test specifications based on feedback from a national curriculum survey; recommend credit-granting scores and standards; develop and select test questions; review statistical data and prepare descriptive material for use by faculty (Test Information Guides) and students planning to take the tests (CLEP Official Study Guide).

The College Board appoints standing committees of college faculty for each test title in the CLEP battery. Committee members usually serve a term of up to four years. Each committee works with content specialists at Educational Testing Service to establish test specifications and develop the tests. Listed below are the current committee members and their institutional affiliations.

College faculty also participate in CLEP in other ways: they convene periodically as part of standard-setting panels to determine the recommended level of student competency for the granting of college credit; they are called upon to write exam questions and to review forms and they help to ensure the continuing relevance of the CLEP examinations through the curriculum surveys.

Frank Bauerle, Chair

University of California — Santa Cruz

Tuncay Aktosun

University of Texas at Arlington

Helen Burn

Highline Community College, Seattle

The primary objective of the committee is to produce tests with good content validity. CLEP tests must be rigorous and relevant to the discipline and the appropriate courses. While the consensus of the committee members is that this test has high content validity for a typical introductory College Mathematics course or curriculum, the validity of the content for a specific course or curriculum is best determined locally through careful review and comparison of test content, with instructional content covered in a particular course or curriculum.

The Curriculum Survey The first step in the construction of a CLEP exam is a curriculum survey. Its main purpose is to obtain information needed to develop test-content specifications that reflect the current college curriculum and to recognize anticipated changes in the field. The surveys of college faculty are conducted in each subject every three to five years depending on the discipline. Specifically, the survey gathers information on: • the major content and skill areas covered in the equivalent course and the proportion of the course devoted to each area • specific topics taught and the emphasis given to each topic • specific skills students are expected to acquire and the relative emphasis given to them • recent and anticipated changes in course content, skills and topics • the primary textbooks and supplementary learning resources used • titles and lengths of college courses that correspond to the CLEP exam

The Committee Meeting The exam is developed from a pool of questions written by committee members and outside question writers. All questions that will be scored on a CLEP exam have been pretested; those that pass a rigorous statistical analysis for content relevance, difficulty, fairness and correlation with assessment criteria are added to the pool. These questions are compiled by test development specialists according to the test specifications, and are presented to all the committee members for a final review. Before convening at a two- or three-day committee meeting, the members have a chance to review the test specifications and the pool of questions available for possible inclusion in the exam.

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At the meeting, the committee determines whether the questions are appropriate for the test and, if not, whether they need to be reworked and pretested again to ensure that they are accurate and unambiguous. Finally, draft forms of the exam are reviewed to ensure comparable levels of difficulty and content specifications on the various test forms. The committee is also responsible for writing and developing pretest questions. These questions are administered to candidates who take the examination and provide valuable statistical feedback on student performance under operational conditions.

developing, administering and scoring the exams. Effective July 2001, ACE recommended a uniform credit-granting score of 50 across all subjects, with the exception of four-semester language exams, which represents the performance of students who earn a grade of C in the corresponding college course. The American Council on Education, the major coordinating body for all the nation’s higher education institutions, seeks to provide leadership and a unifying voice on key higher education issues and to influence public policy through advocacy, research and program initiatives. For more information, visit the ACE CREDIT website at www.acenet.edu/acecredit.

Once the questions are developed and pretested, tests are assembled in one of two ways. In some cases, test forms are assembled in their entirety. These forms are of comparable difficulty and are therefore interchangeable. More commonly, questions are assembled into smaller, content-specific units called testlets, which can then be combined in different ways to create multiple test forms. This method allows many different forms to be assembled from a pool of questions.

CLEP Credit Granting CLEP uses a common recommended credit-granting score of 50 for all CLEP exams. This common credit-granting score does not mean, however, that the standards for all CLEP exams are the same. When a new or revised version of a test is introduced, the program conducts a standard setting to determine the recommended credit-granting score (“cut score”).

