As a subject matter expert in statistical analysis, I specialize in interpreting and applying statistical methods to real-world data. When we talk about a "90% confidence interval," we're delving into the realm of inferential statistics, which is concerned with making inferences about populations based on sample data.

### What is a Confidence Interval?

A confidence interval is a range that is likely to contain an unknown population parameter. It's derived from a sample and is used to estimate the range within which the true population parameter lies. The confidence interval is calculated with a certain level of confidence, which is a measure of how certain we can be that the interval contains the population parameter.

### Understanding the 90% Confidence Level

When we say we have a "90% confidence interval," we're making a statement about the reliability of our estimate. Here's what it means:

- Probability Statement: The statement "There is a 90% probability that the calculated confidence interval from some future experiment encompasses the true value of the population parameter" is a bit of a misnomer. It's not the interval that has the probability; rather, it's the method that, if repeated many times, would yield intervals that contain the true population parameter in 90% of the cases.

- Long-Run Frequency: The 90% confidence level is based on the long-run frequency with which the confidence interval method will capture the true population parameter. If we were to take many samples from the same population and calculate a 90% confidence interval from each, we would expect that about 90% of these intervals would contain the true value of the parameter.

- Not a Guarantee: It's important to note that a 90% confidence interval is not a guarantee that the true parameter is within the interval for a single sample. It's a statement about the long-term accuracy of the method, not about the success of any individual interval.

- Margin of Error: The confidence interval includes a margin of error, which is the amount by which the interval may miss the true population parameter. The margin of error is often denoted as E and is part of the formula for calculating the confidence interval.

- Sample Size and Variability: The width of the confidence interval is influenced by the sample size and the variability within the data. Larger sample sizes and less variability typically result in narrower confidence intervals, indicating a more precise estimate.

### Calculating a Confidence Interval

The calculation of a confidence interval typically involves the following steps:


1. Select the Confidence Level: Choose the level of confidence, such as 90%, which corresponds to a z-score (for a normal distribution) or a t-score (for a t-distribution, often used with smaller sample sizes).


2. Determine the Sample Statistic: Calculate the sample mean or proportion, depending on what parameter you're estimating.


3. Calculate the Standard Error: The standard error is a measure of the variability of the sample statistic and is calculated as the standard deviation of the sample divided by the square root of the sample size.


4. Determine the Margin of Error: Multiply the standard error by the z-score or t-score to find the margin of error.


5. Construct the Interval: The confidence interval is constructed by adding and subtracting the margin of error from the sample statistic.

### Interpreting the Results

When you report a 90% confidence interval, you're saying that you are fairly confident—based on the data and the method used—that the interval captures the true population parameter. However, there is still a 10% chance that the interval does not include the true value. This is a fundamental concept in statistical inference and is crucial for making informed decisions based on data.

Now, let's proceed to the translation.

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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 >>