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Which of the following statements are true of the graph of $f$ below?www.apstudy.net I. $f' \geq 0$ on the interval from D to F II. $f'' = 0$ at points B, D, and F III. $f'' > 0$ on the interval from A to B IV. $f'' > 0$ on the interval from D to F A. I and II B. I and III C. II and IV D. II, III, and IV
A： Statement I is correct since $f$ is strictly increasing on the interval from D to F. Statement II is true since there is a horizontal tangent to the curve at points B, D, and F, implying $f' = 0$. Statement III is false since $f$ is concave downward on the interval from A to B. Statement IV is false since $f$ is concave upward only on the interval from D to E. The function is concave downward from E to F, so $f'' < 0$ for that interval.
Math/101
Math
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Given the table below, find the slope of the tangent line of $\frac{f(x)}{g(x)}$ at the point $x = 2$. A. $\frac{5}{7}$ B. $\frac{19}{9}$ C. $-\frac{55}{9}$ D. Undefined
C: At $x = 2$, $\left(\frac{f}{g}\right)'$ is $\frac{f'(2)g(2) - f(2)g'(2)}{(g(2))^2}$, which equals $\frac{5(3) - 10(7)}{9}$, which simplifies to $-\frac{55}{9}$.
Math/102
Math
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Find the area of the region bounded by the graphs of $f(x) = x^3$ and $g(x) = 3x^2 - 2x$ shown below.The graph of the acceleration function of a moving particle is shown below. On what intervals in $[0, 20]$ does the particle have a positive change in velocity, and what is the change in velocity of the particle over the entire interval $[0, 20]$? A. $[5, 15]$ and $-\frac{25}{2}$ m/s B. $[5, 15]$ and $\frac{25}{2}$ m/s C. $[0, 10]$ and $1$ m/s D. $[0, 10]$ and $\frac{125}{2}$ m/s
A: Recall that $a(t) = \int v(t) dt$ and integrals represent net change, so the area under the curve represents change in velocity. This area is positive for $[5, 15]$. To find the total change in velocity you calculate the area under the curve. Area above the x-axis is positive, and area below the x-axis is negative. Total change in velocity is $= -(5)(10) \left(\frac{1}{2}\right) + (10)(5) \left(\frac{1}{2}\right) - (5)(5) \left(\frac{1}{2}\right) = -\frac{25}{2}$ m/s.
Math/103
Math
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The graph of the acceleration function of a moving particle is shown below. On what intervals in $[0, 20]$ does the particle have a positive change in velocity, and what is the change in velocity of the particle over the entire interval $[0, 20]$? A. $[5, 15]$ and $-\frac{25}{2} \text{ m/s}$ B. $[5, 15]$ and $\frac{25}{2} \text{ m/s}$ C. $[0, 10]$ and $1 \text{ m/s}$ D. $[0, 10]$ and $\frac{125}{2} \text{ m/s}$
A: Recall that $a(t) = \int v(t) , dt$ and integrals represent net change, so the area under the curve represents change in velocity. This area is positive for $[5, 15]$. To find the total change in velocity you calculate the area under the curve. Area above the x-axis is positive, and area below the x-axis is negative. Total change in velocity is $= -(5)(10) \left(\frac{1}{2}\right) + (10)(5) \left(\frac{1}{2}\right) - (5)(5) \left(\frac{1}{2}\right) = -\frac{25}{2} \text{ m/s}$.
Math/104
Math
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A particle is moving along a straight line, and its acceleration is shown below. Assuming the particle started at rest, calculate the speed of the particle after 3 s. A. $-3$ m/s B. $\frac{3}{2}$ m/s C. $0$ m/s D. $3$ m/s
E: The integral of acceleration is velocity. The area under the curve in the interval $[0,3]$ is triangular and below the x-axis: $-\left(\frac{3}{2}\cdot 2.5\right) = -3$ m/s Since speed is the absolute value of velocity, the correct answer is $3$ m/s.
Math/105
Math
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The graph of the function f is shown above. For how many values of x in the open interval (-3, 6) is f discontinuous? A. two B. three C. four D. five
B A function is continuous at a value of x (e.g., x = 2) if https://img.apstudy.net/ap/calculus-bc/kp21/390ef1efbadf4286bccf10941a7dbb44.png . Graphically, this means that a function is continuous if you can draw its graph without lifting up your pencil. Here, you would have to lift your pencil at x = -1.5, at x = 3, and again at x = 4. Thus, the function is not continuous (and therefore discontinuous) for three values of x in the interval (-3, 6), making (B) the correct answer.
Math/106
Math
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Let g be defined as shown above. If g is continuous for all real numbers, what is the value of k ? A. -6 B. -10 C. 0 D. 4
A You are given a piecewise function g and told that it is continuous for all real numbers. Both individual "pieces" of the function are continuous on their own, so all you need to check is where the function breaks, which is at x = 3. To be continuous at x = 3, https://img.apstudy.net/ap/calculus-bc/kp21/3533b661ae60464eb24f291bb8f73ee5.png must equal https://img.apstudy.net/ap/calculus-bc/kp21/6d035923cc8d461493a40c448da2968a.png
Math/107
Math
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Part of the graph of f on the closed interval [-5, 5] is shown above. If f is discontinuous for only one value of x in the open interval (-5, 5), then what must f(1) equal? A. -4 B. 4 C. f(1) can take on any value. D. f(1) cannot be determined.
A The graph shown has a gap at x = -2 and a hole at x = 1. For there to be only one discontinuity, one of the apparent discontinuities needs to be continuous. It's not possible to make the gap continuous, but if f(1) = -4, the discontinuity is eliminated, leaving f with only one discontinuity. Thus, (A) is correct.
Math/108
Math
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A. 0 B. 10 C. 25 D. nonexistent
B Memorize the limit definition of the derivative before Test Day: https://img.apstudy.net/ap/calculus-bc/kp21/e7acb89e5c1040249ad62f7f9e17b1a4.png . Notice that both terms in the numerator are being squared, so https://img.apstudy.net/ap/calculus-bc/kp21/5bdc8a4e04ee4f4389deaf873a8c8eb3.png (B) is correct. Note that you could also use L'Hôpital's rule to find the limit.
Math/109
Math
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The graph of f(x) is given above. Which of the following statements is true about f"(x) ? A. f"(x) > 0 for all values of x B. f"(x) < 0 for all values of x C. f"(x) < 0 for -2 < x < -1 and f"(x) > 0 for -1 < x < 1 and x > 2 D. f"(x) > 0 for -2 < x < 0 and x > 2, and f"(x) < 0 for 0 < x < 2
D This question is all about knowing what the second derivative tells you about a graph: f"(x) > 0 means the second derivative is positive, which tells you the graph of f is concave up (a cup that opens up). Similarly, f"(x) < 0 means the second derivative is negative, which tells you the graph of f is concave down (a cup that opens down). The concavity of this graph changes, so eliminate (A) and (B). The graph is concave up in the range -2 < x < 0 and x > 2, and concave down in the range 0 < x < 2, so (D) is correct.
