Lower Bound Upper Bound Confidence Interval Calculator with Pooled Variance

This calculator helps you determine the lower and upper bounds of a confidence interval when using pooled variance. Confidence intervals provide a range of values that are likely to contain the true population parameter with a specified level of confidence.

Introduction

When comparing two independent samples, the pooled variance approach is often used to estimate the standard error of the difference between the two means. This method assumes that the two populations have equal variances.

The confidence interval for the difference between two means with pooled variance is calculated using the following formula:

Lower Bound = (X̄₁ - X̄₂) - t*(√(Sₚ²*(1/n₁ + 1/n₂))) Upper Bound = (X̄₁ - X̄₂) + t*(√(Sₚ²*(1/n₁ + 1/n₂)))

Where:

X̄₁ and X̄₂ are the sample means
n₁ and n₂ are the sample sizes
Sₚ² is the pooled variance
t is the critical t-value from the t-distribution

How to Use This Calculator

Enter the sample means for both groups (X̄₁ and X̄₂)
Enter the sample sizes for both groups (n₁ and n₂)
Enter the pooled variance (Sₚ²)
Select the confidence level (typically 90%, 95%, or 99%)
Click "Calculate" to get the confidence interval bounds

The calculator will display the lower and upper bounds of the confidence interval, along with a visualization of the interval.

Formula

The confidence interval for the difference between two means with pooled variance is calculated using the following formula:

Lower Bound = (X̄₁ - X̄₂) - t*(√(Sₚ²*(1/n₁ + 1/n₂))) Upper Bound = (X̄₁ - X̄₂) + t*(√(Sₚ²*(1/n₁ + 1/n₂)))

Where:

X̄₁ and X̄₂ are the sample means
n₁ and n₂ are the sample sizes
Sₚ² is the pooled variance
t is the critical t-value from the t-distribution

The pooled variance is calculated as:

Sₚ² = [( (n₁ - 1)*S₁² + (n₂ - 1)*S₂² ) / (n₁ + n₂ - 2)]

Where S₁² and S₂² are the sample variances.

Worked Example

Suppose we have two independent samples:

Sample 1: n₁ = 20, X̄₁ = 75, S₁² = 100
Sample 2: n₂ = 25, X̄₂ = 70, S₂² = 120

First, calculate the pooled variance:

Sₚ² = [( (20 - 1)*100 + (25 - 1)*120 ) / (20 + 25 - 2)] = [1900 + 2880] / 43 = 4780.47

Next, calculate the standard error:

SE = √(Sₚ²*(1/n₁ + 1/n₂)) = √(4780.47*(1/20 + 1/25)) ≈ √(4780.47*0.095) ≈ √454.14 ≈ 21.31

For a 95% confidence level with degrees of freedom (n₁ + n₂ - 2 = 43), the critical t-value is approximately 2.02.

Now calculate the confidence interval:

Lower Bound = (75 - 70) - 2.02*21.31 ≈ 5 - 43.08 ≈ -38.08 Upper Bound = (75 - 70) + 2.02*21.31 ≈ 5 + 43.08 ≈ 48.08

The 95% confidence interval for the difference between the two means is approximately (-38.08, 48.08).

Interpreting Results

The confidence interval provides a range of values that is likely to contain the true population parameter. For example, if the confidence interval for the difference between two means is (-38.08, 48.08), we can be 95% confident that the true difference between the population means lies within this range.

If the confidence interval includes zero, it suggests that there is no statistically significant difference between the two groups at the specified confidence level. If the interval does not include zero, it suggests a statistically significant difference.

Note: The confidence level you select affects the width of the confidence interval. Higher confidence levels result in wider intervals, while lower confidence levels result in narrower intervals.

FAQ

What is the difference between a confidence interval and a confidence level?

The confidence level is the percentage that the interval is expected to contain the true population parameter. The confidence interval is the range of values calculated from the sample data.

When should I use pooled variance?

Pooled variance should be used when the two samples have equal variances. If the variances are not equal, you should use separate variances instead.

How does sample size affect the confidence interval?

Larger sample sizes result in narrower confidence intervals, providing more precise estimates of the population parameter. Smaller sample sizes result in wider intervals, indicating less precision.