Concept explanation of interval estimation
In the vast field of statistics, interval estimation is an extremely important technical means. It works based on samples drawn from the population. When we are faced with a large population, it is often extremely difficult to accurately know the true values of the unknown parameters of its distribution or the functions of the parameters. At this time, interval estimation plays a huge role.
It will operate according to certain requirements for accuracy and precision. Accuracy can be understood as the reliability of the interval we construct containing the true parameter value, while precision is related to the width of this interval. By comprehensively considering these two key factors, we can construct an appropriate interval. This interval is like a "encirclement", and we believe that the true value of the unknown parameter of the population distribution or the function of the parameter is likely to fall within this range.
Let's take a simple example. In daily life, we often hear such expressions: having a certain percentage of certainty that a certain value falls within a certain range. This is actually the simplest and most common application of interval estimation. For instance, a weather forecaster says there is a 90% certainty that tomorrow's temperature will be between 20°C and 25°C. This is the use of interval estimation to estimate the parameter of temperature.
The establishment of the theory of interval estimation
In the continuous development and progress of the statistics community, the year 1934 is of great significance. In this year, the statistician J. Neyman made a pioneering contribution - he founded a strict theory of interval estimation. The birth of this theory laid a solid foundation for the application of interval estimation in statistics. It made interval estimation no longer a vague and empirical method, but instead, it was supported by a rigorous theoretical system. It made the process of interval estimation more scientific and standardized, and also enabled us to more accurately evaluate the reliability of the estimation results.
Common methods for finding confidence intervals
Utilize the known sampling distribution
In the process of finding the confidence interval, using the known sampling distribution is a commonly used method. The sampling distribution refers to the distribution formed by the statistics calculated from a certain number of samples drawn from the population. In many cases, we have conducted in - depth research and understanding on some common sampling distributions, such as the normal distribution, t - distribution, F - distribution, etc. When we are faced with a specific interval estimation problem, if we can determine that the involved statistic follows a certain known sampling distribution, we can use the properties of this distribution to determine the confidence interval. For example, when the population variance is known, the sample mean follows a normal distribution. We can then calculate the confidence interval of the population mean according to the characteristics of the normal distribution and in combination with the given confidence level.
Utilize the connection between interval estimation and hypothesis testing
Interval estimation and hypothesis testing are two closely related concepts in statistics. Hypothesis testing is to test a certain hypothesis about the population parameter to determine whether the hypothesis holds. Interval estimation, on the other hand, is to estimate the range of values of the population parameter. There is an inherent connection between them. We can use this connection to find the confidence interval. When conducting hypothesis testing, we calculate a test statistic based on the sample data and determine the rejection region and acceptance region according to the distribution of the statistic. The confidence interval can be regarded as the set of all parameter values that are not rejected in the hypothesis testing. Through this connection, we can use the methods and ideas of hypothesis testing to solve for the confidence interval. For example, when conducting a two - sided hypothesis test, we can adjust the significance level of the test so that the parameter range corresponding to the acceptance region is the confidence interval we need.
Utilize the large sample theory
The large-sample theory is also an important method for finding confidence intervals. When the sample size is large enough, according to the central limit theorem, regardless of the distribution of the population, the sample mean approximately follows a normal distribution. This provides convenience for us to conduct interval estimation. In practical applications, when the sample size is large, we can ignore the specific form of the population distribution and directly use the approximate normal distribution of the sample mean under the large-sample condition to calculate the confidence interval. Moreover, the large-sample theory allows us to avoid accurately modeling the population distribution when dealing with some complex population distributions. We only need to ensure that the sample size is large enough to obtain relatively accurate confidence intervals. This method is very practical in many real problems because in many cases, it is difficult for us to have an accurate understanding of the population distribution, but we can use the large-sample theory for interval estimation by increasing the sample size.