Math/110
Math
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Consider the graph of f(x) shown above. Which of the following statements is true about f ? A. f"(x) > 0 for 1 < x < 2 B. f"(x) < 0 for 1 < x < 2 C. f'(x) > 0 for 0 < x < 1 and 2 < x < 3 D. f'(x) < 0 for 0 < x < 1 and 2 < x < 3
A This question is all about knowing what the first and second derivatives tell you about a graph. Examine each statement, one at a time. Cross out false statements as you go. (A): The given graph is of the function itself. If f"(x) > 0, the second derivative is positive, and the graph is concave up (a cup that opens up). This graph is indeed concave up for 1 < x < 2, so (A) is correct. There is no need to check the other statements unless you're not sure of your answer, but just in case you're curious ... (B): f"(x) < 0 means the graph is concave down (a cup that opens down). This graph is concave up, not down, between 1 < x < 2, so this statement is false. (C): f'(x) > 0 means the graph is increasing. This graph increases for approximately the first half of the interval 0 < x < 1, but then decreases for the second half of the interval, so this statement is false. (D): f'(x) < 0 means the graph is decreasing. This graph increases for approximately the first half of the interval 0 < x < 1, so this statement is false.
Math/111
Math
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The graph of a piecewise-defined function f is shown above. The graph has a vertical tangent at x = 3 and horizontal tangent lines at x = 1.5 and x = 4.5. What are all values of x, for -4 < x < 6, at which f is continuous but not differentiable? A. x = 0 B. x = -2 and x = 3 C. x = 1.5 and x = 4.5 D. x = -2, x = 0, and x = 3
B The question is asking for points at which f is continuous but not differentiable. That means you need to identify the points where you would not lift up your pencil if you were drawing the graph (so f is continuous), but where the derivative is undefined. This occurs at sharp turns or corners, like those at the vertex of an absolute value function, and at points where the tangent line is vertical. Here, you have two such points: a corner point at x = -2 and a vertical tangent at x = 3. This means (B) is the correct answer. Be careful of (D). While the function is not differentiable at x = 0, it is also not continuous there.
Math/112
Math
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The graph of a function g is shown above. At which of the following points is g'(x) positive and decreasing? A. a B. b C. c D. d
A This question is asking where the first derivative is positive and decreasing. A positive first derivative, or g'(x) > 0, is represented graphically by a positive slope. Only the points at a and c have positive slopes, so eliminate (B) and (D). For the first derivative to be decreasing, g"(x) must be negative. In other words, the graph must be concave down. This is true at a, but not at c, so (A) is the correct answer.
Math/113
Math
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The graph of y = f(x) is shown above. For which of the following values of x is f'(x) negative and increasing? A. a B. b C. c D. d
D This problem is asking where the first derivative of f is negative and increasing. A negative first derivative, or f'(x) < 0, is represented graphically by a negative slope. The points at a, c, and d have negative slopes, so eliminate (B). For the first derivative to be increasing, f"(x) must be positive. In other words, the graph must be concave up. This is true at d, but not at a or c, so (D) is the correct answer.
Math/114
Math
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The graph of the derivative of f, f′, is shown above. If f(0) = 7, what is f(6) ? A. 9 B. 12 C. 14 D. 15
D To find f(6), compute f(6) = f(0) + total change in f on 0 ≤ x ≤ 6. The area under the graph of f′ from x = 0 to x = 6 gives the total change in f on 0 ≤ x ≤ 6. Express this as the definite integral: https://img.apstudy.net/ap/calculus-bc/kp21/02e8c4b4a8dd45aa8fb2f6f81feb177e.png The definite integral https://img.apstudy.net/ap/calculus-bc/kp21/93a029f7799b4d32b4b0439c8608aa1c.png is the area under the graph of f′. To compute this area, break it into three regions, as shown below: https://img.apstudy.net/ap/calculus-bc/kp21/chapter14_ch22_figure_text_un418.png The area under the curve is therefore Area of rectangle A + Area of triangle B - Area of triangle C. Note that the area of triangle C is subtracted because the definite integral considers the area beneath the x-axis to be negative. This gives: https://img.apstudy.net/ap/calculus-bc/kp21/dfbc5d1d557a49d39339539847427da6.png Adding this to f(0) = 7 yields (D). https://img.apstudy.net/ap/calculus-bc/kp21/538c1ff8dd2a44919391339c3f1aba6d.png
Math/115
Math
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The acceleration of an airplane from the moment of liftoff (t = 0) to 20 minutes into the flight is shown above. If the speed at liftoff is 900 ft/min, what is the speed of the plane after 20 minutes? A. 610 ft/min B. 900 ft/min C. 1,510 ft/min D. 1,800 ft/min
C Think of this as an accumulation problem. The speed of the airplane 20 minutes after liftoff is the initial speed plus the change in speed on 0 ≤ t ≤ 20. The change in speed is given by the definite integral https://img.apstudy.net/ap/calculus-bc/kp21/f61cdc87ac3d436fbca818bac205266d.png Therefore: https://img.apstudy.net/ap/calculus-bc/kp21/9fb433ccddc1470abd4e1345eff78aa3.png The definite integral https://img.apstudy.net/ap/calculus-bc/kp21/4587a96ef2a449eb866dadf87b9542d4.png is the area under the acceleration curve on 0 ≤ t ≤ 20. Break this area into two regions: trapezoid A and rectangle B: https://img.apstudy.net/ap/calculus-bc/kp21/chapter14_ch22_figure_text_un421.png https://img.apstudy.net/ap/calculus-bc/kp21/06373c3cd4654018b18eaa7828668217.png Therefore, the speed at t = 20 is s(20) = 900 + 610 = 1,510 ft/min, which is (C).
Math/116
Math
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The graph of a function f is shown above. Which of the following statements about f is true? A. An infinite discontinuity occurs at x = 4. B. A jump discontinuity occurs at x = 3. C. A removable discontinuity occurs at x = 2. D. A jump discontinuity occurs at x = 1.
D Examine the discontinuity at each value of x. The graph is continuous at x = 4, so (A) is not correct. The discontinuity at x = 3 is a removable discontinuity (a hole), so (B) is not correct. The discontinuity at x = 2 is an infinite discontinuity (a vertical asymptote), so (C) is not correct. That leaves (D) as the correct choice. The discontinuity at x = 1 is a jump discontinuity because there is a gap in the graph there.
Math/117
Math
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Let g be the function given above. If g is continuous for all real numbers, what is the value of k ? A. -6 B. -3 C. 0 D. 2
A You are given a piecewise function g and told that it is continuous for all real numbers. Both individual functions, or "pieces," are continuous on their own, so all you need to check is where the function potentially splits, which is at x = 2. To be continuous at x = 2, https://img.apstudy.net/ap/calculus-bc/kp21/286a8054549e49a492a6109d34629fec.png must equal https://img.apstudy.net/ap/calculus-bc/kp21/f49b21b013d14fdeb79c48cf223216ee.png
Math/118
Math
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The graph of f(x) appears above. Which of the following could be the zeros of the derivative function f′(x) ? A. x = -2, 2.5 B. x = -3, 1, 4 C. x = -2, -1, 2.5 D. x = -4, -1, 3, 5
B The zeros of the derivative function f′(x) are the values of x at which f(x) has a horizontal tangent, in this case at the relative minimums and maximums. Here, this occurs at x = -3 (a relative minimum), x = 1 (a relative maximum), and x = 4 (a relative minimum). That's (B).
Math/119
Math
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The position of a particle, s, is given above for 0 ≤ t ≤ 10. On what interval(s) is the particle slowing down? A. 2 ≤ t ≤ 6 B. 0 ≤ t ≤ 4 C. 0 ≤ t ≤ 2 UU 4 ≤ t ≤ 6 D. 2 ≤ t ≤ 6 U 6 ≤ t ≤ 10
C A particle is slowing down when the velocity and acceleration have opposite signs. This means you need to take the derivative of s to find velocity, v, and the second derivative to find acceleration, a. The result is v(t ) = s′(t) = t2 - 8t + 12 and a(t) = v′(t) = 2t - 8. The easiest way to determine where v and a have opposite signs is to graph their equations on your calculator. You're only interested in the interval from 0 to 10, so adjust your window accordingly. The graph should look like this: https://img.apstudy.net/ap/calculus-bc/kp21/chapter23_pt1_q83_a_e.png Examine the graph to find that the velocity and acceleration have opposite signs (one is above the x-axis while the other is below) on the intervals 0 ≤ t ≤ 2 and 4 ≤ t ≤ 6, which is (C). Note that you could also find where v and a are equal to zero and where they are positive and negative by hand, but using your calculator is the more efficient method.
