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® TEST GUIDE
MATHEMATICS SUBTEST II Sample Questions and Responses and Scoring Information Copyright © 2006 by National Evaluation Systems, Inc. (NES®) “California Subject Examinations for Teachers,” “CSET,” and the “CSET” logo are registered trademarks of the California Commission on Teacher Credentialing and National Evaluation Systems, Inc. (NES®). “NES®” and its logo are registered trademarks of National Evaluation Systems, Inc.™ CSTGQR111X03 Sample Test Questions for CSET: Mathematics Subtest II
Below is a set of multiplechoice questions and constructedresponse questions that are similar to the questions you will see on Subtest II of CSET: Mathematics. The term "enhanced" is used in this test guide to identify complex multiplechoice questions that require 2–3 minutes each to complete. Please note that enhanced mathematics questions will not be identified on the actual test form. You are encouraged to respond to the questions without looking at the responses provided in the next section. Record your responses on a sheet of paper and compare them with the provided responses. A calculator will be needed and will be only allowed for Mathematics Subtest II: Geometry; Probability and Statistics. You must bring your own graphing calculator to the test administration, and it must be one of the approved models listed in the current version of the CSET registration bulletin. Since the approved calculator brands and models are subject to change, the list of approved graphing calculators will be updated as necessary. Test administration staff will clear the memory of your calculator before and after the test. Be sure you back up the memory on your calculator, including applications, to an external device before arriving at the test site. 1. Use the diagram below to answer the question that follows.
y B θ m β x In the diagram above, θ and β are complementary angles and lines B and m are given by a1x + b1y = c1 and a2x + b2y = c2, respectively. Which of the following must be true? A. B. C. D. a1c1 = a2c2 b1c2 = b2c1 a1b1 = a2b2 a1b2 = a2b1 California Subject Examinations for Teachers Test Guide 1 Mathematics Subtest II (ENHANCED) 2. Use the statement below to answer the question that follows. If a transversal intersects two parallel lines, then the alternate interior angles are congruent. If the above statement is false, which of the following is also false? A. B. C. D. If two angles are supplements of congruent angles (or the same angle), then the two angles are congruent. Vertical angles are congruent. The base angles of an isosceles triangle are congruent. The angle sum of every triangle is 180°. 3. Use the diagram below to answer the question that follows.
B C A E D If AB is parallel to CD , which of the following is true? m m A. B. C. D. AC = BD
EC BE = AB CD ED = EC
ED BE = AE EC 2 California Subject Examinations for Teachers Test Guide Mathematics Subtest II 4. An artist has been commissioned to build a statue equidistant from the three sides of a triangular park. If the artist has a scale drawing of the triangular park, which of the following constructions could the artist use in determining the location of the statue? A. B. C. D. bisecting a given angle bisecting a given line segment constructing an angle congruent to a given angle constructing a segment congruent to a given segment (ENHANCED) 5. A regular hexagon has a perimeter of 72 cm. Which of the following is the approximate area of the hexagon in cm2? A. B. C. D. 144 339 374 452 (ENHANCED) 6. S(4, 0) and T(–2, 0) are two fixed points in a coordinate axis system, and point P has coordinates (x, y). If the length of PS is twice the length of PT , which of the following correctly m m describes the locus of points satisfying these conditions? A. B. C. D. an ellipse with foci (–2, 0) and (4, 0) an ellipse with foci (–4, 0) and (2, 0) a circle with center (2, 0) a circle with center (–4, 0) California Subject Examinations for Teachers Test Guide 3 Mathematics Subtest II 7. Use the diagram below to answer the question that follows.
