Question: Determine whether the following statement is true or false, and explain why: “A positive correlation coefficient between two variables means that an increase in one causes an increase in the other.”
Answer Choices:
False, a positive correlation coefficient means that an increase in one variable causes a decrease in the other.
True, correlation implies causation.
False, correlation does not imply causation.
True, a positive correlation coefficient means that an increase in one variable causes an increase in the other.
Answer: False, correlation does not imply causation.
Question: Determine whether the following pair of events are dependent or independent. Two cards are drawn from separate decks. A = {first card is a jack}, B = {second card is a nine}.
Answer Choices:
The events are dependent because event B does not depend on the outcome of event A, and vice versa.
The events are independent because event B does not depend on the outcome of event A, and vice versa.
The events are independent because event B depends on the outcome of event A, and vice versa.
The events are dependent because event B depends on the outcome of event A, and vice versa.
Answer: The events are independent because event B does not depend on the outcome of event A, and vice versa.
Question: Which of the following is not one of the three basic axioms of probability?
Answer Choices:
The probability of any event is always between 0 and 1, inclusive.
The probability of the entire sample space is 1.
The probability of the union of any two mutually exclusive events is the sum of their individual probabilities.
The probability of an event is always equal to the number of favorable outcomes divided by the total number of possible outcomes.
Answer: The probability of an event is always equal to the number of favorable outcomes divided by the total number of possible outcomes.
Question: Decide whether the statement is true or false. To find the x-intercept of the graph of a linear function, we solve y = f(x) = 0, and to find the y-intercept, we evaluate f(0).
Answer Choices:
The statement is false. To find both intercepts, we find the value of x for which f(a) = 0, and the value of f(a). The x-intercept and the y-intercept are both a.
The statement is false. To find the x-intercept we evaluate f(0), and to find the y-intercept, we solve y = f(x) = 0.
The statement is false. To find the x-intercept we solve y = f(x) = x, and to find the y-intercept, we evaluate f(-1).
The statement is true. To find the x-intercept we solve y = f(x) = 0, and to find the y-intercept, we evaluate f(0).
Answer: The statement is true. To find the x-intercept we solve y = f(x) = 0, and to find the y-intercept, we evaluate f(0).
Question: What is the sample space for an experiment where a coin is tossed and a die is rolled?
Answer Choices:
S = {H1,H2,H3,H4,H5,H6}
S = {H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6}
S = {HH,HT,TH,TT,1,2,3,4}
None of the above.
Answer: S = {H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6}
Question: Determine whether the statement is true or false, and explain why: “The sum of a 2 by 3 matrix and a 4 by 5 matrix is a 6 by 8 matrix.”
Answer Choices:
The statement is false because the size of a sum matrix is that of the largest addend matrix, in this case, 4 by 5.
The statement is true because 2 + 4 = 6 and 3 + 5 = 8.
The statement is false because matrix addition is only defined for matrices of the same size.
The statement is true because 6 − 2 = 4 and 8 − 3 = 5.
Answer: The statement is false because matrix addition is only defined for matrices of the same size.
Question: In probability theory, when a random variable has a continuous distribution, its probability density function (PDF) f(x) is linked to integrals. Which of the following ideas is correct?
Answer Choices:
The total probability must equal the area under the entire PDF, which is always 1.
The probability of the variable falling in an interval (like between 2 and 5) is the area under the curve over that interval.
The probability of the variable being exactly one value (like exactly 3) is 0.
All of the above.
Answer: All of the above.
Question: Which of the following statements about the expected value of a random variable is true?
Answer Choices:
The expected value is a long-run average of many repetitions of the random process.
The expected value must always be a value that the random variable can actually take.
The expected value is the same as the median of the distribution.
None of the above.
Answer: The expected value is a long-run average of many repetitions of the random process.
Question: How do consumers harness the sun’s power and use it as an energy source?
Answer Choices:
A. by exposing themselves to sunlight, sunbathing
B. by eating producers at the bottom of the food chain
C. absorption through the retina
D. by eating tertiary consumers at the top of the food chain
E. by producing vitamin D from sun exposure
Answer: B. by eating producers at the bottom of the food chain
Question: Which of the following are accurate descriptions of the properties of water? (select all that apply)
Answer Choices:
Water molecules are polar, and thus dissolve other polar or ionic substances.
Water molecules are nonadhesive, tending to repel each other forcefully, resulting in a low surface tension.
Water shrinks when it crystallizes, and ice is more dense than liquid water.
The slow response of water to temperature change helps moderate global temperatures.
Water’s high heat of vaporization makes evaporation an effective way for many organisms to shed excess heat.
Answer: Water molecules are polar, and thus dissolve other polar or ionic substances.
The slow response of water to temperature change helps moderate global temperatures.
Water’s high heat of vaporization makes evaporation an effective way for many organisms to shed excess heat.
Question: Which of the following statements about systems and feedback are true? (select all that apply)
Answer Choices:
Both positive and negative feedback can lead to system instability and collapse.
A system is essentially a network, with materials and energy flowing from one component to another.
With positive feedback, an increase in a state variable leads to further increases in that same variable.
With negative feedback, an increase in a state variable leads to further increases in that same variable.
All systems are closed.
Answer: A system is essentially a network, with materials and energy flowing from one component to another.
With positive feedback, an increase in a state variable leads to further increases in that same variable.
Question: Decide whether the events are mutually exclusive: Winning a baseball game and losing the same baseball game.
Answer Choices:
No. It is possible for both events to occur simultaneously.
No. Neither event can ever occur.
Yes. It is not possible for both events to occur simultaneously.
Yes. In order for one of the events to occur, both events must occur simultaneously.
Answer: Yes. It is not possible for both events to occur simultaneously.