As a domain expert in statistics, I often encounter questions about confidence levels, particularly the concept of being "90% confident." In statistics, confidence levels are a way to express the reliability of an estimate. When statisticians say they are "90% confident" about a certain parameter, they are referring to the confidence interval, which is a range of values likely to contain the true population parameter.

### Understanding Confidence Intervals

A confidence interval is a statistical concept that provides an estimated range of values that likely includes an unknown population parameter. It is constructed from a data sample where we calculate a range (interval) that, with a certain level of confidence, should contain the true value of the population parameter.

### What Does "90% Confident" Mean?

When we say we are "90% confident," we are making a statement about the method of our estimation, not the actual parameter itself. The statement implies that if we were to take many different samples from the same population and calculate a confidence interval for each sample, then in the long run, 90% of these intervals would contain the true value of the population parameter.

### The Role of Probability

The concept of "90% confidence" is inherently probabilistic. It does not mean that there is a 90% chance the true value is within the calculated interval for a single sample. Instead, it means that if we could repeat the sampling process an infinite number of times, 90% of the intervals calculated from those samples would include the true population parameter.

### Misinterpretations

A common misunderstanding is to conflate the confidence level with the likelihood of the parameter being in the interval for a single sample. It's crucial to note that the 90% confidence level applies to the process of interval estimation across many samples, not to any single interval.

### Factors Influencing Confidence Levels

Several factors can affect the width of a confidence interval and, by extension, the level of confidence we can have in it:


1. Sample Size: Larger samples generally lead to narrower intervals, indicating more precision.

2. Variability in the Data: Greater variability in the data can result in wider intervals.

3. Level of Confidence: Higher confidence levels (e.g., 95% vs. 90%) typically result in wider intervals to account for the increased certainty.

4. Margin of Error: The margin of error is the range within which the true value is expected to lie. A smaller margin of error can increase confidence but may require a larger sample size.

### Practical Implications

In practical terms, a 90% confidence interval is often used when a slightly wider interval is acceptable in exchange for the simplicity of calculations or when the cost of data collection is a concern. It's a balance between the precision of the estimate and the resources required to achieve that precision.

### Conclusion

To be "90% confident" means that you are using a statistical tool that, if applied correctly and repeatedly, should capture the true population parameter 90% of the time. It's a measure of the reliability of the statistical method rather than a statement about the certainty of any single interval.

Now, let's proceed with the translation into Chinese.

read more >>

The confidence interval can be expressed in terms of a single sample: "There is a 90% probability that the calculated confidence interval from some future experiment encompasses the true value of the population parameter." Note this is a probability statement about the confidence interval, not the population parameter.read more >>