s t r α If line s is perpendicular to lines r and t and both r and t lie in plane α, which of the following is true? A. B. C. D. Line s is perpendicular to any line in plane α that passes through the intersection of r and t. Lines r and t are perpendicular to each other. Lines s and t define a plane that forms an acute dihedral angle with plane α. Lines r, s, and t are coplanar. (ENHANCED) 8. A cylindrical can is 10 inches in height and has a surface area of about 245 square inches, including the top and bottom. Which of the following is the approximate volume of the can in cubic inches? A. B. C. D. 283 385 785 1131 4 California Subject Examinations for Teachers Test Guide Mathematics Subtest II (ENHANCED) 9. Use the diagram below to answer the question that follows. 8 cm 8 cm 8 cm 8c 8c 8c m m m 2m 666 cm cm cm A beam of wood in the shape of a rectangular prism 2 meters long is cut into 6 pieces, as shown in the diagram above. What is the volume of the shaded piece? A. B. C. D. 14,400 cm3 14,600 cm3 14,800 cm3 15,000 cm3 California Subject Examinations for Teachers Test Guide 5 Mathematics Subtest II 10. Use the diagram below to answer the question that follows.
y A' B' C' B C A m n x The diagram above shows the transformation of ABC under the composition R2R1, where R1 is a reflection over line n, followed by a reflection R2 over line m. Transforming ABC under the composition R1 R2 would demonstrate that: A. B. C. D. the set of reflections does not include inverse elements. the composition of transformations is not commutative. the composition of two reflections is not an isometry. the composition of transformations is not associative. 6 California Subject Examinations for Teachers Test Guide Mathematics Subtest II 11. A hexagon undergoes a coordinate transformation given by T:(x, y) → (3x, 2y). What is the ratio of the area of the original polygon to the area of the transformed hexagon? A. B. C. D.
2 3 9 4 7 2 37 4 1 12. A test contains 50 multiplechoice questions and each question has five possible answers. If a student answers every question, how many different ways can the student answer the questions on the test? A. B. C. D. 550 505 5! × 50 5 × 50! 13. The volume of liquid in soda cans is normally distributed with a mean of 12 fl. oz. and a standard deviation of 0.05 fl. oz. What is the approximate percentage of cans of this brand of soda that contain less than 11.9 fl. oz.? A. B. C. D. 0.5% 1% 2.5% 5% California Subject Examinations for Teachers Test Guide 7 Mathematics Subtest II 14. A scientist who has been weighing birds collected from the wild discovers that the laboratory scale has been reading out weights that are 1.5 ounces heavier than the birds' actual weights. Which of the following descriptive statistics is likely to be most affected by this error? A. B. C. D. standard deviation range variance quartiles 15. If the regression line for a data set is y = – 2.7x + 4.8, which of the following is the most reasonable value for the correlation coefficient of the data set? A. B. C. D. –2.7 –0.8 4.8 0.7 8 California Subject Examinations for Teachers Test Guide Mathematics Subtest II 16. Use the diagram below to complete the exercise that follows.
y S U V R T x Use techniques of coordinate geometry to prove that the segment joining the midpoints of two sides of a triangle is parallel to the third side of the triangle and one half its length. California Subject Examinations for Teachers Test Guide 9 Mathematics Subtest II 17. Complete the exercise that follows. Find the equation of the curve traced by a point that moves so the sum of its distances to the points (0, 0) and (0, 4) is 12. 10 California Subject Examinations for Teachers Test Guide Mathematics Subtest II 18. Use the diagram below to complete the exercise that follows.