Math/120
Math
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The graph of a function f is given above. Based on the graph, which of the following statements is true? A. f is continuous and differentiable everywhere and f′(2) = 0. B. f is continuous everywhere and differentiable everywhere except at x = 2. C. f is discontinuous at x = 2 and f is differentiable everywhere except at x = 2. D. f is discontinuous at x = 2, f is differentiable everywhere, and f′(2) = 0.
B This question is testing your knowledge of what continuity and differentiability mean graphically. Continuity is the more intuitive concept: You don't need to take your pencil off the page to trace the graph where it is continuous. This means you can immediately eliminate (C) and (D). Next, recall that the derivative of a function at a point corresponds to the slope of the tangent line to the graph at that point. There is no tangent line to the function at the point where x = 2 (because there is a sharp corner), but there is a tangent line at every other point. From this, you can eliminate (A), leaving (B) as the correct answer. Careful-don't be fooled into thinking that f′(2) = 0 just because the graph has a minimum there. It would have to be differentiable there too, and it is not.
Math/121
Math
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The graph above shows the polar curve r = 3θ + sin θ for 0 ≤ θ ≤ π . Let R be the region bounded by the polar curve and the x-axis. What is the area of R? A. 16.804 B. 20.923 C. 56.720 D. 113.439
C The equation for finding a region bounded by a polar curve is https://img.apstudy.net/ap/calculus-bc/kp21/79539ab614b74eb89e67f41e0a480ba5.png https://img.apstudy.net/ap/calculus-bc/kp21/403a2365da3640efa865d151aca09780.png Hence, (C) is correct.
Math/122
Math
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Question: 17 What is the purpose of the chart shown below? A. To identify the standard deviation of revenue compared to profit B. To identify whether the mean profit is larger than the mode revenue C. To make a multivariate projection of sales based on revenue and profit D. To determine whether there is any correlation between revenue and profit E. To observe whether there is any causation going from revenue to profit
(D) The chart shown is a scatter plot, and its function is to show the values of up to two variables. It can also be used to observe whether there is any correlation if two variables are plotted.
Math/123
Math
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Question: 21 Describe the shape of the histogram below. A. Normal B. Bimodal C. Skewed right D. Skewed left E. Uniform
(E) Uniform describes the shape of this histogram because the bars are approximately the same height, which represents approximately equal frequencies.
Math/124
Math
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Question: 22 Two college professors are grading several papers and are required to draw histograms to show the grade distribution at the end of the semester. The test scores are graded on a range of 0 to 100 points. Which of the following claims is correct? A. The exam results graded by the first professor have a right-skewed distribution because the median is different from the standard deviation. B. The exam results graded by the second professor have a normal distribution because the mode and mean are equal. C. The exam results graded by the first professor have a normal distribution because the mean, median, and average are the same. D. The exam results for the first professor cannot be visually represented because two of the students achieved all of the points available. E. The exam results for the second professor have a right-skewed distribution because the mode and mean are larger than the median.
(C) The exam results for the first professor, as can be seen from the table, follow a normal distribution where the median, mode, and mean are all the same.
Math/125
Math
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Question: 24 Find the mean and the median in the dot plot below (n = 20). A. Mean: 12.9; median: 13 B. Mean: 12.4; median: 13 C. Mean: 12.9; median: 13.5 D. Mean: 12.9; median: 14 E. Mean: 12.4; median: 14
(C) Recall that in a dot plot, each dot represents a single data value. Thus the mean is (5 + 7 + 8 + 9 + 10 + 10 + 11 + 11 + 12 + 13 + 14 + 14 + 14 +15 + 16 + 17 + 17 +18 + 18 19)/20 = 12.9. The median is 13.5 = (13 + 14)/2, since the number of data points is even.
Math/126
Math
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Question: 25 Find the number of data values represented and the median for the stem plot below. A. n = 13; median: 36.5 B. n = 20; median: 36.56 C. n = 24; median: 32 D. n = 13; median: 34 E. n = 20; median: 34
(E) In a stem plot, each leaf represents a single value. Counting the number of leaves, there are a total of 20, so n = 20.; The median is 34 = (34 + 34)/2 since the number of data points is even.
Math/127
Math
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Question: 26 Data values represented by the bar labeled "10" in the histogram below fall into which range? A. 7.5 up to 12.5 B. 7.25 up to 12.75 C. 8.5 up to 11.5 D. 8.75 up to 11.75 E. 8.75 up to 11.25
(E) The boundaries are midway between the bar labels: left boundary: (7.5 + 10)/2 = 8.75; right boundary: (10 + 12.5)/2 = 11.25. Values represented by the bar therefore lie in the range 8.75 up to 11.25.
Math/128
Math
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Question: 27 The variable marked with x in the table below is of an unknown value. If the mean of the dataset is 15, what is the value of x? A. The value of x is 1. B. The value of x is 3. C. The value of x is 21. D. The value of x is 7.5. E. The value of x is 5.
(C) To calculate the value of x, we must understand that the mean is calculated by the sum of all of the figures divided by the number of figures. The sum of the values shown in the table is 104. As there are 6 known values and x, making a total of 7, this means that: With a simple multiplication of both sides of the equation with 7, we can conclude that: 104 + x = 105 Meaning that x has the value of 1.
Math/129
Math
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Question: 28 The variable marked with y has the smallest value of the dataset. If the range of the dataset is 23, what is the value of y and what is the mean? A. The value of y is 11, while the mean is 19. B. The value of y is 57, while the mean is 28.2. C. The value of y is 23, while the mean is 21.4. D. The value of y is 26, while the mean is 22. E. The value of y is 29, while the mean is 22.75.
(A) The range is defined as the maximum minus the minimum value of the dataset. As the largest value is 34 and y is defined as the lowest value, meaning the minimum, this means that x = 34 - 23, which is equal to 11. Calculating the mean simply requires summing all of the values and dividing them by 5, which yields a mean value of 19.
Math/130
Math
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Question: 33 Find the five-number summary from the box plot and calculate the interquartile range. A. Min: 4; Q1: 5.75; Med: 9.5; Q3: 13.75; Max: 16; IQR: 8 B. Min: 3; Q1: 5.75; Med: 10.5; Q3: 13.75; Max: 16; IQR: 8 C. Min: 3; Q1: 5.75; Med: 9.5; Q3: 13.75; Max: 15; IQR: 8 D. Min: 3; Q1: 5.75; Med: 9.5; Q3: 13.75; Max: 16; IQR: 8 E. Min: 3; Q1: 5.75; Med: 9.5; Q3: 13.75; Max: 16; IQR: 9
(D) Reading from the graph, Min: 3; Q1: 5.75; Med: 9.5; Q3: 13.75; Max: 16. Calculating the interquartile range = Q3 - Q1 = 13.75 - 5.75 = 8.
Math/131
Math
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Question: 34 Which statement is true of the dataset summarized by this five-number summary? A. There are no outliers. B. 3 is a mild outlier and 20 is an extreme outlier. C. 3 and 20 are mild outliers. D. 20 is an extreme outlier. E. 20 is a mild outlier. There are no extreme outliers.