Test Specifications Test content specifications are determined primarily through the curriculum survey, the expertise of the committee and test development specialists, the recommendations of appropriate councils and conferences, textbook reviews and other appropriate sources of information. Content specifications take into account: • the purpose of the test • the intended test-taker population • the titles and descriptions of courses the test is designed to reflect • the specific subject matter and abilities to be tested • the length of the test, types of questions and instructions to be used

A standard-setting panel, consisting of 15–20 faculty members from colleges and universities across the country who are currently teaching the course, is appointed to give its expert judgment on the level of student performance that would be necessary to receive college credit in the course. The panel reviews the test and test specifications and defines the capabilities of the typical A student, as well as those of the typical B, C and D students.* Expected individual student performance is rated by each panelist on each question. The combined average of the ratings is used to determine a recommended number of examination questions that must be answered correctly to mirror classroom performance of typical B and C students in the related course. The panel’s findings are given to members of the test development committee who, with the help of Educational Testing Service and College Board psychometric specialists, make a final determination on which raw scores are equivalent to B and C levels of performance.

Recommendation of the American Council on Education (ACE) The American Council on Education’s College Credit Recommendation Service (ACE CREDIT) has evaluated CLEP processes and procedures for

*Student performance for the language exams (French, German and Spanish) is defined only at the B and C levels.

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College Mathematics Description of the Examination

10%

Logic Truth tables The College Mathematics examination covers Conjunctions, disjunctions, implications material generally taught in a college course for and negations nonmathematics majors and majors in fields not Conditional statements requiring knowledge of advanced mathematics. Necessary and sufficient conditions The examination contains approximately Converse, inverse and contrapositive 60 questions to be answered in 90 minutes. Hypotheses, conclusions and Some of these are pretest questions that will not counterexamples be scored. Any time candidates spend on tutorials 20% Real Number System and providing personal information is in addition Prime and composite numbers to the actual testing time. Odd and even numbers The examination places little emphasis on arithmetic Factors and divisibility calculations, and it does not contain any questions Rational and irrational numbers that require the use of a calculator. However, an Absolute value and order online scientific calculator (nongraphing) is available Open and closed intervals to candidates during the examination as part of the 20% Functions and Their Graphs testing software. Properties and graphs of functions It is assumed that candidates are familiar with Domain and range currently taught mathematics vocabulary, symbols Composition of functions and inverse and notation. functions Simple transformations of functions: Knowledge and Skills Required translations, reflections, symmetry Questions on the College Mathematics examination 25% Probability and Statistics require candidates to demonstrate the following Counting problems, including permutations abilities in the approximate proportions indicated. and combinations Computation of probabilities of simple and â€˘ Solving routine, straightforward problems compound events (about 50 percent of the examination) Simple conditional probability â€˘ Solving nonroutine problems requiring an Mean, median, mode and range understanding of concepts and the application Concept of standard deviation of skills and concepts (about 50 percent of Data interpretation and representation: tables, the examination) bar graphs, line graphs, circle graphs, pie The subject matter of the College Mathematics charts, scatterplots, histograms examination is drawn from the following topics. The percentages next to the main topics indicate the 15% Additional Topics from Algebra and Geometry approximate percentage of exam questions on that Complex numbers topic. Logarithms and exponents 10% Sets Applications from algebra and geometry Union and intersection Perimeter and area of plane figures Subsets, disjoint sets, equivalent sets Properties of triangles, circles and rectangles Venn diagrams The Pythagorean theorem Cartesian product Parallel and perpendicular lines Algebraic equations, systems of linear equations and inequalities Fundamental Theorem of Algebra, Remainder Theorem, Factor Theorem

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M A T H E M A T I C S

The following sample questions do not appear on an actual CLEP examination. They are intended to give potential test-takers an indication of the format and difficulty level of the examination and to provide content for practice and review. Knowing the correct answers to all of the sample questions is not a guarantee of satisfactory performance on the exam. Directions: An online scientific calculator will be available for the questions in this test. Some questions will require you to select from among four choices. For these questions, select the BEST of the choices given. Some questions will require you to type a numerical answer in the box provided. Some questions refer to a table in which statements appear in the first column. For each statement, select the correct properties by check-marking the appropriate cell(s) in the table.

