Sampling Confidence Interval Calculation

What is a Confidence Interval?

A confidence interval is a range of values that is likely to contain an unknown population parameter. For example, if you want to estimate the average height of all students in a school, you might take a sample of 100 students and calculate a confidence interval around your sample mean.

The confidence level (usually 90%, 95%, or 99%) represents the probability that the interval contains the true population parameter. A 95% confidence interval means that if you took many samples and calculated intervals each time, 95% of those intervals would contain the true parameter.

How to Calculate a Confidence Interval

To calculate a confidence interval, you need three key pieces of information:

• The sample mean (x̄)
• The sample standard deviation (s)
• The sample size (n)

You also need to choose a confidence level. Common choices are 90%, 95%, and 99%. The confidence level determines the critical value (z-score or t-score) used in the calculation.

The calculation involves these steps:

1. Calculate the standard error of the mean (SEM)
2. Determine the critical value based on your confidence level
3. Calculate the margin of error (MOE)
4. Add and subtract the margin of error from the sample mean

The Formula

The general formula for a confidence interval is:

Where:

• Sample Mean (x̄) = Sum of all sample values / Sample size (n)
• Standard Error (SE) = Sample Standard Deviation (s) / √n
• Critical Value = The z-score or t-score corresponding to your confidence level

For large samples (n > 30), you typically use the z-score from the standard normal distribution. For smaller samples, you use the t-score from the t-distribution with n-1 degrees of freedom.

Worked Example

Let's calculate a 95% confidence interval for the average height of students in a school.

Given:

• Sample mean height (x̄) = 165 cm
• Sample standard deviation (s) = 8 cm
• Sample size (n) = 50

Since n > 30, we'll use the z-score for 95% confidence (1.96).

1. Calculate standard error: SE = 8 / √50 ≈ 1.131
2. Calculate margin of error: MOE = 1.96 × 1.131 ≈ 2.222
3. Calculate confidence interval: 165 ± 2.222 → (162.778, 167.222)

We can be 95% confident that the true average height of all students in the school is between 162.78 cm and 167.22 cm.

Interpreting Results

When interpreting a confidence interval, remember:

• The confidence level (e.g., 95%) refers to the method, not the interval itself
• A 95% confidence interval means that if you took many samples, 95% of the intervals would contain the true parameter
• The interval provides a range of plausible values, not a probability that the true parameter is within the interval
• Wider intervals indicate more uncertainty in the estimate

Confidence intervals are particularly useful when comparing different groups or treatments, as they provide a range of expected differences rather than just point estimates.