(E) Interquartile range (IQR) is 10.5 - 4.5 = 6. To find the boundary for mild outliers, multiply the IQR by 1.5:6(1.5) = 9. Any values that are less than Q1 - 9 or greater than Q3 + 9 are considered mild outliers (but not more than 3(IQR), which is considered extreme). 20 is a mild outlier because it is greater than 10.5 + 9 = 19.5 and less than 10.5 + 18 = 28.5.
Math/132
Math
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Question: 39 The range of a dataset is 20 and x has the highest value of the variables provided. What is the value of y and the standard deviation of the dataset if rounded to two decimals? A. The value of y is 32, and the standard deviation is 7. B. The value of y is 30, and the standard deviation is 6.41. C. The value of y is 35, and the standard deviation is 7.92. D. The value of y is 20, and the standard deviation is 4.03. E. The value of y is 2, and the standard deviation is 5.76.
(B) Because the minimal value in the dataset is 10, the maximum can be obtained from the formula for the range y - 20 = 10. Through this equation, it is clear that y = 30. After calculating the mean by calculating the sum and dividing it by the number of values, we can see that the mean is 13. The standard deviation is then calculated by subtracting the mean from each value, and this new value then needs to be squared. After calculating the mean of the squared values, we see that the standard deviation is the root of this value and is 6.41.
Math/133
Math
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Question: 41 For the probability density curve below, which of the following statements is true? I. The area is exactly 1 underneath it. II. It does not model the distribution of the data. III. It is a function that is always positive. A. I only B. I and III only C. II and III only D. II only E. III only
(B) All probability density curves are mathematical models of probability distributions. Probability distributions are proportional distributions of data, and the area underneath them sums to 1. They are also always above the horizontal axis because probability is nonnegative.
Math/134
Math
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Question: 42 Why is the following not an example of a valid probability density curve? A. Because the curve does not follow a normal distribution B. Because such a function is by definition either constantly positive or negative C. Because such a function has to have at least two maximum values and one minimum value D. Because the function cannot be negative E. Because the function cannot be positive
(D) A valid probability density curve has to be nonnegative, and the area between the curve and the x-axis should be equal to 1.
Math/135
Math
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Question: 44 The following box plot illustrates the movement of two variables, A1 and B1. Which of the following assumptions is correct? A. A1 has smaller outliers compared to B1. B. The value of the median is higher for B1. C. The range of the data is larger in A1. D. The two variables have medians of the same value. E. A1 has problems with outliers that are larger compared to most of the dataset, while B1 has outliers that are smaller than most of the dataset.
(C) It is clear that the values for the dataset labeled A1 have a larger range and a larger median compared to B1; the outliers are more pronounced than in B1.
Math/136
Math
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Question: 46 The histograms below show the number of cat and dog owners based on the number of each animal owned. Choose the best description from the statements below. A. There are more cat owners than dog owners. There are more cats than dogs. B. There are more cat owners than dog owners. There are more dogs than cats. C. There are more dog owners than cat owners. There are more cats than dogs. D. There are more dog owners than cat owners. There are more dogs than cats. E. The cat owners are equal to dog owners. There are more cats than dogs.
(A) There are more cat owners than dog owners. There are more cats than dogs. This can be estimated by the relative heights of the bars labeled 1, 2, 3, 4, and 5. Notice that there are more households that own greater quantities of cats.
Math/137
Math
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Question: 52 The cumulative frequency distribution of airfares is displayed in the ogive. What airfares correspond to the 20th percentile, the 50th percentile, and the 80th percentile? A. P20 = 95; P50 = 102; P80 = 106 B. P20 = 96; P50 = 103; P80 = 106 C. P20 = 96; P50 = 102; P80 = 106 D. P20 = 96; P50 = 102; P80 = 100 E. P20 = 96; P50 = 102; P80 = 107
(C) Reading from the vertical axis the percentages and down to the horizontal axis, P20 = 96; P50 = 102; P80 = 106.
Math/138
Math
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Question: 53 The histogram represents the heights of males in the United States between the ages 20–29. The mean is 69.6 inches, and the standard deviation is 2.6 inches. Since the heights are approximately normal, find the height that represents the 20th percentile. How should this be interpreted? A. 60 inches. 20% of the population of males between the ages 20–29 are shorter than approximately 60 inches. B. 68.6 inches. 20% of the population of males between the ages 20–29 are shorter than approximately 68 inches. C. 67.4 inches. 20% of the population of males between the ages 20–29 are shorter than approximately 68 inches. D. 67.4 inches. 20 males between the ages 20–29 are shorter than approximately 67 inches. E. 60 inches. 20 males between the ages 20–29 are shorter than approximately 60 inches tall.
(C) The z-score associated with 20% to the left of it is -0.84. To find the height associated with that z-score: z σ + μ = - 0.84(2.6) + 69.6 = 67.4. That means that 20% of the population is shorter than 68 inches tall.
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Question: 54 The histogram represents the heights of males in the United States between the ages 20–29. The mean is 69.6 inches, and the standard deviation is 2.6 inches (approximately normal). Three adult males are selected from the group. Their heights are 75 inches, 63 inches, and 79 inches. Determine which heights, if any, are unusual. A. The 79-inch height is the only one that is unusual because it is more than three standard deviations above the mean. The z-scores, respectively, are: 2.08, –2.54, and 3.62. B. The 79-inch and 63-inch heights are the only ones that are unusual because they are more than two standard deviations from the mean. The z-scores, respectively, are: 2.00, –2.54, and 3.62. C. All heights are unusual since they are more than two standard deviations from the mean. The z-scores, respectively, are: –2.08, –2.54, and 3.62. D. All heights are unusual since they are more than two standard deviations from the mean. The z-scores, respectively, are: 2.08, –2.54, and 3.62. E. All heights are unusual since they are more than two standard deviations from the mean. The z-scores, respectively, are: 2.08, 2.54, and 3.62.
(D) http://img.apstudy.net/ap/statistics/a500/Ans_Eq_015.jpg Calculating for each height: 2.08, -2.54, and 3.62, respectively, shows that all are greater than two standard deviations from the mean.
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Question: 55 The dataset shown below has a mean value of 15. Based on this information, which of the following correctly identifies the unknown value x and the range of the dataset? A. The value of x is 3, and the range is 27. B. The value of x is 9, and the range is 25. C. The value of x is 5, and the range is 25. D. The value of x is 11, and the range is 19. E. The value of x is 18, and the range is 10.
(B) By understanding that the mean is the sum of all of the elements divided by the number of the elements, it is possible to determine the value of x. The sum of the values shown in the table is 81. As there are 5 known values and x, making a total of 6, this means that: which when multiplying both sides of the equation leads to: 81 + x = 90 meaning that x has a value of 9. Because the range is the maximum minus the minimum, with the largest value being 30 and the smallest 5, the range is 25.
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Question: 58 The frequency distribution shows the number of magazine subscriptions per household in a random sample (n = 50). Find the mean number of subscriptions per household. A. Mean is 2.0 B. Mean is 1.8 C. Mean is 1.5 D. Mean is 3.0 E. Mean is 92
(B) http://img.apstudy.net/ap/statistics/a500/Ans_Eq_019.jpg
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Question: 59 The table below shows the correct values of the mean and the standard deviation for four different datasets. Which of the following assumptions is most likely accurate? A. Dataset A has, on average, the smallest number of values that are different from the mean compared to the other datasets in the table. B. The maximum value of dataset B is 17.3. C. The maximum value of dataset D is 71.57. D. Dataset C has, on average, the smallest number of values that are different from the mean compared to the other datasets in the table. E. Dataset A has a variance of 2.53.
