I85PREP
⌘ K
Extremadura PAU Applied Mathematics for Social Sciences II study materials
Extremadura PAU Applied Mathematics for Social Sciences II study resources

Extremadura PAU Applied Mathematics for Social Sciences II practice questions

One question at a time. Build your Extremadura PAU Applied Mathematics for Social Sciences II understanding.

First 50 of 190 questions. Each one lists the choices, the correct answer, and an explanation. The checker at the top of this page saves progress in this browser when it is available.

Continue in the practice tool above to load more questions, 50 at a time.

Question 1Análisis, límites, derivadas y optimización

Evaluate the limit lim_(x->inf) (4x^2 - 3x + 1) / (2x^2 + 5x - 7).

  1. 0
  2. 4
  3. 2 Correct answer.
  4. Infinity

Explanation

Since both numerator and denominator are polynomials of degree 2, the limit as x -> inf equals the ratio of leading coefficients: 4 / 2 = 2.

Question 2Análisis, límites, derivadas y optimización

Find the derivative of the polynomial function f(x) = 3x^4 - 5x^2 + 7x - 9.

  1. 12x^3 - 10x^2 + 7
  2. 3x^3 - 5x + 7
  3. 12x^3 - 5x + 7
  4. 12x^3 - 10x + 7 Correct answer.

Explanation

Apply the power rule to each term: d/dx(3x^4) = 12x^3 d/dx(-5x^2) = -10x d/dx(7x) = 7 d/dx(-9) = 0. Thus, f'(x) = 12x^3 - 10x + 7.

Question 3Análisis, límites, derivadas y optimización

If f(x) = x^3 - 4x, calculate the value of f'(2).

  1. 8 Correct answer.
  2. 12
  3. 4
  4. 0

Explanation

f'(x) = 3x^2 - 4. Substitute x = 2: f'(2) = 3(2)^2 - 4 = 3(4) - 4 = 12 - 4 = 8.

Question 4Análisis, límites, derivadas y optimización

Calculate the indefinite integral int (6x^2 - 4x + 3) dx.

  1. 6x^3 - 4x^2 + 3x + C
  2. 2x^3 - 2x^2 + 3x + C Correct answer.
  3. 12x - 4 + C
  4. 3x^3 - 2x^2 + 3x + C

Explanation

Integrate term by term using power rule int x^n dx = x^(n+1)/(n+1): int 6x^2 dx = 6(x^3/3) = 2x^3 int -4x dx = -4(x^2/2) = -2x^2 int 3 dx = 3x. Result: 2x^3 - 2x^2 + 3x + C.

Question 5Análisis, límites, derivadas y optimización

Evaluate the definite integral int_1^3 (2x + 1) dx.

  1. 8
  2. 12
  3. 10 Correct answer.
  4. 14

Explanation

Find antiderivative F(x) = x^2 + x. Apply Fundamental Theorem of Calculus: F(3) = 3^2 + 3 = 9 + 3 = 12. F(1) = 1^2 + 1 = 1 + 1 = 2. F(3) - F(1) = 12 - 2 = 10.

Question 6Análisis, límites, derivadas y optimización

Find the equation of the tangent line to the curve f(x) = x^2 - 3x + 2 at x = 4.

  1. y = 5x - 6
  2. y = 3x - 6
  3. y = 5x - 20
  4. y = 5x - 14 Correct answer.

Explanation

1) Point of tangency: f(4) = 4^2 - 3(4) + 2 = 16 - 12 + 2 = 6. Point (4, 6). 2) Slope m = f'(4): f'(x) = 2x - 3 => f'(4) = 2(4) - 3 = 5. 3) Line equation: y - 6 = 5(x - 4) => y = 5x - 20 + 6 => y = 5x - 14.

Question 7Análisis, límites, derivadas y optimización

A revenue function is given by R(x) = 100x - 0.5x^2. Find the marginal revenue MR(x) = R'(x) when x = 40 units.

  1. 60 Correct answer.
  2. 80
  3. 100
  4. 40

Explanation

MR(x) = R'(x) = 100 - 2(0.5x) = 100 - x. At x = 40: MR(40) = 100 - 40 = 60.