3. Triangle DEF (not shown) is similar to 䉭 ABC shown, and the length of side DE is 6 cm. If the area of 䉭 ABC is 5 square centimeters, what is the area of 䉭DEF ? (A) (B) (C) (D)

10 cm2 12 cm2 18 cm2 45 cm2

4. m is an odd integer. For each of the following numbers, indicate whether the number is odd or even. Number 2m – 1 2m + 1 m2 – m m2 + m + 1

Odd

Even

Click on your choices.

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M A T H E M A T I C S 3x − 4 ≥ 0 10. Which of the following subsets of the real numbers best describes the solution set of the inequality above?

⎡ 4⎞ (A) ⎢0, ⎟ ⎣ 3⎠ ⎡4 ⎞ (B) ⎢ , ∞ ⎟ ⎣3 ⎠ (C) (D) 11.

8. x is the standard deviation of the set of numbers {a , b, c, d, e}. For each of the following sets, indicate which sets must have a standard deviation equal to x . Set

Must Have Standard Deviation Equal to x

{a + 2, b + 2, c + 2, d + 2, e + 2} {a − 2, b − 2, c − 2, d − 2, e − 2} {2a, 2b, 2c, 2d, 2e} ⎧a b c d e ⎫ ⎨ , , , , ⎬ ⎩2 2 2 2 2⎭

12.

Click on your choices.

13.

7

( −∞, ∞ ) ( −∞, −4] ∪ [3, ∞ )

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M A T H E M A T I C S

14. On an exam for a class with 32 students, the mean score was 67.2 points. The instructor rescored the exam by adding 8 points to the exam score for every student. What is the mean of the scores on the rescored exam?

18.

15. In the truth table below, T and F are used to indicate that statements are true and false, respectively. In the fourth column, click on each box for which the statement is true. (Note: p means not p) p

q

p

T T F F

T F T F

F F T T

19. A scientist estimated the number of bacteria in a sample every hour and recorded the estimates in the table above. Then the scientist used the data to create the scatterplot above. Based on the information, which of the following equations best models the number of bacteria, f ( t ), at time t, in hours?

pÂťq

(A) (B) (C) (D)

Click on your choices.

16.

f ( t ) = 100 ( 2t ) f ( t ) = 100 + 2t f ( t ) = t 2 + 220t + 100 f ( t ) = 120 log ( t + 1)

20. Which of the following integers is a prime number? (A) (B) (C) (D) 17. The faces of a fair cube are numbered 1 through 6; the probability of rolling any number from 1 through 6 is equally likely. If the cube is rolled twice, what is the probability that an even number will appear on the top face in the first roll or that the number 1 will appear on the top face in the second roll?

21.

8

104 105 109 111

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M A T H E M A T I C S

25. The graph above shows the closing price of one share of stock of Company Y for each of the five business days last week. Which of the following is closest to the percent change in the closing price of one share of stock from Tuesday to Wednesday?

22. In country I when a house is sold to a first-time buyer, a purchase tax is paid based on the tax rate table above. Last week Liam, a first-time buyer, purchased a house for $275,000. How much purchase tax did he pay?

(A) 40% (B) 55% (C) 67% (D) 150%

(A) $8,250 (B) $11,000 (C) $12,375 (D) $13,750 23.

m

4 and n

26. Let A be a nonempty set and let B and C be any two subsets of A. Which of the following statements must be true?

2

For each of the following expressions, indicate whether the value will be a rational or irrational number.

Expression 3

Rational

(A) (B) (C) (D)

Irrational

mn

27. The area of a rectangular field is the product of its length and width. If each dimension of a certain field is multiplied by 3, then the area of the new field is how many times the area of the original field?

m3 3

B âˆŞC = A B âˆŠ C = { } , the empty set BâŠ†CâŠ† A B âˆŞC âŠ† A

m n

m n2

Click on your choices. 24.

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28.

M A T H E M A T I C S 32. The results of a survey of 200 college students showed that some students who were business majors were women and all students who were business majors took calculus. Which of the following is a valid conclusion from the survey? (A) All students who were women took calculus. (B) Some students who were women took calculus. (C) Some students who were women did not take calculus. (D) Some students who were women were not business majors.

