probability and statistics homework help

probability and statistics homework help

Enhancing Understanding of Probability and Statistics: A Comprehensive Guide for Homework Success

1. Introduction to Probability and Statistics

Traditionally, the study of statistics usually involves first a one-semester course in the fundamentals of probability and combination of class lectures and homework to learn the basic concepts and techniques necessary. Then, a second course in statistics is taken. In most schools and colleges, the study of statistics is popularly based on various disciplines of the educational system. Some of the statistical techniques help in understanding and enlightening students when some statistical research is conducted through the students’ own field of study. The statistics and probability thus are important for greater emphasis in helping the students to understand their basic principles, which would enable them to know and make various kinds of useful statistical and probability inferences in their life in the future.

2. Fundamental Concepts and Definitions

We commonly make use of sample information to guide our conclusions about populations. However, collecting data on an entire population is generally infeasible. Samples are observed in practice because they are surrogates for the entire population. Data from a sample are used to compute statistics. Statistics are used to perform inference on population parameters. Common examples of population parameters are: the population mean, μ, the population variance, σ^2, and the population proportion, p. The corresponding specific sample statistics are: the sample mean, x̄, the sample variance, s^2, and the sample proportion, p̂. To close this section, we provide several other key definitions necessary for upcoming content.

In this lesson, we discuss a few basic concepts in probability and statistics. Prior to defining any of the core concepts of probability or statistics, we begin with a brief discussion on the difference between these two fields. Probability can be considered as an abstract theory of uncertainty and chance. It serves as a theoretical foundation for the study and development of various statistical methods and models. In applied work, statistical methods are used to answer questions or obtain conclusions about populations based on information in samples.

3. Key Probability Distributions

If an experiment can be described by an SRS of n trials, on each of which can result in only a “success” or “failure”, with the probability of success on each trial equal to p, then the number of successes in the n trials results in a random variable. This random variable is called a binomial random variable, and its distribution, called the binomial distribution, models the chance process. The formula for the binomial distribution gives the probability of x successes in n trials, and is in terms of the total number of trials n, the number of successes x, and the probability of success on any one trial p. By considering the binomial distribution formula, we shall be able to study the basic properties and likelihood of occurrence of the probabilities that make up the binomial distribution, as well as graph the distribution, study how increasing (or decreasing) n or p changes the shape and location of the distribution, and find the mean and standard deviation that summarize the probability model.

In this chapter, we shall consider several probability distributions which have significance in various types of problems. In particular, we shall discuss the characteristics of an experiment which enable one to choose the appropriate distribution to model it. By studying the basic properties and likelihood of occurrence characteristics of these distributions, students will begin to understand situations in which these distributions play a role in problems. We also provide web applets to simulate selected discrete distributions so that students will gain a better comprehension of these topics.

4. Statistical Inference and Hypothesis Testing

These responses sound reasonable, but if your friend asks, “Which test should we use?” you may not be as quick with your retort. You respond hesitantly, “Let me take a look at some other examples, and I will tell you.” This is the dilemma of statistical inference. In most situations of real-world studies and surveys, the analyst has essentially no background knowledge on the problem at hand. In the face of such uncertainty, the traditional approach in statistics is hypothesis testing, a concept pivotal to the use of statistics to draw conclusions about particular problems.

In our continuing case about testing a new methodology, we have finally conducted our trial with the new methodology and collected the data. We are now ready to analyze this data and reach some conclusion. A friend asks, “How do we decide if the new methodology works?” You answer, with some thought, “We should compare its performance with the old methodology” or “We could compare it to the current best known method.”

5. Practical Applications and Problem-Solving Strategies

1. Matching Pennies. One person holds out a coin hidden in his hand, either heads or tails. He then asks his partner to call heads or tails. The hider wins the game if his partner calls a second head if the hider has heads hidden, or tails if the hider has tails hidden. On the other hand, the partner wins the game if his call is the same as the coin in the hider’s hand. (a) Does either player have a winning strategy? (b) What are your chances of winning in a single game? In a series of games?

The number of applications of probability and statistics is almost infinite. Here, we discuss several common applications, those that in all likelihood you are familiar with. Most of these problems can be solved using the counting and rules of probability we learned, in addition to those yet to be introduced. It is also important to note that we can enhance our understanding of probability and statistics by solving problems related to other fields in mathematics. The second part of this chapter is devoted to problem-solving strategies.