Question 8Análisis, límites, derivadas y optimización

Find the critical points of the function f(x) = x^3 - 3x.

  1. x = 0 and x = 3
  2. x = -1 and x = 1 Correct answer.
  3. x = 0 and x = sqrt(3)
  4. x = -3 and x = 3

Explanation

Set first derivative to zero: f'(x) = 3x^2 - 3 = 0. 3(x^2 - 1) = 0 => x^2 = 1 => x = 1 and x = -1.

Question 9Análisis, límites, derivadas y optimización

Determine the vertical asymptote of the rational function f(x) = (2x + 1) / (x - 3).

  1. x = -3
  2. y = 2
  3. x = 3 Correct answer.
  4. y = 3

Explanation

Vertical asymptotes occur where the denominator is zero and numerator is non-zero: x - 3 = 0 => x = 3. Since numerator 2(3) + 1 = 7 != 0, x = 3 is a vertical asymptote.

Question 10Análisis, límites, derivadas y optimización

Calculate the area of the region bounded by the curve y = 3x^2, the x-axis, and lines x = 0 and x = 2.

  1. 6
  2. 12
  3. 4
  4. 8 Correct answer.

Explanation

Area = int_0^2 3x^2 dx = [x^3]_0^2 = 2^3 - 0^3 = 8 square units.

Question 11Análisis, límites, derivadas y optimización

For what value of parameter 'a' is the piecewise function continuous at x = 2? f(x) = 2x + a for x <= 2, and x^2 - 1 for x > 2.

  1. a = -1 Correct answer.
  2. a = 1
  3. a = 3
  4. a = -3

Explanation

For continuity at x = 2, left-hand limit equals right-hand limit: Left-hand limit: lim_(x->2-) (2x + a) = 2(2) + a = 4 + a. Right-hand limit: lim_(x->2+) (x^2 - 1) = 2^2 - 1 = 3. Set 4 + a = 3 => a = 3 - 4 = -1.

Question 12Análisis, límites, derivadas y optimización

Evaluate the limit lim_(x->3) (x^2 - 9) / (x - 3).

  1. 0
  2. 6 Correct answer.
  3. 3
  4. Undefined

Explanation

Direct substitution gives 0/0 (indeterminate form). Factor numerator: x^2 - 9 = (x - 3)(x + 3). lim_(x->3) [(x - 3)(x + 3)] / (x - 3) = lim_(x->3) (x + 3) = 3 + 3 = 6.

Question 13Análisis, límites, derivadas y optimización

Find the derivative of f(x) = x^2 * e^x.

  1. 2x e^x
  2. x^2 e^x
  3. (x^2 + 2x) e^x Correct answer.
  4. (2x + 1) e^x

Explanation

Apply the product rule (u*v)' = u'*v + u*v': u = x^2 => u' = 2x v = e^x => v' = e^x. f'(x) = 2x e^x + x^2 e^x = (x^2 + 2x) e^x.

Question 14Análisis, límites, derivadas y optimización

Find the derivative of f(x) = (2x + 1) / (x + 2).

  1. 2 / (x+2)^2
  2. 1 / (x+2)^2
  3. (4x + 5) / (x+2)^2
  4. 3 / (x+2)^2 Correct answer.

Explanation

Apply quotient rule (u/v)' = (u'v - uv') / v^2: u = 2x + 1 => u' = 2 v = x + 2 => v' = 1. f'(x) = [2(x + 2) - (2x + 1)(1)] / (x + 2)^2 = [2x + 4 - 2x - 1] / (x + 2)^2 = 3 / (x + 2)^2.

Question 15Análisis, límites, derivadas y optimización

A firm's profit function is P(x) = -x^2 + 80x - 1200, where x is the production quantity in units. What production quantity x maximizes total profit?

  1. x = 40 Correct answer.
  2. x = 80
  3. x = 20
  4. x = 50

Explanation

To maximize profit, set P'(x) = 0: P'(x) = -2x + 80 = 0 => 2x = 80 => x = 40. Check second derivative: P''(x) = -2 < 0, confirming a local maximum at x = 40 units.