A B F D C E In the diagram above, B and D are points on segment AC and segment AE, respectively. AB = AD and BC = DE. Prove that BF = DF. California Subject Examinations for Teachers Test Guide 11 Mathematics Subtest II 19. Complete the exercise that follows. Housing units in U.S. suburban areas in 1999 were distributed as follows: 69.1% owner occupied, 24.6% renter occupied, and 6.3% vacant. A researcher, wishing to determine whether this distribution is currently the same, takes a random sample of 500 current housing units in U.S. suburban areas and obtains the following data: 375 owneroccupied units, 100 renteroccupied units, and 25 vacant units. • • • • State the null hypothesis that the researcher should use to determine if these data suggest that the distribution of current yearround housing differs from the 1999 distribution. Based on the 1999 distribution, determine the number of housing units, out of a sample of 500, expected to be owner occupied, renter occupied, and vacant. Determine the value of the chisquare (χ2) test statistic for these sample data. The table below gives the probability corresponding to given values on a χ2distribution with two degrees of freedom. Use the table to determine if, at the 5% significance level, the data suggest that the distribution of current housing units in U.S. suburban areas differs from the 1999 distribution. Probability χ2
0.10 4.605 0.05 5.991 0.025 7.378 0.01 9.210 0.005 10.597 12 California Subject Examinations for Teachers Test Guide Sample Written Response Sheets for CSET: Mathematics Subtest II
For questions 16–19, examinees would record their written response to each question on a twopage response sheet located in their answer document. The length of their response to each question is limited to the lined space available on the response sheet. A sample of the response sheet is provided below. California Subject Examinations for Teachers Test Guide 13 Mathematics Subtest II 14 California Subject Examinations for Teachers Test Guide Annotated Responses to Sample MultipleChoice Questions for CSET: Mathematics Subtest II
Geometry
1. Correct Response: D. (SMR Code: 2.1) Since θ and β are complementary angles, the third angle in the right triangle in the diagram equals θ. The angle vertical to this third angle θ also equals θ. Therefore, since the alternate exterior angles of lines B and m both equal θ, lines B and m are parallel. The equations – a1 c1 of lines B and m can be converted to slopeintercept form: a1x + b1y = c1 ⇒ y = b x + b and 1 1 – a2 c2 – a1 – a2 a2x + b2y = c2 ⇒ y = b x + b . Since B and m are parallel, b = b , so a1b2 = a2b1. 2 2 1 2 2. Correct Response: D. (SMR Code: 2.1) In Euclidean geometry, if two parallel lines are cut by a transversal, then the alternate interior angles formed are congruent. This statement is used to prove that the angles of every triangle sum to 180°. If the given statement is false, then the sum of angles in a triangle is not necessarily 180°, as is the case in nonEuclidean geometries. 3. Correct Response: D. (SMR Code: 2.2) If AB is parallel to CD , then ∠A ≅ ∠C and ∠B ≅ ∠D, since m m they are alternate interior angles. Then by AA, K ABE ∼ K CDE, so the ratios between corresponding pairs of sides are equal. Hence BE = AE . 4. Correct Response: A. (SMR Code: 2.2) Constructing an angle bisector in a triangle displays a locus of points that are equidistant from the two sides of the triangle that form the angle. The point at which the three angle bisectors intersect is equidistant from the three sides of a triangle. Thus, bisecting one of the angles in the triangular park is an appropriate first step. 5. Correct Response: C. (SMR Code: 2.2) If the perimeter of a regular hexagon is 72 cm, then
ED EC each edge has a length of 12 cm. A regular hexagon can be split into six equilateral triangles. The height of each triangle is 6 3 cm, by the Pythagorean theorem, and the area of each triangle equals
1 2(12)( 6 3) = 36 3 cm2. Thus the area of the hexagon is 216 3 cm2, or approximately 374 cm2. 6. Correct Response: D. (SMR Code: 2.2) Use the fact that PS = 2PT to write the equation (x – 4)2 + (y – 0)2 = 2 (x + 2)2 + (y – 0)2 . Then squaring both sides gives x2 – 8x + 16 + y2 = 4(x2 + 4x + 4 + y2), which simplifies to 3x2 + 24x + 3y2 = 0. This is the equation of a circle. To find the center, complete the square: 3(x2 + 8x) + 3y2 = 0 ⇒ 3(x2 + 8x + 16) + 3y2 = 48 ⇒ (x + 4)2 + y2 = 16. The circle's center is at (–4, 0). 7. Correct Response: A. (SMR Code: 2.3) If a line is perpendicular to two distinct lines in a plane, then it is perpendicular to the plane. This means that it is perpendicular to any line in the plane that passes through its foot. Since the intersection of r and t is its foot, the statement given in response choice A is true. California Subject Examinations for Teachers Test Guide 15 Mathematics Subtest II 8. Correct Response: A. (SMR Code: 2.3) The surface area of a cylinder is 2πr2 + 2πrh. If the height is 10 in. and the surface area is 245 in.2, then 2πr2 + 20πr = 245. This equation approximately equals r2 + 10r – 39 = 0, which factors into (r + 13)(r – 3) = 0. Solving for r gives r = – 13 or 3, so the radius is approximately 3 cm. The volume of a cylinder is πr2h, so this cylinder has a volume of approximately 90π in.3, or about 283 in.3. 9. Correct Response: A. (SMR Code: 2.3) The shaded section is a trapezoidal prism, so its volume is the product of the area of the trapezoid and the height of the prism (200 cm).