(D) Because the standard deviation shows the dispersion of data within a certain dataset, it is most likely that the dataset with the smallest standard deviation compared to the size of the mean will, on average, be the dataset with the most consistent values. Answer choices (B) and (C) show incorrect calculations of the range, which is not calculated in such a manner. Answer choice (E) is not correct because variance is standard deviation squared, not the root of the standard deviation.
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Question: 63 Five students had five tests in one semester, and every test was graded on a scale of 0 to 20 points. Based on information from the table, which of the following students had the largest variance in his or her results, and which student had the largest mean value? A. James had the largest variance, while Mike had the largest mean score. B. Mike had the largest mean score and the largest variance. C. Tina had the largest variance, while James had the highest mean score. D. Tina had the largest variance, while Mark had the largest mean score. E. Angela had the largest variance, while Mike had the largest mean score.
(A) The dataset of the test scores for James had by far the highest variance, with a value of 40.3, because his results had the largest dispersion of data. Mike had the largest mean score because the sum of his results divided by the number of tests, which was 5, was the highest, with a mean score of 14.8.
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Question: 65 What data point is indicated on the distribution? A. 61.9 B. 64.5 C. 67.0 D. 72.1 E. 77.2
(D) The data point is at the point of inflection on the bell curve which is one standard deviation above the mean or X = 67 + 5.1 = 72.1.
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Question: 66 Which of the following statements best describes the scatter plot? I. A linear model would fit the data well. II. The bivariate data are positively correlated. III. The linear relationship is weak A. I only B. I and III only C. I and II only D. II only E. III only
(C) The data points are trending linearly with a positive slope.
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Question: 67 The scatter plot of study time in hours versus test scores in a statistics class is shown below. What can be said about the relationship between time spent studying and test scores? A. More time spent studying is related to lower test scores. B. Less time spent studying is related to higher test scores. C. More time spent studying is related to higher test scores. D. Higher test scores are caused by more time spent studying. E. There is no discernible relationship between study time and test scores.
(C) The more time spent studying is related to a higher test scores since there is a positive linear relationship.
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Question: 70 Why does Chart 1 show a better fitting line than Chart 2? A. Because it has a negative correlation B. Because it shows a positive correlation C. Because it shows proof of causality D. Because it has a lower value of the sum of squares E. Because the value of the slope is negative
(D) The best-fitting line is the least-squares regression line. The line in Chart 1 has a sum of squares of 3.6, which is smaller than 4.171.
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Question: 71 Based on the data shown in the chart below, what is the most likely correlation between the two datasets shown? A. There is a correlation between the variables lower than -0.9. B. There is a correlation between the variables of -0.7. C. There is a correlation between the variables that is close to 0. D. There is a correlation between the variables that is around 0.65. E. There is a correlation between the variables higher than 0.9.
(E) As can be seen, the two datasets tend to move in a highly similar manner. This indicates that there is a very strong positive correlation; the actual correlation between the two datasets is 0.97.
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Question: 72 Why does the following dataset most likely have a correlation of roughly 0.5? A. Because the two datasets move in opposite directions B. Because there is no connection between the datasets C. Because, at certain times, the increase in the value of one dataset is associated with the increase of the value of the second dataset D. Because, at certain times, as the value of one dataset increases, the other significantly decreases E. Because it is clear that the change of one variable causes a change in the other variable
(C) The connection between the two datasets indicates that there is a positive correlation of roughly 0.5. This indicates that as the value of one variable increases, the value of the other variable is expected to increase as well.
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Question: 74 In the scatter plot, what effect does the indicated point have on the correlation coefficient and the slope of the least-squares regression line? A. The point is influential; r decreases and the slope increases. B. The point is influential; r decreases and the slope decreases. C. The point is influential; r decreases and the slope is unaffected. D. The point is not influential; therefore, r and the slope are unaffected. E. The point is influential; therefore, r and the slope are unaffected.
(B) The point is influential; therefore, r decreases and slope decreases. Since this point is an outlier, r is reduced; because the outlier is above and to the left of the center of the data, the slope is tilted and flattened (smaller positive value).
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Question: 75 In the scatter plot, how could the data be transformed in order to do a linear regression? A. By squaring each data point y-value B. By taking the natural logarithm of each data point y-value C. By raising each data point y-value to the base 10 D. By taking the square root of each data point y-value E. By cubing each data point y-value
(D) The appropriate regression would be quadratic since the points are in a parabolic shape. Taking the square root of each data point y-value would be the appropriate transformation in order to perform a linear regression.
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Question: 77 Using the dataset, what can be said about the relationship between earnings and dividends? A. More earnings are related to more dividends. B. There is no discernible relationship between earnings and dividends. C. More earnings are related to less dividends. D. More dividends are caused by more earnings. E. Fewer earnings are related to more dividends.
(B) The scatter plot and coefficient of determination indicate that earnings and dividends are unrelated. http://img.apstudy.net/ap/statistics/a500/03xAns82b.jpg Regression Analysis: Dividends versus Earnings The regression equation is: Dividends = 0.949 + 0.258 Earnings http://img.apstudy.net/ap/statistics/a500/f0234-02.jpg
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Question: 79 From the regression output below, which type of variation does R-squared (R-Sq) pertain to? A. Total variation B. Unexplained variation C. Explained variation D. Least squares variation E. Error variation
(C) R-squared is a measure of the explained variation in the response variable by the predictor variable.
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Question: 81 Find the slope from the least-squares regression line for the data below of final grade as a percent and days absent. What does it tell us about the relationship? A. Slope is approximately 94.695. For each additional day absent, the student's final grade increases by about 0.947%. B. Slope is approximately -94.695. For each additional day absent, the student's final grade decreases by about 0.947%. C. Slope is approximately 6.767. For each additional day absent, the student's final grade increases by about 6.767%. D. Slope is approximately 6.767. For each additional day absent, the student's final grade decreases by about 6.767%. E. Slope is approximately -6.767. For each additional day absent, the student's final grade decreases by about 6.767%.
(E) Slope is approximately -6.767 from the linear regression model for these data http://img.apstudy.net/ap/statistics/a500/Ans_Eq_039.jpg . For each day additional absent, the student's final grade decreases by about 6.767%.
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Question: 84 Compare the two scatter plots. Which of the following statements is true about r ? A. r1 indicates a highly negative correlation, while r2 indicates a moderate positive correlation. B. The r value shows that there is causation between the datasets. C. r2 indicates a stronger correlation than r1. D. r cannot be used to find the percentage of variation that has been explained. E. The r value in this example is sensitive to outliers.
(E) Due to the significant decrease from r1 to r2, it is clear that the value of r in this case is highly sensitive to outliers.
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Question: 87 Compare the two regression outputs. Which regression equation explains more of the variation in y ? A. Regression 1, since R-Sq, unexplained variation, is the smaller. B. Regression 2, since R-Sq, total variation, is the smaller. C. Regression 1, since R-Sq, explained variation, is the greater. D. Regression 2, since R-Sq, explained variation, is the greater. E. Regression 1, since R-Sq, total variation, is the greater.
(C) In Regression 1, R-sq, the explained variation by the predictor variable, is 81.5% while in Regression 2 the R-Sq is 80.7%. More of the variation in the response variable y is explained by x in Regression 1.
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Question: 88 Find and interpret r for the dataset. A. r = 0.0903. There is a weak positive linear correlation. B. r = -0.0903. There is a weak negative linear correlation. C. r = 0.903. There is a strong positive linear correlation. D. r = -0.903. There is a strong negative linear correlation. E. r = 0.0903. There is no linear correlation.