29.

33.

30. 34.

31. 35.

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M A T H E M A T I C S

36.

39.

37.

40.

38.

41.

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M A T H E M A T I C S

42.

47.

, where a, b, and c are distinct numbers. 43.

48. 44. Each number in data set A is increased by 3 to form data set B. Which of the following is the same for sets A and B ? (A) (B) (C) (D)

Mean Median Mode Range

49.

45. 50. If x â€“ 3 is a factor of x4 â€“ 3x3 + kx + 3, what is the value of k ?

46. In a group of 33 students, 15 students are enrolled in a mathematics course, 10 are enrolled in a physics course, and 5 are enrolled in both a mathematics course and a physics course. How many students in the group are not enrolled in either a mathematics course or a physics course?

(A) â€“1 (B) 0 (C) 1 3 (D) 1 51. Which of the following is NOT a subset of the set {2, 4, 6, 8} ?

(A) 3 (B) 8 (C) 13 (D) 20

(A) The empty set (B) {2} (C) (D)

12

{2, 8} {2, {4, 6} }

C O L L E G E

M A T H E M A T I C S

1 52. If f ( x ) = , where x | 2, and g ( x ) = 2 x for x2 all values of x, then f ( g ( 0 ) ) is

x2 1 54. For all nonzero numbers, if f ( x ) = 2 x +1 1 g f x = and g ( x ) = 2 , then ( ( ) ) x

(A) 1 1 (B) 2 (C) 0 (D) undefined 53. The six students, P, Q, R, S, T, and U in a class took four exams, and the scores for the four exams were recorded in the following graphs. In which graph do the scores shown have the least standard deviation? (A)

Exam 1

Score

100 90 80 70

(B)

P

Q

R S Students

T

U

T

U

T

U

T

U

Exam 2

Score

100 90 80 70

(C)

P

Q

R S Students Exam 3

Score

100 90 80 70

(D)

P

Q

R S Students Exam 4

Score

100 90 80 70

P

Q

R S Students

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(A)

(x (x

(B)

(x (x

(C)

x4 + 1 x4 1

(D)

x4 1 x4 + 1

2

+ 1)

2

2

1)

2

1)

2

2

+ 1)

2 2

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M A T H E M A T I C S

55. Which of the following best represents the graph ¯ x 2 for x < 1 of the function f ( x ) = ° ±2 for x v 1 in the xy-plane?

56. A rectangular flat-screen computer monitor has a diagonal that measures 20 inches. The ratio of the length of the screen to the width of the screen is 4 to 3. What is the perimeter of the screen, in inches?

(A)

(A) (B) (C) (D)

48 56 64 192

57. Which of the following is an equation of the line in the xy-plane that passes through the point ( 2, − 1) and that is parallel to the line with equation 2 x − 3 y = 5 ? (B)

(C)

(D)

14

3 2

(A)

y = −2 x −

(B)

3 y =− x+2 2

(C)

y=

2 7 x− 3 3

(D)

y=

2 x −1 3

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M A T H E M A T I C S

58. In triangle ABC shown above, what is the length of line segment AB ? (A) (B) (C) (D)

17 3 16 14 2 34

59. A circular pizza with a 16-inch diameter is cut into 12 equal slices. What is the area, in square inches, of each slice?

16 π 3 8 (B) π 3 4 (C) π 3 2 (D) π 3 (A)

60. A square tablecloth lies flat on top of a circular table with area π square feet. If the four corners of the tablecloth just touch the edge of the circular table, what is the area of the tablecloth, in square feet?

square feet

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M A T H E M A T I C S 4.

Study Resources Most textbooks used in college-level mathematics courses cover the topics in the outline given earlier, but the approaches to certain topics and the emphases given to them may differ. To prepare for the College Mathematics exam, it is advisable to study one or more introductory college-level mathematics textbooks, which can be found in most college bookstores. Elementary algebra textbooks also cover many of the topics on the College Mathematics exam. When selecting a textbook, check the table of contents against the knowledge and skills required for this test.