Question 16Análisis, límites, derivadas y optimización

Using the profit function P(x) = -x^2 + 80x - 1200, what is the maximum profit value achievable?

  1. 200
  2. 400 Correct answer.
  3. 800
  4. 1600

Explanation

Substitute the optimal production quantity x = 40 into P(x): P(40) = -(40)^2 + 80(40) - 1200 = -1600 + 3200 - 1200 = 400 monetary units.

Question 17Análisis, límites, derivadas y optimización

A total cost function is C(x) = 2x^2 + 500x + 1800. What quantity x minimizes average cost AC(x) = C(x)/x?

  1. x = 20
  2. x = 25
  3. x = 30 Correct answer.
  4. x = 45

Explanation

Average cost AC(x) = (2x^2 + 500x + 1800) / x = 2x + 500 + 1800/x. Set AC'(x) = 0: AC'(x) = 2 - 1800/x^2 = 0 => 2 = 1800/x^2 => 2x^2 = 1800 => x^2 = 900 => x = 30 units.

Question 18Análisis, límites, derivadas y optimización

Find the inflection point of the cubic function f(x) = x^3 - 6x^2 + 9x + 5.

  1. (1, 9)
  2. (3, 5)
  3. (2, 5)
  4. (2, 7) Correct answer.

Explanation

1) First derivative: f'(x) = 3x^2 - 12x + 9. 2) Second derivative: f''(x) = 6x - 12. Set f''(x) = 0 => 6x = 12 => x = 2. 3) Y-coordinate: f(2) = 2^3 - 6(2^2) + 9(2) + 5 = 8 - 24 + 18 + 5 = 7. Inflection point is (2, 7).

Question 19Análisis, límites, derivadas y optimización

Evaluate the indefinite integral int 2x (x^2 + 3)^4 dx using substitution.

  1. (x^2 + 3)^5 / 5 + C Correct answer.
  2. (x^2 + 3)^5 + C
  3. 2x (x^2 + 3)^5 / 5 + C
  4. (x^2 + 3)^3 / 3 + C

Explanation

Let u = x^2 + 3, so du = 2x dx. The integral becomes int u^4 du = u^5 / 5 + C = (x^2 + 3)^5 / 5 + C.

Question 20Análisis, límites, derivadas y optimización

Find the area of the region bounded between the curves f(x) = x^2 and g(x) = 4.

  1. 16/3
  2. 32/3 Correct answer.
  3. 8
  4. 64/3

Explanation

Intersection points: x^2 = 4 => x = -2 and x = 2. Area = int_-2^2 (4 - x^2) dx = [4x - x^3/3]_-2^2 = [4(2) - 8/3] - [4(-2) - (-8)/3] = (8 - 8/3) - (-8 + 8/3) = (16/3) - (-16/3) = 32/3 square units.

Question 21Análisis, límites, derivadas y optimización

Calculate the area of the region bounded by the curve y = x^2 - 4x and the x-axis.

  1. 16/3
  2. 64/3
  3. 32/3 Correct answer.
  4. 8

Explanation

Roots of y = x^2 - 4x: x(x - 4) = 0 => x = 0 and x = 4. Between x = 0 and x = 4, the curve lies below the x-axis (y <= 0). Area = - int_0^4 (x^2 - 4x) dx = - [x^3/3 - 2x^2]_0^4 = - [64/3 - 32] = - [64/3 - 96/3] = - (-32/3) = 32/3 square units.

Question 22Análisis, límites, derivadas y optimización

Find parameters 'a' and 'b' so that f(x) = a x^2 + b for x <= 1, and 3x + 1 for x > 1 is both continuous and differentiable at x = 1.

  1. a = 3, b = 1
  2. a = 2, b = 2
  3. a = 1, b = 3
  4. a = 1.5, b = 2.5 Correct answer.

Explanation

1) Differentiability at x = 1: Left derivative d/dx(ax^2 + b) = 2ax => at x = 1 is 2a. Right derivative d/dx(3x + 1) = 3. Set 2a = 3 => a = 1.5. 2) Continuity at x = 1: a(1)^2 + b = 3(1) + 1 => 1.5 + b = 4 => b = 2.5.