(D) r = -0.903. This indicates strong negative linear correlation.
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Question: 90 From the time series data, find r. What does r tell us about the regression? A. r = 0.238. There is a very weak linear relationship. B. r = -0.238. There is a very weak linear relationship. C. r = 0.488. There is a very weak positive linear relationship. D. r = -0.488. There is a very weak negative linear relationship. E. r = -0.488. There is no discernible relationship.
(C) r = 0.488 and is a measure of the strength and direction of a linear relationship; therefore, it indicates a weak positive linear relationship. The scatter plot indicates that a transformation would be appropriate before fitting a linear model to these data.
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Question: 91 Based on the residuals shown in the chart below, what kind of transformation should be applied to better fit the dataset? A. The first difference should be used. B. A square transformation should be used. C. All of the values should be divided by 2. D. The root transformation should be used. E. The values should be transformed by multiplying them with an instrumental variable.
(D) Based on the values in the chart, there are signs of a pattern that could be discovered by performing a root transformation.
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Question: 95 Compare the two datasets below. Which statements are true? I. Absences are a better predictor of the final grade. II. The sum of the residuals is the same for both. III. The unexplained variation in the final is much greater than in the midterm. A. I only B. I and III only C. I and II only D. II only E. III only
(C) The coefficients of determination are 92.4% and 48.6%, respectively. That shows that absences explain more of the variation in the final then the midterm. Because the regression is a least-squares linear regression, the sum of the residuals is always zero, the same for both.
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Question: 96 From the regression output below, calculate the coefficient of determination. What does it tell you about the relationship? A. r2 = 38.9%; that 38.9% of the variation of the residual error is explained by the regression B. r2 = 44.0%; that 44.0% of the variation of the difference of the regression and the residual error is explained by the total C. r2 = 28.0%; that 28.0% of the variation of the response variable is explained by the residuals D. r2 = 56.3%; that 56.3% of the variation of the explanatory variable is explained by the response variable E. r2 = 72.0%; that 72.0% of the variation of the response variable is explained by the explanatory variable
(E) http://img.apstudy.net/ap/statistics/a500/Ans_Eq_047.jpg or 72.0%. The coefficient of determination is a measure of the variation of the dependent (response) variable that is explained by the regression line and the independent (explanatory) variable.
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Question: 97 From the regression output below, calculate the percentage of unexplained variation in the dependent (response) variable. A. 1.8% B. 1.9% C. 96.4% D. 98.1% E. 98.2%
(E) The percentage of unexplained variation in the dependent (response) variable is http://img.apstudy.net/ap/statistics/a500/Ans_Eq_048.jpg ⋅ http://img.apstudy.net/ap/statistics/a500/Ans_Eq_049.jpg
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Question: 98 Use the dataset to find the correlation coefficient. What would be the best explanation of the strength of this relationship? A. Correlation coefficient is -0.730. That higher grades are earned by more efficient study techniques. B. Correlation coefficient is 0.730. That students expecting higher grades study more hours per week. C. Correlation coefficient is -0.730. That less time spent studying results in better grades. D. Correlation coefficient is -0.730. That a higher expected GPA predicts a lower study time. E. Correlation coefficient is 0.730. That more time spent studying results in better grades.
(D) r = -0.730. The best explanation for the relationship is that students expecting higher grades study fewer hours.
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Question: 109 What statement is true about the two models for the dataset? Correlations: Distress, Brain 1 Pearson correlation of Distress and Brain 1 = 0.878 P-Value = 0.000 A. Under both models, the correlation between the variables is the same. B. Model II is more appropriate since it has a larger residual error. C. Model I suggests a negative linear correlation between the variables. D. The residuals for the different data values will be the same under both models. E. The correlation should not be observed, as it is not statistically significant.
(A) The correlation between the variables is the same. The only difference between Model I and Model II is what the dependent variable is. Otherwise, the strength of the correlation is identical.
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Question: 165 The probability of Bill serving an ace in tennis is 0.15, and the probability that he double faults is 0.25. What is the probability that Bill does not serve an ace or a double fault? A. 0.5 B. 0.15 C. 0.4 D. 0.9 E. 0.6
(E) The probability distribution for Bill's serving in tennis is: P (not an Ace or Double Fault) = 1 - (0.15 + 0.25) = 0.6
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Question: 170 Calculate the expected value and the variance of the following probability distribution. A. E(X ) = 2.0, V(X ) = 1.4 B. E(X ) = 2.1, V(X ) = 1.0 C. E(X ) = 2.1, V(X ) = 1.2 D. E(X ) = 2.1, V(X ) = 1.4 E. E(X ) = 2.0, V(X ) = 1.0
(D) http://img.apstudy.net/ap/statistics/a500/f0247-01.jpg
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Question: 171 The probability distribution for the number of incidents occurring in a year is shown below. Calculate the expected value and the variance. A. E(X ) = 2.0, V(X ) = 2.0 B. E(X ) = 1.9, V(X ) = 1.8 C. E(X ) = 2.0, V(X ) = 1.8 D. E(X ) = 2.0, V(X ) = 1.4 E. E(X ) = 1.9, V(X ) = 1.4
(C) http://img.apstudy.net/ap/statistics/a500/f0247-02.jpg
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Question: 172 The probability distribution for the frequency of an event occurring in five trials is shown below. Calculate the expected value and the standard deviation. A. E(X ) = 1.25, SD = 1.0 B. E(X ) = 1.5, SD = 1.05 C. E(X ) = 1.5, SD = 1.1 D. E(X ) = 1.5, SD = 1.0 E. E(X ) = 1.25, SD = 1.05
(D) http://img.apstudy.net/ap/statistics/a500/f0247-03.jpg
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Question: 173 The probability distribution for the number of defects found in a sample is shown below. Calculate the expected value and the standard deviation. A. E(X ) = 0.9, SD = 0.8 B. E(X ) = 0.9, SD = 0.6 C. E(X ) = 0.9, SD = 0.7 D. E(X ) = 1.0, SD = 0.6 E. E(X ) = 1.0, SD = 0.7
(A) http://img.apstudy.net/ap/statistics/a500/f0248-01.jpg
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Question: 175 What should be the height of the missing bar in the histogram? A. 75 B. 0.7 C. 70 D. 60 E. 50
(A) The scale of the vertical axis is in percents. The sum of the heights of the bars must equal 100. Let x be the unknown bar height. Then 100 = x + 15 + 5 + 5 and solving for x, x = 75.
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Question: 176 Suppose a cell phone company manufactures cell phones and tablets in the following four states: Arizona, California, Washington, and Minnesota. The table below shows the percentage of total output by state, and within each state, the percentage output of each product. What is the probability that a randomly selected product will be a tablet manufactured in Minnesota? (Assume that all products are shipped from one distribution center.) A. 0.05 B. 0.08 C. 0.004 D. 0.5 E. 0.04
(C) P (Minnesota & tablet) = 0.05(0.08) = 0.004
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Question: 198 Students in a biomedical statistics course take two tests before a final. The statistics from the first test for three different classes (May, June, and July) are shown. What would be the average and the standard deviation of the sum of the test scores? A. Mean of the sum: 51. Standard deviation: 10.51. B. Mean of the sum: 51. Standard deviation: 5.45. C. Mean of the sum: 51. Standard deviation: 3.24. D. Mean of the sum: 51. Standard deviation: 2.33. E. Mean of the sum: 50. Standard deviation: 3.24.
