CI for Given Mean and Variance

Confidence Intervals: Unveiling the Range of Plausible Values

In civil engineering, dealing with uncertainty is inevitable. Material properties, traffic flow, and even soil strength can vary. Confidence intervals help us quantify this uncertainty for two key statistical measures: mean and variance.

Confidence Interval for the Mean (µ)

Imagine you want to estimate the average compressive strength (MPa) of concrete cylinders produced by a batch plant. You take a random sample of n cylinders and calculate the sample mean (x̄). However, this might not perfectly reflect the entire batch's true mean (µ).

A confidence interval for the mean provides a range of values where µ is likely to lie, with a certain level of confidence (usually 1 - α, expressed as a percentage). It's like saying, "We're X% confident the true mean falls within this interval."

Here's the formula for a confidence interval for the mean, assuming a normally distributed population (often applicable in engineering):

µ ≈ x̄ ± (z_α/√n) * σ

• x̄: Sample mean (calculated from your sample)
• z_α: Critical value from the standard normal distribution table (based on chosen confidence level 1 - α)
• σ: Population standard deviation (often estimated by sample standard deviation, s)
• n: Sample size

Confidence Interval for the Variance (σ^2)

Similarly, you might be interested in the variability of concrete compressive strength within the batch. The population variance (σ^2) reflects this spread. However, you can only estimate it using the sample variance (s^2).

A confidence interval for the variance helps you pinpoint a range where σ^2 is likely to reside. Here's the formula for a chi-square distribution-based confidence interval (assuming a normal population):

(n - 1) * s^2 / χ²_(α/2, n-1)  < σ^2 <  (n - 1) * s^2 / χ²_(1 - α/2, n-1)

• s^2: Sample variance (calculated from your sample)
• χ²: Chi-square critical values from the chi-square distribution table (based on chosen confidence level 1 - α and degrees of freedom n - 1)
• n: Sample size

Example: Confidence Interval for Compressive Strength

A civil engineer tests 10 (n = 10) concrete cylinders from a batch and finds a sample mean compressive strength (x̄) of 30 MPa and a sample standard deviation (s) of 2 MPa. Let's construct a 95% confidence interval for the mean compressive strength (α = 0.05).

1. Find the critical value (z_α) for a 95% confidence level (1 - α = 0.95). Using a standard normal distribution table, z_α ≈ 1.96.
2. Substitute the values: µ ≈ 30 ± (1.96 / √10) * 2 ≈ 30 ± 1.26 MPa.

Therefore, we can be 95% confident that the true mean compressive strength (µ) of the batch lies between 28.74 MPa and 31.26 MPa.

Remember:

• These are just examples. The specific formulas and critical values might differ depending on the underlying distribution and chosen confidence level.
• Always consult relevant engineering codes and practices for appropriate statistical procedures in your field.

By understanding confidence intervals, you can make more informed decisions in civil engineering, accounting for the inherent variability in materials and processes.

 

Let's build on the previous example and find the confidence interval for the variance (σ^2) of the concrete compressive strength, assuming a 95% confidence level (α = 0.05).

We have the following information from the previous example:

• Sample size (n) = 10
• Sample variance (s^2) = 2 MPa^2

We need to find the chi-square critical values based on the chosen confidence level (1 - α = 0.95) and degrees of freedom (n - 1).

1. Degrees of Freedom (df): df = n - 1 = 10 - 1 = 9
2. Chi-Square Critical Values:
• Lower critical value (χ²_(α/2, df)): We need the value at α/2 = 0.05/2 = 0.025 with 9 degrees of freedom. You can find this using a chi-square distribution table or a statistical software package. Generally, chi-square tables provide values for the upper tail (1 - α). Look for the value in the table with 9 degrees of freedom such that the cumulative area to the right is 0.975 (1 - 0.025). This value is approximately χ²_(0.025, 9) ≈ 17.28.
• Upper critical value (χ²_(1 - α/2, df)): This corresponds to 1 - α/2 = 1 - 0.025 = 0.975 with 9 degrees of freedom. Look for the value in the chi-square table with 9 degrees of freedom such that the cumulative area to the right is 0.025. This value is approximately χ²_(0.975, 9) ≈ 2.00.
3. Confidence Interval Calculation:
• Lower limit: (n - 1) * s^2 / χ²_(α/2, df) = (10 - 1) * 2 MPa^2 / 17.28 ≈ 1.16 MPa^2
• Upper limit: (n - 1) * s^2 / χ²_(1 - α/2, df) = (10 - 1) * 2 MPa^2 / 2.00 ≈ 10.00 MPa^2

Therefore, based on the sample data and with 95% confidence, we can say that the true population variance (σ^2) of the concrete compressive strength likely falls between 1.16 MPa^2 and 10.00 MPa^2. This indicates that the individual concrete cylinders might have a variance in compressive strength ranging from roughly 1.1 MPa to 3.2 MPa (square root of the variance limits).