6 8 6 The triangle in the top left corner has legs of 8 cm and
10 6 cm, so its area is ⎛2⎞(8)(6) = 24 cm2. ⎝⎠ The area of the larger triangle is ⎛2⎞(16)(12) = 96 cm2. ⎝⎠
1 1 8 The trapezoid's area is the difference between the area of the large triangle (96 cm2) and the area of the small triangle (24 cm2), so the area of the trapezoid is 72 cm2. Multiplying the area of the trapezoid by the height of the prism gives a volume of 14,400 cm3. 10. Correct Response: B. (SMR Code: 2.4) If the composition of these two transformations were commutative, it would not matter which reflection was done first. However, reflecting K ABC first across line m and then across line n would not result in K A'B'C', so the composition of these two transformations is not commutative. Correct Response: A. (SMR Code: 2.4) Since T is a linear transformation, one need only check how T transforms the area of a square with vertex coordinates (0, 0), (0, 1), (1, 1), and (1, 0). This square has area 1. The image of these points is (0, 0), ⎛0, 2⎞ , ⎛3, 2⎞ , and (3, 0). This is a rectangle of area 2 . Hence, ⎝ ⎠⎝ ⎠
3 11. 1 1 T maps each unit square to a rectangle of area 2 . It follows that if A is the area of the original hexagon,
3 2A 3 is the area of the transformed hexagon, and the ratio of the original hexagon to the transformed
2 hexagon is 3 . Probability and Statistics
12. Correct Response: A. (SMR Code: 4.1) Each question has 5 possible answers, so the number of ways in which a student could answer one question is 5. Since there are 50 questions, the number of possible sets of answers is 5•5•5• . . . •5 (50 times), or 550. 13. Correct Response: C. (SMR Code: 4.1) If the mean of a normally distributed data set is 12 fl. oz. and the standard deviation is 0.05 fl. oz., then approximately 67% of the points in the data set have values within 0.05 fl. oz. of the mean and approximately 95% of the data points are within 0.1 fl. oz. (i.e., two standard deviations) of the mean, that is, between 11.9 fl. oz. and 12.1 fl. oz. This means that 2.5% of the data points are below 11.9 fl. oz., and 2.5% are above 12.1 fl. oz. 16 California Subject Examinations for Teachers Test Guide Mathematics Subtest II 14. Correct Response: D. (SMR Code: 4.2) If each measurement is exactly 1.5 ounces heavier than its true weight, then the histogram depicting the spread of measured weights will have the same shape as a histogram representing the true data, but the graph will be shifted 1.5 units (ounces) to the right. Hence, the standard deviation and variance (which measure the spread of data) will be unchanged, as will the range (difference between lowest and highest value). The quartiles, however, will change, since the values are all shifted up 1.5 units. 15. Correct Response: B. (SMR Code: 4.2) The correlation coefficient is a number between –1 and 1, inclusive. Therefore, the only possible values for the correlation coefficient are –0.8 and 0.7. Since the slope of the regression line for the data set is –2.7, the most reasonable value for the correlation coefficient is –0.8. California Subject Examinations for Teachers Test Guide 17 Examples of Responses to Sample ConstructedResponse Questions for CSET: Mathematics Subtest II
