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Chapter 5: Problem 2

Classify each of the following random variables as discrete or continuous. a. The time left on a parking meter b. The number of bats broken by a major league baseball team in a season c. The number of cars in a parking lot d. The total pounds of fish caught on a fishing trip e. The number of cars crossing a bridge on a given day f. The time spent by a physician examining a patient

Short Answer

Expert verified
a. Continuous b. Discrete c. Discrete d. Continuous e. Discrete f. Continuous

Step by step solution

01

Classify Variable A

a. The time left on a parking meter. This is a continuous random variable as the time left can take any value within a certain range.
02

Classify Variable B

b. The number of bats broken by a major league baseball team in a season. This is a discrete random variable because the number of bats broken can only take certain countable values.
03

Classify Variable C

c. The number of cars in a parking lot. This is a discrete random variable because you can only have a countable number of cars in a parking lot.
04

Classify Variable D

d. The total pounds of fish caught on a fishing trip. This variable is continuous because the total weight can be any value within a certain range, depending on the size and number of fish caught.
05

Classify Variable E

e. The number of cars crossing a bridge on a given day. This is a discrete random variable, as the number of cars can only be a countable whole number.
06

Classify Variable F

f. The time spent by a physician examining a patient. This is continuous because it can take any value within a certain range, depending on the length and complexity of the examination.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability
Probability is a key concept in understanding random variables. It measures the likelihood that a particular event will occur. Imagine you are flipping a coin or rolling a die; these actions are influenced by probability. Probability values range from 0 to 1, where 0 means impossible and 1 means certain. In the context of random variables, probability helps us predict possible outcomes and is used to calculate how likely each outcome is.
  • **Probability of an event**: It is calculated as the number of successful outcomes divided by the total number of possible outcomes.
  • **Random Variable Probability**: For a discrete random variable, the probabilities from all possible outcomes should sum to 1. For a continuous random variable, we use probability density functions.
The concept of probability guides us in making informed decisions by understanding how likely certain events are, particularly in statistics and everyday situations.
Random Variables Classification
Understanding the classification of random variables is crucial in statistics. A random variable can be either discrete or continuous. This classification helps determine the type of statistical methods needed for analysis.
  • **Discrete Random Variables**: These are variables that take on countable values. For example, the number of broken bats or cars in a parking lot. They are often associated with events where outcomes are clearly distinct and countable.
  • **Continuous Random Variables**: These take on an infinite number of possible values. An example would be the time left on a parking meter or the time spent by a doctor with a patient. These variables usually result from measurements and can embrace any value within a specific range.
Recognizing whether a variable is discrete or continuous is critical as it influences the analysis approach, such as whether to use probability mass functions or probability density functions.
Statistics
Statistics is the science of collecting, analyzing, and interpreting data. It gives us tools to understand random variables and probability, enabling us to make sense of data.
  • **Descriptive Statistics**: These involve summarizing or describing a collection of data. Measures such as mean, median, mode, and standard deviation are used to present this data concisely.
  • **Inferential Statistics**: These allow us to make conclusions and predictions about a population based on a sample of data. It often involves probability to determine the likelihood of certain outcomes.
In the context of random variables, statistics helps in making decisions based on data, whether it's predicting future events or analyzing trends. It is an essential skill in numerous fields such as economics, medicine, and engineering, equipping us with the knowledge to handle and interpret various kinds of data.

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Most popular questions from this chapter

In the 2008 Beach to Beacon \(10 \mathrm{~K}\) run, \(27.4 \%\) of the 5248 participants finished the race in \(49 \mathrm{~min}\) utes 42 seconds \((49: 42)\) or faster, which is equivalent to a pace of less than 8 minutes per mile (Source: http://www.beach2beacon.org/b2b_2008_runners.htm). Suppose that this result holds true for all people who would participate in and finish a \(10 \mathrm{~K}\) race. Suppose that two \(10 \mathrm{~K}\) runners are selected at random. Let \(x\) denote the number of runners in these two who would finish a \(10 \mathrm{~K}\) race in \(49: 42\) or less. Construct the probability distribution table of \(x\). Draw a tree diagram for this problem.

According to a 2008 Pew Research Center survey of adult men and women, close to \(70 \%\) of these adults said that men and women possess equal traits for being leaders. Suppose \(70 \%\) of the current population of adults holds this view. a. Using the binomial formula, find the probability that in a sample of 16 adults, the number who will hold this view is \(\begin{array}{ll}\text { i. exactly } 13 & \text { ii. exactly } 16\end{array}\) b. Use the binomial probabilities table (Table I of Appendix C) or technology to find the probability that the number of adults in this sample who will hold this view is i. at least 11 ii. at most 8 iii. 9 to 12

An average of \(6.3\) robberies occur per day in a large city. a. Using the Poisson formula, find the probability that on a given day exactly 3 robberies will occur in this city. b. Using the appropriate probabilities table from Appendix \(C\), find the probability that on a given day the number of robberies that will occur in this city is i. at least 12 ii. at most 3 iii. 2 to \(\overline{6}\)

Uniroyal Electronics Company buys certain parts for its refrigerators from Bob's Corporation. The parts are received in shipments of 400 boxes, each box containing 16 parts. The quality control department at Uniroyal Electronics first randomly selects 1 box from each shipment and then randomly selects 4 parts from that box. The shipment is accepted if at most 1 of the 4 parts is defective. The quality control inspector at Uniroyal Electronics selected a box from a recently received shipment of such parts. Unknown to the inspector, this box contains 3 defective parts. a. What is the probability that this shipment will be accepted? b. What is the probability that this shipment will not be accepted?

Six jurors are to be selected from a pool of 20 potential candidates to hear a civil case involving a lawsuit between two families. Unknown to the judge or any of the attorneys, 4 of the 20 prospective jurors are potentially prejudiced by being acquainted with one or more of the litigants. They will not disclose this during the jury selection process. If 6 jurors are selected at random from this group of 20, find the probability that the number of potentially prejudiced jurors among the 6 selected jurors is a. exactly \(\overline{1}\) b. none c. at most 2
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