(C) These classes are to be considered independent since the students are all different from class to class. With that assumption, http://img.apstudy.net/ap/statistics/a500/Ans_Eq_116.jpg and http://img.apstudy.net/ap/statistics/a500/Ans_Eq_117.jpg
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Question: 202 Given the following probability distribution, find P (1 ≤ X ≤ 4). A. 0.01 B. 0.08 C. 0.20 D. 0.58 E. 0.70
(C) Since this is a discrete random variable, the probability can be found using P(X = 1) + P(X = 2) + P(X = 3).
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Question: 204 The dataset below is based on a survey of 100 randomly selected travelers who were asked, "How many trips per year do you typically take?" What is the probability that a randomly selected survey participant is male or takes two or fewer trips per year? A. 0.17 B. 0.18 C. 0.36 D. 0.69 E. 0.86
(D) P(male or takes fewer than two trips per year) = P(male) + P(takes fewer than two trips per year) - P(male and takes fewer than two trips per year) = 0.50 + 0.36 - 0.17.
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Question: 266 In a recent online survey of 1,700 adults, it was revealed that only 17% felt the Internet was secure. With what degree of confidence can you say that 17% ± 2% of adults believe that online security is secure? A. 72.9% B. 90% C. 95% D. 97.2% E. 98.6%
(D) http://img.apstudy.net/ap/statistics/a500/f0268-03.jpg
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Question: 267 One of the students in the school council election wants to make healthy lunch menus her pitch to gain votes. What size voter sample would she need to obtain a 90% confidence that her support level margin of error is no more than 4%? A. 25 B. 600 C. 423 D. 1691 E. None of the above
(C) http://img.apstudy.net/ap/statistics/a500/f0268-04.jpg
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Question: 279 The researchers conducting an experiment have obtained the results as shown in the table below. Which of the following correctly identifies the minimum and maximum values of the 95% confidence interval? A. 14.2–20.2 B. 11.151–20.258 C. 11.21–23.21 D. 9.652–24.748 E. 8.148–25.184
(D) The sample's size is 5 and the standard deviation can be calculated; it is 6.058. If we are searching for a 95% confidence interval, this means that we need to look at the critical t-values for 4 degrees of freedom at a 5% significance level, which is 2.776. The next step is to divide the standard deviation of the sample with the root of the sample size. This means that 6.058/2.236 = 2.719. This value then needs to be multiplied with the t-value, leading to the value of 7.548. The last step is determining the value of the mean, which is 17.2, and then subtracting and adding 7.548 to get the final confidence interval.
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Question: 282 A team of researchers is testing a new drug. They have obtained the results as shown in the table below. What is the minimum and maximum value of the researchers' 95% confidence interval, rounded to one decimal? A. 8.7–22.2 B. 7.9–23.2 C. 5.4–21.4 D. 6.7–24.7 E. 3.4–28.4
(A) After we determine that the sample size is 9, we know that the degrees of freedom are 8. We need to find the value for the 5% level of statistical significance; this value is 2.306. After this, we need to determine the standard deviation, which is 8.791. By dividing the standard deviation with the root of the sample size, we find that 8.791/3 = 2.9303. By multiplying this value with the t-value, we find the value of 6.757. This value is then added to the value of the mean, which is 15.444.
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Question: 283 While weighing five infants in the hospital we found their weights in kilograms to be 3.1, 3.5, 2.8, 3.2, and 3.4. What is a 90% confidence interval estimate of the infants' weights? A. 3.2 ± 0.202 B. 3.2 ± 0.247 C. 3.2 ± 0.261 D. 4.0 ± 0.202 E. 4.0 ± 0.261
(C) df = 4 http://img.apstudy.net/ap/statistics/a500/f0270-01.jpg
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Question: 284 A poll identifies the proportions of high-income and low-income voters who support a decrease in taxes. If we need to know the answer to be within ± 0.02 at the 95% confidence interval, what sample size should be taken? A. 3383 B. 4803 C. 7503 D. 9453 E. 4503
(B) http://img.apstudy.net/ap/statistics/a500/f0270-02.jpg http://img.apstudy.net/ap/statistics/a500/f0270-03.jpg n ≥ 69.32 4802.5 and hence the poll would use the sample size of 4803.
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Question: 285 A survey conducted on 1,000 Canadians found that 700 of them refused to receive the H1N1 vaccination. Construct a 95% confidence interval of the estimated proportion of Canadians who refused to receive the vaccination. A. (0.723, 0.777) B. (0.686, 0.714) C. (0.728, 0.672) D. (0.986, 0.914) E. (0.672, 0.728)
(D) http://img.apstudy.net/ap/statistics/a500/f0270-04.jpg
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Question: 286 For a 95% confidence interval, you want to create a population proportion with a marginal error of no more than 0.025. How large of a sample would you need? A. 423 B. 1537 C. 385 D. 2237 E. 1082
(B) For our problem, we will use p* = 0.5: http://img.apstudy.net/ap/statistics/a500/f0270-05.jpg
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Question: 287 In a random sample of 100 balloons, 30 of them are blue. Construct a 95% confidence interval for the true proportion of the blue balloons. A. (0.210, 0.399) B. (0.193, 0.407) C. (0.165, 0.334) D. (0.200, 0.380) E. (0.198, 0.378)
(A) http://img.apstudy.net/ap/statistics/a500/f0270-06.jpg
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Question: 325 In a sample of 500 students from statistics courses in the fall semester, 312 earned a grade of C or higher. In the spring semester, 400 of the 525 students sampled earned a grade of C or higher. If we treat the fall semester as "sample 1" and the spring semester as "sample 2," what is the appropriate standardized test statistic to determine whether a smaller proportion of students passed in the fall? A. z = -6.699 B. z = -4.792 C. z = 4.792 D. z = 6.699 E. The z-test statistic depends on the level of significance of the test
(B) We are testing two proportions from two independent samples; therefore, the test statistic is http://img.apstudy.net/ap/statistics/a500/f0275-01.jpg This can also be found by performing a "2-prop-z-test" on the TI84 or 83 calculator as it gives both the test statistic and the p-value.
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Question: 336 A survey of randomly selected visitors to a mall yielded the following data: Based on this sample, what would be the correct decision and conclusion of a test to determine whether the proportion who approve of a new design plan for the seating area in the food court is higher for women than it is for men at the 5% level? A. Reject the null hypothesis concluding that there is evidence the proportion is higher for women. B. Do not reject the null hypothesis concluding that there is no evidence that the proportion is higher for women. C. Reject the null hypothesis concluding that there is no evidence that the proportion is higher for women. D. Do not reject the null hypothesis concluding that there is no evidence that the proportion is higher for women. E. Conclude the proportion is higher for women since 199/301 > 152/254.
(B) This is a two-sample proportion test, and the p-value can be calculated using the TI83 or 84 as 0.06345. (The test statistic is -1.526.) Since this is larger than 0.05, the null hypothesis is not rejected.
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Question: 340 Which of the following is a correct interpretation of the t-test shown below? A. There is a statistically significant difference between the means of the variables. B. There is a statistically significant difference when one is viewing the two-tailed test, but not for the one-tailed test. C. There is a weak negative correlation between the two variables. D. The results of the t-test suggest that the null hypothesis cannot be rejected. E. There is a statistically significant difference between the means of the sample.
(D) As all of the p-values of the t-tests are higher than 0.05, this indicates that it is not possible to reject the null hypothesis.
