Probability and Inferences in Statistics - Term Paper Example

 Abstract In this work, the author explored the role of probability in inferential statistics. Man has discovered several probability distributions. Some of these probability distributions include the binomial, hypergeometric, normal, chi-square, and several other probability distributions. Populations or statistics are assumed to follow one of these several probability distributions and these are used to estimate population parameters as well as test hypotheses on the population. The standard normal distribution, for example, is used to draw inferences on population parameters such as the population mean. The t-statistics is used for inferential purposes involving small samples or where the population variance is unknown. Finally, the chi-square statistics are used for inferential statistics related to correlations and associations. These are only some of the distributions but they indicate nevertheless the value of probability distribution for inferences in statistics. I. Introduction In making inferences on the population based on sample characteristics, man used probability distribution or probability density functions (McClave et al. 172-173; Schaum 55). Probability distributions or density functions associates a numeral to a probability value. Probability density functions are curves represented by mathematical equations that best capture the characteristics of the curve. In turn, the curves or the areas under the curve are associated with probability level or probabilities. Some of the probability density functions that are used in statistics are the binomial, hyper geometric, Poisson, normal and the chi-square distributions (Walpole 10). In using any of the probability density function, a statistician assumes that the distribution represented by the probability density function best captures how the population or the data of the population is distributed (Buglear 197-198). Alternatively, at least, the statistician assumes or believes that the distribution represented by the probability density function is most useful for him or her to infer population characteristics based on sample characteristics. Among several probability density functions used in statistics, some of the more widely used are the normal distribution, the chi-square distribution, the F-distribution, and the binomial distribution. In this work, we focus on three probability density functions: two types of a normal probability density function and the chi-square distribution. The types of normal distribution are the standard normal and the t-distribution. II. Z-statistics Figure 1 captures the mathematical function of a normal distribution. Figure is also known as the normal distribution function. The function associates a value of the variable x with an f(x). In turn, the f(x) is associated with probability values. The association comes through the area under the curve of f(x). The entire area under the curve is 1 associated with probability level 1 or 100%. Thus, the probability that x is between negative infinity to positive infinity is 1 because this represents the entire area under the curve of f(x). In equation form, this is P(-< x < +) = 1 or 100%. Relatedly, the probability that x> is 0.5 or 50% which is expressed mathematically as P(x>)=0.5. On the other hand, the probability that x Read More