Geometry
Question #16 (Score Point 4 Response) Let the coordinates of R = (0, 0), S = (2b, 2c), S (2b, 2c) U (b, c) R (0, 0) V (a + b, c) T (2a, 0) and T = (2a, 0). Since U is the midpoint of RS, its coordinates are ⎜ ⎛0 + 2b 0 + 2c⎞ , ⎟ = (b, c). 2⎠ ⎝2 ⎛2a + 2b 0 + 2c⎞ , ⎟ = (a + b, c). 2⎠ ⎝2 Since V is the midpoint of TS, its coordinates are ⎜ So to show UV ║RT means showing their slopes are the same: mm Slope is ∆y ∆x . So, slope of UV = m slope of RT = m c–c 0 = =0 (a + b) — b a 0–0 0 = =0 2a — 0 2a Since the slopes of UV and RT are the same, the segments are parallel. m m
m m Show the length of UV = 2 length of RT: m m The length of UV = a + b – b = a (since UV is horizontal). m m The length of RT is 2a (RT horizontal). m m Therefore the length of UV = 2 length of RT.
1 1 18 California Subject Examinations for Teachers Test Guide Mathematics Subtest II Question #16 (Score Point 3 Response) R = (0, 0) S = (x, y) T = (z, 0) Using the midpoint formula, S (x, y) U V T (z, 0) ⎛x U=⎜ , ⎝2 y⎞ 2⎠ y⎞ 2⎠ y 2 2 y 2 x 2 =0 ⎟ ⎛x + z V=⎜ , ⎝2 ⎟ R (0, 0) slope of UV = m x+z slope of RT = m 0–0 =0 z—0 Since UV and RT have the same slopes, they are parallel. m m Using the distance formula, UV = ⎛x + z x ⎞2 ⎛ y y ⎞2 ⎜ 2 + 2⎟ + ⎜2 + 2⎟ ⎠⎝ ⎠ ⎝
z⎞2 ⎛ ⎜ x + 2 ⎟ + y2 = ⎠ ⎝ x2 + y2 + xz + z2 4 = RT = If x2 + (z + 0)2 + (0 + 0) 2 = y2 + xz + z2 4 = 1 z2 = z 1 RT. 2 z, then UV = 2 California Subject Examinations for Teachers Test Guide 19 Mathematics Subtest II Question #16 (Score Point 2 Response) Let coordinates be R(0, 0), S(a, b) and T(x, 0) U = midpoint of RS m V = midpoint of ST m U=⎜ S (a, b) U V T (x, 0) R (0, 0) ⎛a – 0 b – 0⎞ ⎛a b⎞ , ⎟=⎜ , ⎟ 2 ⎠ ⎝2 2 ⎠ ⎝2 ⎛a – x b – 0⎞ ⎛a – x b⎞ , ,⎟ ⎟=⎜ 2⎠ ⎝2 2⎠ ⎝2 V=⎜ bb – 2 2 =0 Slope of UV = m a a–x — 2 2
m Slope of RT = 0–0 =0 x—0 The lines have no slopes, so they are parallel. 20 California Subject Examinations for Teachers Test Guide Mathematics Subtest II Question #16 (Score Point 1 Response) y (2, 5) U R S (4, 10) V (6, 5) x T (8, 0) UV = 6 – 2 = 4 RT = 8 – 0 = 8
∴ RT = 2 UV
1 California Subject Examinations for Teachers Test Guide 21 Mathematics Subtest II Question #17 (Score Point 4 Response) Let (x, y) be the point on the curve. The distance from (x, y) to (0, 0) is found by using the distance formula d1 =
(x – 0) 2 + (y – 0) 2 = x 2 + y2 (x – 0) 2 + (y – 4) 2 = x 2 + (y – 4) 2 The distance from (x, y) to (0, 4) is d2 =
We know d1 + d2 = 12, so x 2 + y2 + x 2 + (y – 4) 2 = 12. To square a radical equation, put the radicals on different sides of the equation and square both sides: ( x 2 + (y – 4) 2 ) = (12 –