Number 2m â€“ 1 2m + 1 m2 â€“ m m2 + m + 1

8.

{a âˆ’ 2, b âˆ’ 2, c âˆ’ 2, d âˆ’ 2, e âˆ’ 2} {2a, 2b, 2c, 2d, 2e} âŽ§a b c d e âŽŤ âŽ¨ , , , , âŽŹ âŽŠ2 2 2 2 2âŽ

Answer Key 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60.

C B â€“5 C A D D 4 B D C C B D C C C A D A D A A A D B C C A 2

Must Have Standard Deviation Equal to x

Set

15.

B A D to the right C B A to the right C B C B C 75.2 to the right C B B A C C B to the right C C D 9 D D A

Even

{a + 2, b + 2, c + 2, d + 2, e + 2}

Visit www.collegeboard.org/clepprep for additional math resources. You can also find suggestions for exam preparation in Chapter IV of the Official Study Guide. In addition, many college faculty post their course materials on their schoolsâ€™ websites.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

Odd

p

q

~ p

T T F F

T F T F

F F T T

23. Expression 3

Irrational âˆš

mn m3

âˆš

m n

âˆš

m n2

âˆš

3

16

Rational

âˆź pâˆ§q

C O L L E G E

M A T H E M A T I C S

Test Measurement Overview Format There are multiple forms of the computer-based test, each containing a predetermined set of scored questions. The examinations are not adaptive. There may be some overlap between different forms of a test: any of the forms may have a few questions, many questions, or no questions in common. Some overlap may be necessary for statistical reasons. In the computer-based test, not all questions contribute to the candidate’s score. Some of the questions presented to the candidate are being pretested for use in future editions of the test and will not count toward his or her score.

Scoring Information CLEP examinations are scored without a penalty for incorrect guessing. The candidate’s raw score is simply the number of questions answered correctly. However, this raw score is not reported; the raw scores are translated into a scaled score by a process that adjusts for differences in the difficulty of the questions on the various forms of the test.

Scaled Scores The scaled scores are reported on a scale of 20–80. Because the different forms of the tests are not always exactly equal in difficulty, raw-to-scale conversions may in some cases differ from form to form. The easier a form is, the higher the raw score required to attain a given scaled score. Table 1 indicates the relationship between number correct (raw) score and scaled score across all forms.

The Recommended Credit-Granting Score Table 1 also indicates a recommended credit-granting score, which represents the performance of students earning a grade of C in the corresponding course. The recommended B-level score represents B-level performance in equivalent course work. These scores were established as the result of a Standard-Setting study, the most recent having been conducted in 2006. The recommended credit-granting scores are based upon the judgments of a panel of experts currently teaching equivalent courses at various colleges and universities. These experts evaluate each question in order to determine

the raw scores that would correspond to B and C levels of performance. Their judgments are then reviewed by a test development committee, which, in consultation with test content and psychometric specialists, makes a final determination. The standard-setting study is described more fully in the earlier section entitled “CLEP Credit Granting” on page 4. Panel members participating in this study were: Jeffrey Baumgartner Hesston College Ed Harri Whatcom Community College Ronda Sanders University of South Carolina — Columbia Adriana Aceves University of New Mexico Kate Acks Maui Community College Tuncay Aktosun University of Texas at Arlington Michael Hall Arkansas State University David Hamrick Boston University Jonathan Kalk Kauai Community College Kathleen Kane Community College of Allegheny County James Lapp Truckee Meadows Community College Keith Leatham Brigham Young University Karen Longhart Flathead Valley Community College Ioana Mihaila California State Polytech University — Pomona Stephanie Ogden University of Tennessee — Knoxville Erick Hofacker University of Wisconsin River Falls Benjamin Kennedy Rutgers University Nasser Dastrange Buena Vista University Dennis Reissig Suffolk County Community College Thomas Smotzer Youngstown State University Lola Swint North Central Missouri College Robert Talbert Franklin College To establish the exact correspondences between raw and scaled scores, a scaled score of 50 is assigned to the raw score that corresponds to the recommended credit-granting score for C-level performance. Then a high (but in some cases, possibly less than perfect) raw score will be selected and assigned a scaled score of 80. These two points — 50 and 80 — determine a linear raw-to-scale conversion for the test.

