There are 12 cell phone charging cords available at a store for purchase. The length of each charging cord in inches is reflected as 3, 3, 4, 6, 6, 6, 6, 7, 10, 10, 11, and 12, respectively. What are the mean of these data?
Correct Answer: A
The mean is the arithmetic average of a data set. To find it, add all values and divide by the number of values. The cord lengths are 3, 3, 4, 6, 6, 6, 6, 7, 10, 10, 11, and 12. Their sum is 84. Since there are 12 charging cords, divide 84 by 12 to obtain 7. Therefore, the mean cord length is 7 inches. This value represents the balance point of the distribution, not necessarily a value that must occur most often. In this data set, 6 appears most frequently, so 6 is the mode, but the question asks for the mean. Option B, 8, and option C, 9, are too large because several values are below 7. Option D, 6, identifies the most common value rather than the average. References/topics from the Study Guide: mean, measures of center, quantitative data, arithmetic average.
Question 2
Empirical probability is based on:
Correct Answer: A
Empirical probability is based on observed data collected from experiments, surveys, simulations, or repeated trials. It is calculated as the relative frequency of an event: number of times the event occurs divided by the total number of trials or observations. For example, if a machine produces 12 defective items in a sample of 300, the empirical probability of a defect is 12/300 = 0.04. This differs from classical or theoretical probability, which is based on equally likely outcomes and mathematical structure, such as a fair die having probability 1/6 for each face. It also differs from subjective probability, which is based on personal judgment or expert belief rather than observed frequency. The word "empirical" signals evidence obtained through observation, so observed frequency is the correct basis. Study Guide references/topics: empirical probability, relative frequency, observed data, probability interpretation.
Question 3
Poisson mean = 4. Probability of exactly 2 events?
Correct Answer: A
For a Poisson random variable, the probability of exactly k events is P(X = k) = e^## #^k / k!, where # is the mean number of events per interval. Here, # = 4 and the question asks for exactly 2 events, so k = 2. Substitute into the formula: P(X = 2) = e^#4 × 4² / 2!. Since 4² = 16 and 2! = 2, this becomes e^#4 × 8. Numerically, e^#4 is about 0.0183, and 0.0183 × 8 # 0.1465. Thus, the probability is approximately 0.1465. Options B, C, and D are unsupported by the Poisson probability formula. The Poisson model is appropriate for counts of events occurring independently over a fixed interval at a constant average rate. Study Guide references/topics: Poisson distribution, # parameter, exact event probability, discrete distributions.
Question 4
Two independent events: P(A and B) = ?
Correct Answer: A
For independent events, the probability that both events occur is the product of their individual probabilities. The rule is P(A and B) = P(A) × P(B). Independence means that the occurrence of one event does not change the probability of the other. For example, if a coin is tossed and a die is rolled, the probability of heads and rolling a 4 is 1/2 × 1/6 = 1/12. Option B is an addition expression used in "or" problems, not "and" problems. Option C applies to mutually exclusive events, not independent events. Option D would mean both events are certain to occur together, which is not generally true. The word "and" signals intersection, and independence allows the intersection to be calculated by multiplication. Study Guide references/topics: independent events, multiplication rule, joint probability, compound probability.
Question 5
Which graph displays data in a manner that is accurate?
Correct Answer: B
The accurate display is the bar graph labeled "Number of Feet Traveled" with the vertical axis showing the number of feet and the horizontal axis identifying the student travelers. For discrete observations associated with individual students, a bar graph is appropriate because each bar represents a separate category or individual and its height represents the measured value. Option A is inaccurate because its title states "Number of Miles Traveled," while the vertical axis is labeled in feet; this creates a unit mismatch. Option C uses a pie chart, which is designed to show parts of a whole, not individual measured distances unless the goal is proportional composition. Option D uses a line graph, which is typically reserved for ordered change, especially over time; connecting student traveler values implies continuity or trend that is not justified. The correct graph must use consistent units, clear labels, and an appropriate display type. References/topics from the Study Guide: data displays, bar graphs, categorical identifiers, quantitative measurements, graph accuracy.