2 x 2 + y2 ) 2 x2 + (y  4)2 = 144 – 24 x 2 + y2 + x2 + y2 Isolate the radical on one side again: x2 + (y – 4)2  144 – x2  y2 = –24 x 2 + y2 y2  8y + 16 – 144 – y2 = –24 x 2 + y2 8y – 128 = –24 x 2 + y2 y + 16 = 3 x 2 + y2 Square both sides: y2 + 32y + 256 = 9(x2 + y2) y2 + 32y + 256 = 9x2 + 9y2 8y2 + 32y – 9x2 = 256 continued on next page 22 California Subject Examinations for Teachers Test Guide Mathematics Subtest II Question #17 (Score Point 4 Response) continued Complete the square for –8y2 + 32y: 8 (y2  4y) – 9x2 = 256 8 (y2  4y + 4) – 9x2 = 256 – 32 8 (y – 2)2  9x2 = 288 (y – 2) 2 36 + 32 x2 =1 This is the equation of an ellipse centered at (0, 2). Question #17 (Score Point 3 Response) (x, y) to (0, 0) = d1 = (x, y) to (0, 4) = d2 = x 2 + y2 + x 2 + y2
x 2 + (y – 4) 2 x 2 + (y – 4) 2 = 12 x 2 + (y – 4) 2 = 12  x 2 + y2 x2 + (y  4)2 = 144 – 24 x 2 + y2 + x2 + y2 8y + 16 = 144 – 24 x 2 + y2 y – 2 = 18 + 3 x 2 + y2 y + 16 = 3 x 2 + y2 y2 + 32y + 256 = 3x2 + 3y2 3x2 + 2y2  32y = 256 3x2 + 2(y2  16y) = 256 3x2 + 2(y  8)2 = 384 California Subject Examinations for Teachers Test Guide 23 Mathematics Subtest II Question #17 (Score Point 2 Response) Distance (x, y) to (0, 0) d1 = Distance (x, y) to (0, 4) d2 = x 2 + y2 + x 2 + (y – 4) 2 = 12 x 2 + y2 x 2 + y2 x 2 + (y – 4) 2 x 2 + (y – 4) 2 = 12  x2 + (y – 4)2 = 144 + x2 + y2 x2 + y2  8y + 16 = 144 + x2 + y2 8y = 128 y = 16 Question #17 (Score Point 1 Response) d1 = x 2 + y2 d2 = x 2 + y2 + 16 d1 + d2 = 12 x 2 + y2 + x 2 + y2 + 16 = 12 x2 + y2 + x2 + y2 + 16 = 144 2x2 + 2y2 = 128 x2 + y2 = 64 24 California Subject Examinations for Teachers Test Guide Mathematics Subtest II Question #18 (Score Point 4 Response) A Given: AB = AD BC = DE B F D Prove: BF = DF C E Since AB = AD and BC = DE, AB + BC = AD + DE, so AC = AE (addition property of equal line segments)
∠A = ∠A (reflexive property) Therefore CAD ≅ EAB (S.A.S.) ∠C ≅ ∠E (corresponding angles of congruent triangles are equal) ∠BFC ≅ ∠DFE (vertical angles are congruent) Therefore ∠CBF ≅ ∠EDF (since 2 angles of CBF are congruent to 2 angles of EDF, the remaining angles must also be congruent) Therefore CBF ≅ EDF (A.S.A.) Therefore BF = DF (corresponding sides of congruent triangles) California Subject Examinations for Teachers Test Guide 25 Mathematics Subtest II Question #18 (Score Point 3 Response) AB = AD given BC = DE given AC = AE addition of equal segments DCA ≅
∠C ≅ ∠E A B F D BEA (S.A.S.) corresponding parts of equal congruent triangles are C E ∠BFC ≅ ∠DFE ∠CBF ≅ ∠EDF vertical angles are equal since 2 angles of CBF equal 2 angles of EDF, the remaining angles are equal A.A.A. CBF ≅ BF = DF EDF corresponding parts of congruent triangles are equal Question #18 (Score Point 2 Response) AB = AD BC = DE AC = AE (addition) ACD ≅
∠C ≅ ∠E A B AEB F D C E BF = DF 26 California Subject Examinations for Teachers Test Guide Mathematics Subtest II Question #18 (Score Point 1 Response) A
