The Use of Probability in Inferential Statistics - Term Paper Example

The Use of Probability in Inferential Statistics There are many distribution models used in inferential statistics. The distribution modelsreflect a statistician’s assumption on how variable values are possibly distributed in a population. Based on sample data and assumptions on the population distribution, inferences are made on how the variable values may be actually distributed in a population. The most conventional or usual assumptions on population distribution are the standard normal, t-distribution, and the chi-square distribution. Using assumptions on how variable values are distributed in a population makes it possible for man to anticipate the characteristics of specific populations based on information provided by sample data. The Use of Probability in Inferential Statistics Statistics use assumption on population distribution to estimate population values based on sample values (Kemp & Kemp 257-66). Probability distributions functions or simply distribution functions are used (Soong 39-41). The distribution function associates a variable value with a probability (Soong 39). This can take the form Fx(x) =P(Xx) where the lowercase x refers to a specific value of a variable. Probability distribution functions have shapes represented by the mathematical equations. The areas under the curves or distribution functions are associated with probabilities. In business statistics, some of the distribution functions that are often used are the Z-statistics, t-statistics, chi-square distribution, and the F-statistics (Kemp & Kemp 47-297). There is also an option to use what statisticians call as the non-parametric statistics (Kemp & Kemp 298-315). The choice of what distribution functions to use are determined by convention or typical practice and theory. For example, in estimating the population mean, it is assumed that the sample mean converge to the population mean through repetition of sampling procedures or if the population is large. Thus, in estimating the mean, statistics usually makes the assumption based on a normal distribution. Although several distribution functions are used in statistics, in this work we focus our sights on three: the z-statistics, the t-statistics, and the chi-square statistics. I. Z-statistics Figure 1 captures a standard normal distribution function. The standard normal distribution associates a value of a variable with probability. For example, the probability that the value of the variable is between a very low number and high number can be represented by 100%. In the language of statistics, this take s the form P (-< x < +) = 1 or 100%. In other words, this means that in a standard normal distribution, the area under the curve is 1 or associated with probability value 100%. Similarly, the area from negative infinity to zero is associated with a probability value 0.5 or 50%. Likewise, the areas from zero to positive infinity is associated with the same probability value 0.5 or 50%. This implies that the standard normal divides a population at zero into two equal parts: 50% lies below zero while the other 50% lies above zero. The standard normal distribution “normalized” a distribution or that the mean becomes equalized to zero through the standardization process. Figure 1. Normal Distribution (Schaum 56) The standard normal distribution equalizes the standard deviation to 1. The standard normal distribution was devised so that we do not have to construct a particular normal curve for every distribution. A standardization procedure is used that is represented by Figure 2 Figure 2. Schaum 56 on standardization The probabilities associated with the areas under the standard normal curve are in Figure 3. For example, the probability that Z is from negative infinity to 1 is 0.8413 or 84.13%. This implies that the probability from zero to 1 is 0.8413 minus 0.5 or 0.3413 or 34.13%. Thus, P(-1.01 Read More