QNT 275 Exercise Week 2: Please Answer The Following Questio

Qnt 275 Exercise Week 2please Answer The Following Questions Using You

QNT 275 Exercise Week 2 Please answer the following questions using your own words and using only the attached chapters for reference and in-text citations. Answers must be 200 words each.

Paper For Above instruction

Introduction

The study of probability and statistics provides foundational tools in understanding variability and making informed decisions based on data. This paper addresses key concepts from Chapters 4 and 5 of Mann’s "Introductory Statistics," focusing on the properties of probability, approaches to calculating probability, types of random variables, and an application involving the analysis of household choices in television viewing.

Question 1: Properties of Probability, Impossible and Sure Events

Probability theory is grounded in two fundamental properties: the probability of an event ranges between 0 and 1, inclusive, and the sum of probabilities of all mutually exclusive outcomes in a sample space is 1 (Mann, 2016). An impossible event is one that cannot occur, with a probability of 0; for example, flipping a coin and getting a head on a coin with no head. Conversely, a sure event is one that is certain to occur, having a probability of 1, such as the sun rising tomorrow. These properties reflect the bounds within which all probabilities lie. The probability of an impossible event is 0 because it cannot happen, while the probability of a sure event is 1 due to its certainty. Understanding these extreme cases helps in comprehensively grasping the concept of probability and its application in real-world scenarios.

Question 2: Approaches to Probability

There are three main approaches to calculating probabilities: classical, empirical (relative frequency), and subjective (Mann, 2016). The classical approach relies on the assumption of equally likely outcomes—used when outcomes are symmetric, such as rolling a fair die, which has six equally likely faces. The relative frequency approach involves analyzing past data, where probability is approximated by the ratio of favorable outcomes to total trials, suitable when historical data is available, such as estimating the probability of rain based on past weather records. The subjective approach is based on personal judgment or expert opinion, often used in situations with little or no historical data, like assessing the probability of a new product’s success. Each approach is suited to different experimental contexts, with classical for symmetric experiments and relative frequency for data-driven experiments, aiding in different decision-making processes.

Question 3: Random Variables

A random variable is a numerical description of the outcome of a random experiment (Mann, 2016). Discrete random variables take specific, separate values, often counts, such as the number of cars passing through a toll; continuous variables take any value within a continuum, such as measuring height or temperature. For example, the number of books in a student’s bag is a discrete random variable because it counts items, while the time spent by a physician examining a patient is a continuous random variable, as it can vary over an interval. Classifying variables, the time left on a parking meter (continuous), the number of bats broken by a baseball team (discrete), the number of cars in a parking lot at a given time (discrete), the price of a car (continuous), the number of cars crossing a bridge (discrete), the time spent by a physician (continuous), and the number of books in a bag (discrete). Recognizing the nature of these variables is essential for selecting appropriate statistical methods for analysis.

Question 4: Random Variable in Household News Watching

In the scenario where five randomly selected households choose among ABC, CBS, or NBC to watch the news, the variable x—representing the number of households watching ABC—is a discrete random variable because it can only take specific whole number values (0 to 5). The possible values for x are 0, 1, 2, 3, 4, or 5, which represent the count of households watching ABC out of the five surveyed. Discrete variables like this often arise in count data, where outcomes are finite and countable, making probability distributions such as the binomial suitable for analysis. Understanding whether a variable is discrete or continuous aids in selecting appropriate probabilistic models, enhances the interpretation of data, and informs decision-making processes in statistical studies.

References

• Mann, P. S. (2016). Introductory Statistics (9th ed.). Retrieved from The University of Phoenix eBook Collection database.