selecting a random sample is an example of a statistical experiment, and the sample statistic x is a numerical description of the result of the experiment. therefore, x is a random variable. the probability distribution of x is called the sampling distribution of x. in practice, you select one random sample and use the information from that sample to estimate the population parameter of interest. however, statisticians sometimes perform a procedure called repeated sampling, in which the experiment is run over and over again and the value of the sample statistic from each running of the experiment is recorded. the distribution of the sample statistics from the repeated sampling is an
LectureNotes said selecting a random sample is an example of a statistical experiment, and the sample statistic x is a numerical description of the result of the experiment. therefore, x is a random variable. the probability distribution of x is called the sampling distribution of x. in practice, you select one random sample and use the information from that sample to estimate the population parameter of interest. however, statisticians sometimes perform a procedure called repeated sampling, in which the experiment is run over and over again and the value of the sample statistic from each running of the experiment is recorded. the distribution of the sample statistics from the repeated sampling is an
Answer: The distribution of the sample statistics from the repeated sampling is known as the sampling distribution of the sample statistic.
To elaborate, let’s break down the key concepts mentioned:
1. Statistical Experiment:
A statistical experiment involves a process or procedure that generates a set of data. Selecting a random sample from a population is a prime example.
2. Sample Statistic (x):
The sample statistic, often denoted as ( x ), is a numerical summary of the sample data. Common examples of sample statistics include the sample mean (( \bar{x} )), sample variance (( s^2 )), and sample proportion (( \hat{p} )).
3. Random Variable:
Since the sample statistic ( x ) is derived from a random sample, it is inherently a random variable. Its value can vary depending on the specific sample chosen from the population.
4. Sampling Distribution:
The probability distribution of the sample statistic ( x ) is called the sampling distribution. It describes how the sample statistic would behave if we were to take many samples from the same population and compute the statistic for each sample.
5. Repeated Sampling:
In practice, we usually select one random sample to estimate a population parameter. However, statisticians may perform repeated sampling, where the experiment is conducted multiple times. Each time, a sample is drawn, and the sample statistic is calculated.
6. Distribution of Sample Statistics:
The distribution of the sample statistics obtained from repeated sampling is the sampling distribution of the sample statistic. This distribution provides critical insights into the variability and reliability of the sample statistic as an estimator of the population parameter.
Example:
If we repeatedly draw samples of size ( n ) from a population and compute the sample mean ( \bar{x} ) for each sample, the distribution of these sample means is the sampling distribution of the sample mean. According to the Central Limit Theorem, if the sample size ( n ) is sufficiently large, the sampling distribution of the sample mean will be approximately normally distributed, regardless of the population’s distribution.
Conclusion:
Understanding the sampling distribution is crucial for making inferences about the population parameter. It allows statisticians to estimate the standard error, construct confidence intervals, and perform hypothesis tests, thereby making informed decisions based on sample data.
By grasping these concepts, one gains a deeper appreciation of the role of sampling distributions in statistical inference, enhancing the ability to interpret and apply statistical methods effectively.