1. Describe the role probability information plays in the decision-making process.
Decision making involves selecting best alternative based on personal values and preferences. Therefore, a decision can only be made from some possible alternatives provided. The decision arrived is taken to be rational if it was the one which had the highest probability of success and it serves the decision makers life goals, desires, values and lifestyle. Likewise, in business organization the managers strive to make decisions which can propel an organization towards achieving its goals. They always consider the probability of an event occurring or even not occurring, and then they settle on one of the possible alternatives (McClave, Benson, & Sincich, 2001).
When the probability of an event occurring is zero then an individual or organization does not prepare itself for such alternative. However, if the probability of an event occurring is one then a person or organization must prepare on how to handle the outcome because it is sure that the event will take place. In some complex situations use of probability concepts may require analysis of statistical data and application of models developed by experts. This makes sure that the decision makers look at problems in an objective manner without being influenced by personal opinions. Making use of probability in an organization leads reliable decisions whose results can easily be compared to industry standards.
2. Discuss the concepts of a random variable and a probability distribution
A Radom variable value is usually determined by chance. Therefore, a random variable does not have a fixed value; it takes a set of different possible values where each of them has its probability of occurring. Such variable is also known as stochastic variable. The values obtained may form discrete random variable or continuous random variables. Discrete random variables can only take certain variables within a given rage of numbers (Srinivasan, 2002). For example, in tossing a coin a head or a tail can be obtained, therefore if the number of heads is counted it can only be a whole number i.e. 0, 1 , 2 e.t.c. in this case there is no possibility of getting a value between two consecutive whole numbers.
On the other hand, continuous random variable can take any value within a given range of values.
Mainly, continuous random variables are measurements. Therefore, the variable cannot be defined at a specific value rather it is defined over an interval of values. The probability of observing a single value is therefore zero because the random variable takes infinite number of values within the given range of values e.g. the weight of a person can take any value within a given range of weights
3. Explain the difference between how probabilities are computed for discrete and continuous-random variables.
The probability of distribution in discrete variables is the chance (probability) associated with a possible value. For example, suppose variable Y takes values 2, 3, 4 or 5 and the probability of 2 occurring is 0.1, that of 3 occurring is 0.3, that of four occurring is 0.4 while that of 5 occurring is 0.2. When determining the probability that Y is equal 2 or 3, the sum of probability of 2 occurring is added to that of 3 occurring i.e.
Unlike discrete random variables whose probability can be expressed in tabular form continuous random variables probability is expressed in form of a formulae or equation. Such equation is called probability density function. The graph for the equation is continuous over the given range since a continuous variable can take any value within the given range. The area bounded by the equation and the x axis is one because the probability of all values occurring is one. In addition the probability that the random variable assumes a value between b and c is taken to be equal to the area under the probability function bounded by b and c (Lynch, Padgett, & Durham, 1990).
4. Discuss the importance of sampling and how results from samples can be used to provide estimates of population characteristics such as the population mean, the population standard deviation, and/or the population proportion.
Sampling is a process used in statistics where a few number of observations are taken to represent a larger population. A sample is always convenient to work with because it is small. The mean and standard deviation of unbiased sample is always very close to that of the population. According to the central limit theory if a large number of simple random samples is selected and the mean of each sample is determined. The distribution of the means form normal probability distribution. This theory holds even for the populations which are not normally distributed.
Some samples are representative while others are not representative i.e. they cannot be used to give characteristics of the population. Therefore, when sampling a large sample should be selected because the larger the sample the more it represents the population. However, a very large sample undermines the basic aim of sampling which is to save on cost of data collection and calculations of determining the sample means, variance and other characteristics (Friday, 1967). It is possible to calculate the population mean from a sample mean and standard deviation given a confidence interval.
Works Cited
Friday, F. A. (1967). The elements of probability and sampling. New York: Barnes & Noble.
Lynch, J. D., Padgett, W. J., & Durham, S. D. (1990). Statistical Models and Inference Procedures for Structural and Materials Reliability. Ft. Belvoir: Defense Technical Information Center.
McClave, J. T., Benson, P. G., & Sincich, T. (2001). Statistics for business and economics (8th ed.). Upper Saddle River, NJ: Prentice Hall.
Srinivasan, R. (2002). Importance sampling: applications in communications and detection. Berlin: Springer.