Question 23Análisis, límites, derivadas y optimización

Find the horizontal asymptote of the rational function f(x) = (5x^2 - 3) / (2x^2 + 7x + 1).

  1. y = 5/2 Correct answer.
  2. y = 0
  3. y = 5
  4. No horizontal asymptote

Explanation

As x -> +-inf, the ratio of leading terms is (5x^2) / (2x^2) = 5/2. Therefore, the horizontal asymptote is y = 5/2.

Question 24Análisis, límites, derivadas y optimización

Find the local minimum point of f(x) = x^3 - 3x^2 - 9x + 2.

  1. (-1, 7)
  2. (3, -25) Correct answer.
  3. (0, 2)
  4. (1, -9)

Explanation

1) f'(x) = 3x^2 - 6x - 9 = 3(x^2 - 2x - 3) = 3(x - 3)(x + 1) = 0 => x = 3, x = -1. 2) Second derivative: f''(x) = 6x - 6. For x = 3: f''(3) = 12 > 0 (Local Minimum). 3) f(3) = 3^3 - 3(3^2) - 9(3) + 2 = 27 - 27 - 27 + 2 = -25. Local minimum point is (3, -25).

Question 25Análisis, límites, derivadas y optimización

Evaluate the indefinite integral int (3x^2 + 2) / (x^3 + 2x - 5) dx.

  1. (3x^2 + 2)^2 / 2 + C
  2. 3 ln|x^3 + 2x - 5| + C
  3. ln|x^3 + 2x - 5| + C Correct answer.
  4. 1 / (x^3 + 2x - 5)^2 + C

Explanation

Notice that numerator is the exact derivative of denominator: d/dx(x^3 + 2x - 5) = 3x^2 + 2. Using standard form int (u'/u) dx = ln|u| + C: Result = ln|x^3 + 2x - 5| + C.

Question 26Análisis, límites, derivadas y optimización

A factory has a marginal cost function MC(x) = 6x + 20 and fixed costs of 100 euros. Find the total cost function C(x).

  1. 6x^2 + 20x + 100
  2. 3x^2 + 20x
  3. 3x^2 + 100x + 20
  4. 3x^2 + 20x + 100 Correct answer.

Explanation

C(x) = int MC(x) dx = int (6x + 20) dx = 3x^2 + 20x + C_0. Since fixed costs C(0) = 100, C_0 = 100. Thus C(x) = 3x^2 + 20x + 100.

Question 27Análisis, límites, derivadas y optimización

Evaluate the definite integral int_0^1 (e^(2x) + x) dx.

  1. 0.5 e^2 Correct answer.
  2. e^2 - 1
  3. 0.5 e^2 + 1
  4. e^2

Explanation

Antiderivative F(x) = 0.5 e^(2x) + 0.5 x^2. Evaluate at upper limit 1: F(1) = 0.5 e^2 + 0.5(1)^2 = 0.5 e^2 + 0.5. Evaluate at lower limit 0: F(0) = 0.5 e^0 + 0 = 0.5. F(1) - F(0) = (0.5 e^2 + 0.5) - 0.5 = 0.5 e^2.

Question 28Análisis, límites, derivadas y optimización

A demand function for a product is p(x) = 120 - 2x euros per unit. What unit price p maximizes total revenue R(x) = p(x) * x?

  1. €30
  2. €60 Correct answer.
  3. €90
  4. €120

Explanation

Revenue function: R(x) = (120 - 2x) * x = 120x - 2x^2. Set R'(x) = 0: R'(x) = 120 - 4x = 0 => 4x = 120 => x = 30 units. Substitute x = 30 into price function: p(30) = 120 - 2(30) = 120 - 60 = €60.

Question 29Análisis, límites, derivadas y optimización

The cubic function f(x) = x^3 + a x^2 + b x + 4 has a local extremum at the point (1, 2). Find the values of parameters 'a' and 'b'.

