As a domain expert in statistics, I'm delighted to provide an in-depth explanation of what a sample statistic is and how it plays a pivotal role in statistical inference. Let's dive into the concept with an example that will illustrate the process and importance of sample statistics in understanding a population from a sample.
**Sample Statistics and Their Role in Statistical Inference**
A sample statistic is a numerical value that summarizes some characteristic or feature of a sample data. It is used as an estimate for the corresponding population parameter, which is a numerical value that describes a characteristic of an entire population. The process of making inferences about a population from sample data is known as statistical inference. It is a cornerstone of statistical analysis and is widely used in fields such as biology, economics, psychology, and social sciences, among others.
The Process of Statistical Inference
When we talk about statistical inference, we are essentially discussing the process of using data analysis to infer properties of a population from a sample. Here's a step-by-step breakdown of how this works:

1. Random Sampling: The first step is to draw a random sample from the population. This means that every member of the population has an equal chance of being included in the sample. This is crucial because it ensures that the sample is representative of the population.

2. Computation of the Sample Statistic: Once the sample is obtained, the next step is to compute the sample statistic. This could be the mean, median, mode, variance, standard deviation, or any other measure of central tendency or dispersion.

3. Estimation of the Population Parameter: The computed sample statistic is then used as an estimate for the population parameter. For instance, if the sample mean is 50, we might estimate that the population mean is also around 50.

4. Hypothesis Testing: Often, statistical inference involves testing a hypothesis about the population. For example, we might want to test if the population mean is different from a certain value (like 50 in our example).

5. Confidence Intervals: To account for sampling variability, we often construct confidence intervals around our sample statistic. A confidence interval provides a range within which we expect the population parameter to lie with a certain level of confidence.

6. Drawing Conclusions: Based on the results of our hypothesis test or the width of our confidence interval, we can draw conclusions about the population.
**An Example of Sample Statistics in Action**
Let's consider a hypothetical example to illustrate these concepts. Imagine we are interested in the average income of adults in a large city. It would be impractical to survey every adult, so we take a random sample of 500 adults and find that the average income in the sample is $55,000 with a standard deviation of $10,000.
- Sample Statistic (Mean): The average income of the sample, which is $55,000, is our sample statistic.
- Population Parameter: We want to infer the average income of all adults in the city, which is our population parameter.
- Estimation: We might estimate that the average income of the entire city's adults is around $55,000.
- Hypothesis Testing: We could test the hypothesis that the average income is higher than $50,000.
- Confidence Interval: Suppose we construct a 95% confidence interval and find it to be ($54,000, $56,000). This tells us that we are 95% confident that the true average income of the city's adults lies within this range.
- Conclusion: If our confidence interval does not contain the value $50,000, we would reject the null hypothesis that the average income is $50,000 or less. We would conclude that the average income is significantly higher.
Conclusion
Sample statistics are essential tools in the field of statistics. They allow us to make educated guesses about the properties of a population based on a smaller, more manageable sample. By understanding the principles of statistical inference, we can use sample statistics to draw meaningful conclusions about larger populations, making the process more efficient and cost-effective.

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A statistic is a characteristic of a sample. Inferential statistics enables you to make an educated guess about a population parameter based on a statistic computed from a sample randomly drawn from that population (see Figure 1). ... This example is one of statistical inference.read more >>