Mean Sd T Df Sig 2-Tailed 95 Confidence Interval Calculation

What is a 95% Confidence Interval?

A 95% confidence interval is a range of values that is likely to contain the true population mean with 95% probability. It's calculated using the sample mean, standard deviation, degrees of freedom, and significance level.

In statistical analysis, confidence intervals help quantify the uncertainty associated with estimating population parameters from sample data. A 95% confidence interval means that if you were to take 100 different samples and calculate a 95% confidence interval for each, approximately 95 of those intervals would contain the true population mean.

How to Calculate the Confidence Interval

To calculate the 95% confidence interval for a mean using the t-distribution, you need four key pieces of information:

1. The sample mean (x̄)
2. The sample standard deviation (s)
3. The degrees of freedom (n-1)
4. The significance level (α = 0.05 for 95% confidence)

The calculation involves finding the critical t-value from the t-distribution table, then using this value to determine the margin of error and finally calculating the confidence interval.

The Formula

The formula for the 95% confidence interval is:

Where:

• x̄ = sample mean
• t = critical t-value from t-distribution table
• s = sample standard deviation
• n = sample size

The critical t-value is determined by the degrees of freedom (n-1) and the significance level (α = 0.05 for 95% confidence).

Worked Example

Let's calculate the 95% confidence interval for a sample with the following data:

• Sample mean (x̄) = 50
• Sample standard deviation (s) = 10
• Sample size (n) = 25

Step 1: Calculate the degrees of freedom (df) = n - 1 = 24

Step 2: Find the critical t-value for df = 24 and α = 0.05 (two-tailed test) from the t-distribution table. The critical t-value is approximately 2.064.

Step 3: Calculate the standard error (SE) = s/√n = 10/√25 = 2

Step 4: Calculate the margin of error (ME) = t * SE = 2.064 * 2 = 4.128

Step 5: Calculate the confidence interval = x̄ ± ME = 50 ± 4.128

The 95% confidence interval is (45.872, 54.128).

Interpreting the Results

When you calculate a 95% confidence interval, you're essentially saying that you're 95% confident that the true population mean falls within the calculated range. For the example above, we can be 95% confident that the true population mean is between 45.872 and 54.128.

If the confidence interval includes values that are clinically or practically meaningful, it suggests that there is evidence to support the presence of an effect or difference. If the interval does not include these values, it suggests that there is not enough evidence to support the presence of an effect or difference.