Introduction to statistical Inference is one of the core requirements for obtaining a degree in Statistics. It offers students a solid foundation for building statistical programs and analyzing real-world data. Although most statistics courses focus on discrete methods, many courses also introduce statistical inference as a fundamental concept. This focus on discrete methods is necessary because understanding the results of historical data (the “historical data set” in statistics) can only be accomplished by considering the model(s) of the prior distribution(s).
There are several different types of statistical inference, and students must learn how to appropriately apply them to sample sets so that they can use the appropriate statistical techniques. Inference with discrete time series is perhaps the most common way to model statistical inference, and students learn how to construct sample series and solve problems associated with it. In addition to sampling distributions, some problems associated with statistical inference will also require using lattice models or neural networks.
Probability theory states that a random variable X will exist at time t and will have an effect on the mean value of t when it occurs. Students learn to calculate the probability density function using standard deviation. Standard deviation is often called beta-divergence, because it attempts to identify the existence of “spike” or “normal” behaviors in the distributions of the random variables. These behaviors can be used to characterize the range of possible outcomes, but they cannot be analyzed independently. Therefore, they cannot be measured against an arbitrary or known prior probability scale.
Students learn how to calculate the normal distribution of the random variable beta, as well as how to normalize the distribution of beta to meet the expectations of the normal distribution formula. The standard deviation of a beta distribution is actually a measure of the deviation from the normally distributed shape. Most distributions are normally distributed, but beta distributions lie outside this range, and are therefore termed “normal-shaped” distributions. A few examples of abnormal-shaped distributions are binomial, log normal, Poisson, and kurtzberg- Weaver.
Students also learn how to use confidence intervals to analyze a data set. Interval estimates can provide a large amount of information about a specific interval, and they can be extremely useful in interpreting results from other statistical analysis tools. Confidence intervals are typically drawn from normal distributions, and they are used to indicate the range of values for a variable. They follow a bell-shaped curve, and their shape is most often continuous over the whole range of the distribution. Students should not draw the 95% confidence interval on their own, but instead should rely on a software package to help them calculate the correct interval estimates.
One other type of interval estimate uses the lognormal distribution. This distribution is a special kind of normal distribution that actually uses two types of parameters, with the degrees of freedom representing the sample mean and standard deviation. The degrees of freedom are called the beta parameters, and they can be thought of as estimating parameters. Students can learn more about beta distributions by consulting one of the many books on statistics that are available in bookstores or online. This is a much more complicated concept than sampling distribution, and its exact workings are not yet fully understood by the statistical profession.
Many statistical methods use the mean of a large number of smaller intervals to create a large sample statistic. For example, the t formula utilizes the log-normal curve (also called the exponential curve) and estimates the parameters of the normal distribution by calculating the sample probability function over a large number of discrete samples. It is possible to obtain high precision estimates using this method, which is often necessary when making inferences about trends and averages over a large number of units. Students should understand the concepts behind the t formula, as well as the significance of sample size when analyzing data sets using this technique.
A final topic for an introduction to statistical methods is the Bayesian approach to data analysis. The approach uses an algorithm to automatically detect and extract the most accurate estimates from the many discrete components of the statistical mixture. The main weakness of the Bayesian statistical method is that it has a poor memory of past information. However, when it does find a good match, the results often give very accurate estimates, especially in comparison to traditional frequency or random sampling methods. This is also a good introduction to the more advanced topics associated with statistics and data analysis for business analytics courses.