Interval Estimate Population Variance Calculator

What is an Interval Estimate for Population Variance?

An interval estimate for population variance is a range of values calculated from sample data that is likely to contain the true population variance. This estimate is based on the sample variance and uses statistical theory to determine the confidence level associated with the interval.

The interval estimate is typically expressed as (lower bound, upper bound) and provides a measure of the precision of the estimate. A narrower interval indicates a more precise estimate, while a wider interval suggests greater uncertainty.

How to Calculate Interval Estimate for Population Variance

To calculate the interval estimate for population variance, follow these steps:

1. Collect a sample from the population of interest.
2. Calculate the sample variance (s²).
3. Determine the degrees of freedom (n-1, where n is the sample size).
4. Select a confidence level (common choices are 90%, 95%, or 99%).
5. Find the critical chi-square values from the chi-square distribution table.
6. Calculate the lower and upper bounds of the interval using the formula provided.

The resulting interval will give you a range of values within which the true population variance is likely to fall with the specified confidence level.

Formula for Interval Estimate of Population Variance

The formula for calculating the interval estimate for population variance is:

Where:

• n = sample size
• s² = sample variance
• χ²α/2, n-1 = critical value from the chi-square distribution
• χ²1-α/2, n-1 = critical value from the chi-square distribution
• α = significance level (1 - confidence level)

The critical values are obtained from the chi-square distribution table based on the degrees of freedom (n-1) and the significance level α.

Example Calculation

Let's consider a sample with the following characteristics:

• Sample size (n) = 20
• Sample variance (s²) = 16
• Confidence level = 95%

Using the formula and chi-square distribution tables:

1. Degrees of freedom = n - 1 = 19
2. Significance level (α) = 1 - 0.95 = 0.05
3. Critical values:
   • χ²0.025, 19 ≈ 8.907
   • χ²0.975, 19 ≈ 32.852
4. Lower bound = (19 * 16) / 32.852 ≈ 9.25
5. Upper bound = (19 * 16) / 8.907 ≈ 34.22

The 95% confidence interval for the population variance is approximately (9.25, 34.22).

Interpreting the Results

When you calculate the interval estimate for population variance, consider the following:

• The interval provides a range of plausible values for the population variance.
• A narrower interval indicates a more precise estimate, while a wider interval suggests greater uncertainty.
• The confidence level you select determines the probability that the interval contains the true population variance.
• If the interval is too wide, consider increasing the sample size to improve the precision of the estimate.

Understanding the interval estimate helps you make informed decisions about the variability of the population based on your sample data.