  1. a = -1, b = -2
  2. a = -3, b = 0
  3. a = 0, b = -3 Correct answer.
  4. a = 1, b = -4

Explanation

1) Point condition f(1) = 2: 1^3 + a(1^2) + b(1) + 4 = 2 => 1 + a + b + 4 = 2 => a + b = -3. 2) Extremum condition f'(1) = 0: f'(x) = 3x^2 + 2ax + b => f'(1) = 3 + 2a + b = 0 => 2a + b = -3. Subtract equations: (2a + b) - (a + b) = -3 - (-3) => a = 0. Substitute a = 0 into a + b = -3 => b = -3. Thus, a = 0 and b = -3.

Question 30Análisis, límites, derivadas y optimización

Calculate the total area enclosed between the curves f(x) = x^3 - 3x and g(x) = x.

  1. 4
  2. 16
  3. 6
  4. 8 Correct answer.

Explanation

1) Intersection points: x^3 - 3x = x => x^3 - 4x = 0 => x(x - 2)(x + 2) = 0 => x = -2, 0, 2. 2) By symmetry, Area = 2 * int_0^2 [x - (x^3 - 3x)] dx = 2 * int_0^2 (4x - x^3) dx. 3) 2 * [2x^2 - x^4/4]_0^2 = 2 * [2(4) - 16/4] = 2 * [8 - 4] = 2 * 4 = 8 square units.

Question 31Análisis, límites, derivadas y optimización

Find the absolute minimum and absolute maximum values of f(x) = 2x^3 - 9x^2 + 12x + 1 on the closed interval [0, 3].

  1. Absolute min = 1, Absolute max = 10 Correct answer.
  2. Absolute min = 5, Absolute max = 6
  3. Absolute min = 1, Absolute max = 6
  4. Absolute min = 0, Absolute max = 3

Explanation

1) Critical points inside (0, 3): f'(x) = 6x^2 - 18x + 12 = 6(x^2 - 3x + 2) = 6(x - 1)(x - 2) = 0 => x = 1, 2. 2) Candidate values: f(0) = 1 f(1) = 2(1) - 9(1) + 12(1) + 1 = 6 f(2) = 2(8) - 9(4) + 12(2) + 1 = 16 - 36 + 24 + 1 = 5 f(3) = 2(27) - 9(9) + 12(3) + 1 = 54 - 81 + 36 + 1 = 10. Absolute minimum is 1 (at x = 0) and absolute maximum is 10 (at x = 3).

Question 32Análisis, límites, derivadas y optimización

Compute the indefinite integral int x * e^(3x) dx using integration by parts.

  1. (x - 1) e^(3x) / 3 + C
  2. (3x - 1) e^(3x) / 9 + C Correct answer.
  3. x e^(3x) / 3 + C
  4. (x^2 / 2) e^(3x) / 3 + C

Explanation

Formula int u dv = u v - int v du. Let u = x => du = dx. Let dv = e^(3x) dx => v = (1/3) e^(3x). int x e^(3x) dx = (x/3) e^(3x) - int (1/3) e^(3x) dx = (x/3) e^(3x) - (1/9) e^(3x) + C = (3x - 1) e^(3x) / 9 + C.

Question 33Análisis, límites, derivadas y optimización

An open box is constructed from a 12 cm by 12 cm square piece of cardboard by cutting equal squares of side length x from each corner and folding up the sides. What side length x maximizes the volume V(x) = x (12 - 2x)^2?

  1. x = 3 cm, V = 108 cm^3
  2. x = 4 cm, V = 64 cm^3
  3. x = 2 cm, V = 128 cm^3 Correct answer.
  4. x = 1 cm, V = 100 cm^3

Explanation

1) V(x) = x(144 - 48x + 4x^2) = 4x^3 - 48x^2 + 144x. 2) V'(x) = 12x^2 - 96x + 144 = 12(x^2 - 8x + 12) = 12(x - 2)(x - 6) = 0 => x = 2 or x = 6. 3) Since x < 6 for a valid box, x = 2 cm. 4) Max volume V(2) = 2(12 - 4)^2 = 2(64) = 128 cm^3.