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Question: 354 A teacher is investigating the relationship between hours spent studying and exam scores (scale 0–100). Based on the computer output above, in a test of H0 : β = 0 versus Ha : β ≠ 0, which of the following would be correct? A. Reject H0 concluding that at the 5% level there is a linear relationship between hours spent studying and exam scores. B. Do not reject H0 concluding that at the 5% level there is a linear relationship between hours spent studying and exam scores. C. Reject H0 concluding that at the 5% level there is no evidence of a linear relationship between hours spent studying and exam scores. D. Do not reject H0 concluding that at the 5% level there is no evidence of a linear relationship between hours spent studying and exam scores. E. Based on the large value of the constant, conclude there is a linear relationship between hours spent studying and exam scores.
(A) Based on the computer output, the p-value is zero, which would reject the null hypothesis for all of the usual significance levels. This is strong evidence of a linear relationship between the variables.
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Question: 357 Based on a sample of 20 data values, the least-squares regression line to predict y from x is . If Sb = 0.76, find the t-test statistic for H0 : β = 0. A. 0.09 B. 0.12 C. 0.67 D. 8.44 E. 28.08
(B) The test statistic for any test is always of the form http://img.apstudy.net/ap/statistics/a500/Ans_Eq_231.jpg Since we are testing whether the slope is zero, the test statistic is http://img.apstudy.net/ap/statistics/a500/Ans_Eq_232.jpg
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Question: 363 A scientist is studying the relationship between the percentage of a particular type of nutrient in the soil and the eventual height of a plant. The regression equation is: Height = 5.31 + 3.34 Percentage Which of the following is true? A. Neither the constant nor the slope are statistically significant at the 1% significance level. B. The constant of the regression is 3.345. C. The regression can be interpreted as a 1% increase of the percentage of nutrients causing a 5.31 increase in the height of the plant. D. The lower the R-squared value, the stronger the connection between the two variables. E. The standard error of the slope is 1.316.
(E) The only correct answer is that the standard error of the slope is 1.316, because its value is 3.345.
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Question: 365 As part of a project analyzing the relationship between the square footage of homes and their final sale price in thousands of dollars, a student performed a regression analysis. The regression equation is: Sale Price = 179 + 0.0289 SqFt Based on the computer output above, what is the value for the t-test statistic to test H0 : β = 0? A. 1.53 B. 4.42 C. 3.06 D. 8.84 E. 40.52
(A) Since "square feet" is used as the independent variable, that line provides information about the slope. Thus the t-test statistic is 1.53.
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Question: 375 Consider the regression output provided below that considers the impact of height on weight. Which of the following is correct? A. The 95% confidence interval for the constant is from –0.435 to 1.695. B. The regression output suggests that an increase in height is associated with a 0.219 increase in weight. C. The model is based on the comparison of seven observations. D. The R-squared value of the model suggests that the model may not be a good fit, and this is backed up by none of the coefficients being statistically significant at the 5% level. E. This model seems to be a good fit because it minimizes the value of the adjusted R-squared and because the intercept is not close to being equal to 0.
(D) Both the value of the R-squared and the adjusted R-squared seem to indicate that the model has a limited ability to predict the value of the independent variable. As both the constant and the coefficient are not statistically relevant at the 5% level of statistical significance, it is clear that the model may not be properly specified. -0.435 to 1.695 is the confidence interval for the coefficient, not the constant.
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Question: 379 The plot below shows how well the unemployment rate predicts the result of the largest political bloc in Germany, CDU/CSU. The final equation is that votes for CDU/CSU = 0.0126 unemployment rate + 0.279. What can we infer from the results of this prediction? A. The regression line is not really accurate in predicting the results and there seem to be numerous outliers, some of which are not even captured by the 95% confidence interval. B. The regression model has a very high R-squared value, likely above 0.9, as almost all of the variables are within the 95% confidence interval. C. The fact that both the constant and the intercept have a value less than 1 means that they are both not statistically significant. D. The regression model is the best-fitting model because the intercept and the slope have similar values. E. The model is an excellent fit because all of the results are close.
(A) This model, which has an R-squared value of 0.267, clearly does not fully capture the relevance of the dependent variable. The fact that the intercept and the slope have similar values does not hold any statistical significance.
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Question: 386 Which of the following datasets was used to gain the regression output below? A. Datasets A and B B. Datasets C and D C. Datasets E and F D. Datasets G and H E. Datasets C and G
(A) Based on the value of the degrees of freedom, the only possible answers are (A) and (C). It is clear that the correlation is quite strong in (C), while the output shows a weak connection between the variables. Therefore, the correct answer is (A) because it shows the correct sample size and no strong correlation between the datasets.
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Question: 388 A researcher is trying to determine the impact of the unemployment rate on the interest rate. Based on the values shown below, what can we conclude about the model? A. Based on the value of the Durbin-Watson coefficient and the R-squared value, it is clear that there is a strong predictability value of the model. B. Due to the fact that the R-squared value is very low, the predictability value of the model is high. C. Due to the fact that the value of the constant is larger than the intercept, the model is not statistically significant. D. Despite the fact that both the constant and the intercept are statistically significant at the 5% level, the predictability value of the model is low. E. Despite the fact that the value of the Akaike criterion is high, the low value of the Durbin-Watson statistic indicates that the model is stable.
(D) Both the constant and the slope are statistically relevant at the 5% significance level, but the R-squared value indicates that the model does not have a high predictability value.
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Question: 399 A student is studying the relationship between the number of hours students work and their final exam scores in statistics courses. A regression analysis resulted in the computer output below. What is the p-value that should be used to test H0 : β = 0 versus Ha : β > 0? A. 0 B. 0.146 C. 0.292 D. 0.708 E. 0.854
(B) The predictor, or independent variable, in this situation is "Hours." On the computer output, the p-value for this test can be found on the line for "Hours."
Math/196
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Question: 400 A student is studying the relationship between the number of hours students work and their final exam scores in statistics courses. A regression analysis resulted in the computer output below. What is the t-test statistic that should be used to test H0 : β = 0 versus Ha : β ≠ 0? A. -3.22 B. -1.61 C. -0.805 D. 5.655 E. 11.31
(B) The predictor, or independent variable, in this situation is "Hours." On the computer output, the t-test statistic for this test can be found on the line for "Hours."
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Question: 401 A student is studying the relationship between the number of hours students work and their final exam scores in statistics courses. A regression analysis resulted in the computer output below. If the student used a sample of 10, which of the following is the 95% confidence interval for the true slope of the regression line based on this dataset? A. -0.4195 ± 10.1695 B. -0.4195 ± 23.451 C. -0.4195 ± 0.2605 D. -0.4195 ± 0.601 E. -0.4195 ± 11.957
(D) The critical value to find this interval is 2.306. From the output, the standard error of the estimate is 0.2605. Thus the margin of error is (2.306)(0.2605).
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Question: 413 A public opinion poll surveyed a simple random sample of voters. Respondents were classified by gender and by voting preference (Republican, Democratic, or independent). Results are shown below. If you conduct a chi-squared test for independence, what is the expected frequency count of male independents? A. 40 B. 50 C. 60 D. 180 E. 270
(A) Using the formula for the expected frequency of this cell, http://img.apstudy.net/ap/statistics/a500/Ans_Eq_237.jpg
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Question: 420 The following data have been collected on the respondents' favorite type of sport: If the expected frequencies rule for chi-square had been violated by the data, which categories could be combined together usefully in an attempt to increase the expected frequencies? A. Individual sport, team sport, and neither B. Males and females C. Individual sport and team sport D. Males and individual sport E. Team sport and neither
(C) While possible, combining all of the levels of one category would possibly affect our analysis. For instance, if we were interested in gender differences, combining male and female would no longer allow us to do this. We could, however, combine the team and contact sport categories, since these are the most related, and still perform a similar analysis.
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