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Table 1: College Mathematics Interpretive Score Data American Council on Education (ACE) Recommended Number of Semester Hours of Credit: 6 Course Grade

B

C

Scaled Score

Number Correct

80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50* 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20

50 49 48 47 46 46 45 44 43 43 42 41 40 39-40 39 38 37 36 35-36 35 34 33 32 31-32 31 30 29 28 27-28 26-27 26 25 24 23-24 22-23 22 21 20 19-20 18-19 18 17 16 15-16 15 14 13 12-13 12 11 10 10 9 8 7 7 6 5 5 0-4

*Credit-granting score recommended by ACE. Note: The number-correct scores for each scaled score on different forms may vary depending on form difficulty.

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M A T H E M A T I C S

Validity

Reliability

Validity is a characteristic of a particular use of the test scores of a group of examinees. If the scores are used to make inferences about the examinees’ knowledge of a particular subject, the validity of the scores for that purpose is the extent to which those inferences can be trusted to be accurate.

The reliability of the test scores of a group of examinees is commonly described by two statistics: the reliability coefficient and the standard error of measurement (SEM). The reliability coefficient is the correlation between the scores those examinees get (or would get) on two independent replications of the measurement process. The reliability coefficient is intended to indicate the stability/consistency of the candidates’ test scores, and is often expressed as a number ranging from .00 to 1.00. A value of .00 indicates total lack of stability, while a value of 1.00 indicates perfect stability. The reliability coefficient can be interpreted as the correlation between the scores examinees would earn on two forms of the test that had no questions in common.

One type of evidence for the validity of test scores is called content-related evidence of validity. It is usually based upon the judgments of a set of experts who evaluate the extent to which the content of the test is appropriate for the inferences to be made about the examinees’ knowledge. The committee that developed the CLEP examination in College Mathematics selected the content of the test to reflect the content of the general College Mathematics curriculum and courses at most colleges, as determined by a curriculum survey. Since colleges differ somewhat in the content of the courses they offer, faculty members should, and are urged to, review the content outline and the sample questions to ensure that the test covers core content appropriate to the courses at their college. Another type of evidence for test-score validity is called criterion-related evidence of validity. It consists of statistical evidence that examinees who score high on the test also do well on other measures of the knowledge or skills the test is being used to measure. Criterion-related evidence for the validity of CLEP scores can be obtained by studies comparing students’ CLEP scores to the grades they received in corresponding classes, or other measures of achievement or ability. At a college’s request, CLEP and the College Board conduct these studies, called Admitted Class Evaluation Service, or ACES, for individual colleges that meet certain criteria. Please contact CLEP for more information.

Statisticians use an internal-consistency measure to calculate the reliability coefficients for the CLEP exam. This involves looking at the statistical relationships among responses to individual multiple-choice questions to estimate the reliability of the total test score. The formula used is known as Kuder-Richardson 20, or KR-20, which is equivalent to a more general formula called coefficient alpha. The SEM is an index of the extent to which students’ obtained scores tend to vary from their true scores.1 It is expressed in score units of the test. Intervals extending one standard error above and below the true score for a test-taker will include 68 percent of the test-taker’s obtained scores. Similarly, intervals extending two standard errors above and below the true score will include 95 percent of the test-taker’s obtained scores. The standard error of measurement is inversely related to the reliability coefficient. If the reliability of the test were 1.00 (if it perfectly measured the candidate’s knowledge), the standard error of measurement would be zero. Scores on the computer-based CLEP examination in College Mathematics are estimated to have a reliability of 0.90. The standard error of measurement is 3.69 scaled-score points. 1

True score is a hypothetical concept indicating what an individual’s score on a test would be if there were no errors introduced by the measuring process. It is thought of as the hypothetical average of an infinite number of obtained scores for a test-taker with the effect of practice removed.

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