Given: AB = AD and BC = DE B F D C E Draw CE CFE is isosceles so CF = FE now CFB ≅ EFD, so BF ≅ DF since they are corresponding parts of congruent triangles. California Subject Examinations for Teachers Test Guide 27 Mathematics Subtest II Probability and Statistics
Question #19 (Score Point 4 Response) • Null Hypothesis: The current distribution is the same as that in 1999. Alternative Hypothesis: The current distribution is not the same as that in 1999. • Based on the 1999 distribution, one expects: 69.1% of 500 = 345.5 to be owneroccupied 24.6% of 500 = 123 to be renteroccupied 6.3% of 500 = 31.5 to be vacant • X  statistic = Sum of
2 (Observed Frequency  Expected Frequency)2 Expected y F r equen c Frequency Frequency Owner Renter Vacant Total 375 100 25 500 345.5 123 31.5 500
X2 = Sum of Observed Expected (Obs  Exp)2 Exp (375  345.5)2 345.5 123 (100  123)2 (25  31.5)2 31.5 = 2.5188 = 4.3008 = 1.3413 (Obs Freq – Exp Freq ) 2 Exp Freq = 8.1609 • For a X2  statistic of 8.1609, the probability lies between 0.01 and 0.025, i.e., the Pvalue is between 1% and 2.5%. Therefore, at the 5% significance level, the null hypothesis is rejected and one in U.S. suburban areas differs from the 1999 distribution. concludes that the data suggest that the distribution of current housing units 28 California Subject Examinations for Teachers Test Guide Mathematics Subtest II Question #19 (Score Point 3 Response) • Null Hypothesis: The current distribution is the same as that in 1999. Alternative Hypothesis: The current distribution is not the same as that in 1999. • 345.5  owneroccupied 123  renteroccupied 31.5  vacant Observed Owner Renter Vacant 375 100 25 Expected 345.5 123 31.5
X2 2.5188 4.3008 1.3413 • The Pvalue is between 0.05% and 1%. At the 5% significance level, the null hypothesis is rejected. California Subject Examinations for Teachers Test Guide 29 Mathematics Subtest II Question #19 (Score Point 2 Response) • Null Hypothesis: The current distribution is the same as that in 1999. Alternative Hypothesis: The current distribution is different from that in 1999. • Based on the 1999 distribution, one expects: 69.1% of 500 = 345.5 to be owneroccupied 24.6% of 500 = 123 to be renteroccupied 6.3% of 500 = 31.5 to be vacant • X  statistic = Sum of
2 (Observed  Expected)2 Expected Observed Owner Renter Vacant 75% 20% 5% Expected 69.1% 24.6% 6.3% (Obs  Exp)2 Exp (75  69.1)2 69.1 (20  24.6)2 24.6 6.3 (5  6.3)2 = 0.5 = 0.9 = 0.3 = 1.7 X2 = Sum of (Obs  Exp) 2 Exp • For a X2  statistic of 1.7, the probability is less than 0.10. Therefore, at the 5% significance level, the null hypothesis is rejected and one concludes that the data suggest that the distribution of current housing units in U.S. suburban areas is different from the 1999 distribution. 30 California Subject Examinations for Teachers Test Guide Mathematics Subtest II Question #19 (Score Point 1 Response) • Null Hypothesis: The current distribution is different from that in 1999. Alternative Hypothesis: The current distribution is the same as that in 1999.