Question 34Análisis, límites, derivadas y optimización

Find parameters 'a' and 'b' so that the 3-branch function f(x) is continuous everywhere: f(x) = x + 1 for x < 0, a x + b for 0 <= x <= 2, and x^2 - 1 for x > 2.

  1. a = 2, b = 1
  2. a = 1, b = 0
  3. a = 0, b = 1
  4. a = 1, b = 1 Correct answer.

Explanation

1) Continuity at x = 0: lim_(x->0-) (x + 1) = 1. Value at 0 is a(0) + b = b. Set b = 1. 2) Continuity at x = 2: Value at 2 is a(2) + b = 2a + 1. Right limit lim_(x->2+) (x^2 - 1) = 3. Set 2a + 1 = 3 => 2a = 2 => a = 1. Thus a = 1 and b = 1.

Question 35Análisis, límites, derivadas y optimización

Evaluate the definite integral int_0^1 (1 / (x + 1) + 2x) dx.

  1. ln(2) + 1 Correct answer.
  2. ln(2)
  3. 2 ln(2)
  4. ln(2) + 2

Explanation

Antiderivative F(x) = ln|x + 1| + x^2. Evaluate at upper limit 1: F(1) = ln(2) + 1^2 = ln(2) + 1. Evaluate at lower limit 0: F(0) = ln(1) + 0^2 = 0. F(1) - F(0) = ln(2) + 1.

Question 36Probabilidad, distribuciones e inferencia

Given two events A and B with P(A) = 0.6, P(B) = 0.5, and P(A n B) = 0.3, find P(A u B).

  1. 1.1
  2. 0.8 Correct answer.
  3. 0.7
  4. 0.5

Explanation

By the inclusion-exclusion principle: P(A u B) = P(A) + P(B) - P(A n B) = 0.6 + 0.5 - 0.3 = 0.8.

Question 37Probabilidad, distribuciones e inferencia

If the probability of event A occurring is P(A) = 0.35, what is the probability of its complement P(A')?

  1. 0.35
  2. 0.55
  3. 0.65 Correct answer.
  4. 0.75

Explanation

The sum of probabilities of an event and its complement is 1: P(A') = 1 - P(A) = 1 - 0.35 = 0.65.

Question 38Probabilidad, distribuciones e inferencia

If events A and B are independent with P(A) = 0.4 and P(B) = 0.5, find P(A n B).

  1. 0.90
  2. 0.10
  3. 0.30
  4. 0.20 Correct answer.

Explanation

For independent events, P(A n B) = P(A) * P(B) = 0.4 * 0.5 = 0.20.

Question 39Probabilidad, distribuciones e inferencia

Given P(A n B) = 0.18 and P(B) = 0.60, calculate the conditional probability P(A|B).

  1. 0.30 Correct answer.
  2. 0.108
  3. 0.42
  4. 0.78

Explanation

By definition of conditional probability: P(A|B) = P(A n B) / P(B) = 0.18 / 0.60 = 0.30.

Question 40Probabilidad, distribuciones e inferencia

In how many different orders can 5 candidates be listed on an election ballot?

  1. 25
  2. 120 Correct answer.
  3. 60
  4. 24

Explanation

The number of permutations of 5 distinct items is 5!: 5! = 5 * 4 * 3 * 2 * 1 = 120 ways.

Question 41Probabilidad, distribuciones e inferencia

How many different 3-person committees can be selected from a group of 8 people?

  1. 336
  2. 24
  3. 56 Correct answer.
  4. 112

Explanation

Since committee member order does not matter, use combinations C(n, k): C(8, 3) = 8! / (3! * 5!) = (8 * 7 * 6) / (3 * 2 * 1) = 336 / 6 = 56.

Question 42Probabilidad, distribuciones e inferencia

For a standard normal random variable Z ~ N(0,1), find P(Z <= 1.96).

  1. 0.9500
  2. 0.9800
  3. 0.9900
  4. 0.9750 Correct answer.

Explanation

Using standard normal distribution tables, the cumulative probability up to Z = 1.96 is Phi(1.96) = 0.9750.