• Observed Owner Renter Vacant
X2 = • Expected 345.5 123 31.5 375 100 25 2.88 For a X2  statistic of 2.88, the probability is more than 0.10. Therefore, at the 5% significance level, the null hypothesis is rejected and one concludes that the data suggest that the distribution of current housing units in U.S. suburban areas differs from the 1999 distribution. California Subject Examinations for Teachers Test Guide 31 Scoring Information for CSET: Mathematics Subtest II
Responses to the multiplechoice questions are scored electronically. Scores are based on the number of questions answered correctly. There is no penalty for guessing. There are four constructedresponse questions in Subtest II of CSET: Mathematics. Each of these constructedresponse questions is designed so that a response can be completed within a short amount of time— approximately 10–15 minutes. Responses to constructedresponse questions are scored by qualified California educators using focused holistic scoring. Scorers will judge the overall effectiveness of your responses while focusing on the performance characteristics that have been identified as important for this subtest (see below). Each response will be assigned a score based on an approved scoring scale (see page 33). Your performance on the subtest will be evaluated against a standard determined by the California Commission on Teacher Credentialing based on professional judgments and recommendations of California educators. Performance Characteristics for CSET: Mathematics Subtest II
The following performance characteristics will guide the scoring of responses to the constructedresponse questions on CSET: Mathematics Subtest II. PURPOSE SUBJECT MATTER KNOWLEDGE SUPPORT DEPTH AND BREADTH OF UNDERSTANDING The extent to which the response addresses the constructedresponse assignment's charge in relation to relevant CSET subject matter requirements. The application of accurate subject matter knowledge as described in the relevant CSET subject matter requirements. The appropriateness and quality of the supporting evidence in relation to relevant CSET subject matter requirements. The degree to which the response demonstrates understanding of the relevant CSET subject matter requirements. 32 California Subject Examinations for Teachers Test Guide Mathematics Subtest II Scoring Scale for CSET: Mathematics Subtest II
Scores will be assigned to each response to the constructedresponse questions on CSET: Mathematics Subtest II according to the following scoring scale. SCORE POINT SCORE POINT DESCRIPTION The "4" response reflects a thorough command of the relevant knowledge and skills as defined in the subject matter requirements for CSET: Mathematics. • The purpose of the assignment is fully achieved. • There is a substantial and accurate application of relevant subject matter knowledge. • The supporting evidence is sound; there are highquality, relevant examples. • The response reflects a comprehensive understanding of the assignment. The "3" response reflects a general command of the relevant knowledge and skills as defined in the subject matter requirements for CSET: Mathematics. • The purpose of the assignment is largely achieved. • There is a largely accurate application of relevant subject matter knowledge. • The supporting evidence is adequate; there are some acceptable, relevant examples. • The response reflects an adequate understanding of the assignment. The "2" response reflects a limited command of the relevant knowledge and skills as defined in the subject matter requirements for CSET: Mathematics. • The purpose of the assignment is partially achieved. • There is limited accurate application of relevant subject matter knowledge. • The supporting evidence is limited; there are few relevant examples. • The response reflects a limited understanding of the assignment. The "1" response reflects little or no command of the relevant knowledge and skills as defined in the subject matter requirements for CSET: Mathematics. • The purpose of the assignment is not achieved. • There is little or no accurate application of relevant subject matter knowledge. • The supporting evidence is weak; there are no or few relevant examples. • The response reflects little or no understanding of the assignment. The "U" (Unscorable) is assigned to a response that is unrelated to the assignment, illegible, primarily in a language other than English, or does not contain a sufficient amount of original work to score. The "B" (Blank) is assigned to a response that is blank. 4 3 2 1 U B California Subject Examinations for Teachers Test Guide 33 ...
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This note was uploaded on 03/05/2009 for the course CS 5 taught by Professor Prof during the Spring '09 term at UCSB.
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