Question 43Probabilidad, distribuciones e inferencia

For Z ~ N(0,1), if P(Z <= 1.645) = 0.9500, find P(Z >= 1.645).

  1. 0.0500 Correct answer.
  2. 0.0250
  3. 0.1000
  4. 0.9500

Explanation

By the complement rule: P(Z >= 1.645) = 1 - P(Z <= 1.645) = 1 - 0.9500 = 0.0500.

Question 44Probabilidad, distribuciones e inferencia

A normal distribution has mean mu = 100 and standard deviation sigma = 15. Standardize X = 130 into a Z-score.

  1. 1.5
  2. 2.0 Correct answer.
  3. 2.5
  4. 3.0

Explanation

Z-score formula Z = (X - mu) / sigma: Z = (130 - 100) / 15 = 30 / 15 = 2.0.

Question 45Probabilidad, distribuciones e inferencia

A population has standard deviation sigma = 20. If random samples of size n = 100 are taken, calculate the standard error of the sample mean sigma_xbar.

  1. 0.2
  2. 5
  3. 2 Correct answer.
  4. 20

Explanation

Standard error of the sample mean formula: sigma_xbar = sigma / sqrt(n) = 20 / sqrt(100) = 20 / 10 = 2.

Question 46Probabilidad, distribuciones e inferencia

Find the margin of error E for a 95% confidence interval of the mean (z = 1.96) when population standard deviation sigma = 10 and sample size n = 100.

  1. 0.196
  2. 19.6
  3. 0.98
  4. 1.96 Correct answer.

Explanation

Margin of error E = z * (sigma / sqrt(n)) = 1.96 * (10 / sqrt(100)) = 1.96 * (10 / 10) = 1.96.

Question 47Probabilidad, distribuciones e inferencia

In a factory, Machine A produces 60% of items with a 2% defect rate, and Machine B produces 40% with a 5% defect rate. What is the overall probability P(D) that a randomly chosen item is defective?

  1. 0.032 Correct answer.
  2. 0.035
  3. 0.070
  4. 0.028

Explanation

By the Total Probability Theorem: P(D) = P(A) * P(D|A) + P(B) * P(D|B) = (0.60 * 0.02) + (0.40 * 0.05) = 0.012 + 0.020 = 0.032 (3.2%).

Question 48Probabilidad, distribuciones e inferencia

Using the data from Q077 (Machine A: 60%, 2% defect; Machine B: 40%, 5% defect; total P(D) = 0.032), if a randomly selected item is defective, what is the probability P(B|D) that it came from Machine B?

  1. 0.375
  2. 0.625 Correct answer.
  3. 0.500
  4. 0.400

Explanation

By Bayes' Theorem: P(B|D) = P(B n D) / P(D) = (0.40 * 0.05) / 0.032 = 0.020 / 0.032 = 5 / 8 = 0.625 (62.5%).

Question 49Probabilidad, distribuciones e inferencia

If test scores follow a normal distribution X ~ N(mu = 70, sigma = 10), find P(60 <= X <= 80). Given Phi(1.0) = 0.8413.

  1. 0.9544
  2. 0.5000
  3. 0.6826 Correct answer.
  4. 0.3413

Explanation

Standardize bounds: Z1 = (60 - 70) / 10 = -1.0 Z2 = (80 - 70) / 10 = +1.0. P(-1.0 <= Z <= 1.0) = Phi(1.0) - Phi(-1.0) = 0.8413 - (1 - 0.8413) = 0.8413 - 0.1587 = 0.6826.

Question 50Probabilidad, distribuciones e inferencia

For a normal distribution X ~ N(mu = 50, sigma = 8), calculate P(X >= 66). Given Phi(2.0) = 0.9772.

  1. 0.9772
  2. 0.0456
  3. 0.1587
  4. 0.0228 Correct answer.

Explanation

Standardize X = 66: Z = (66 - 50) / 8 = 16 / 8 = 2.0. P(X >= 66) = P(Z >= 2.0) = 1 - Phi(2.0) = 1 - 0.9772 = 0.0228.