Python How to Calculate 95 Percent Confidence Interval

What is a 95% Confidence Interval?

A 95% confidence interval (CI) is a range of values that likely contains the true population parameter with 95% probability. For sample means, it's calculated using the sample mean, standard error, and critical t-value from the t-distribution.

The 95% confidence interval means that if we took many samples and calculated a 95% CI for each, about 95% of these intervals would contain the true population mean.

Python Method Using scipy.stats

The scipy.stats module provides the t.interval() function to calculate confidence intervals. Here's how to use it:

Step-by-Step Code

1. Import the required functions from scipy.stats
2. Calculate the sample mean and standard deviation
3. Use t.interval() to compute the confidence interval

from scipy import stats
import numpy as np

# Sample data
data = [23, 25, 28, 22, 27, 24, 26, 29, 21, 25]

# Calculate sample mean and standard deviation
sample_mean = np.mean(data)
sample_std = np.std(data, ddof=1)  # ddof=1 for sample standard deviation
n = len(data)

# Calculate 95% confidence interval
confidence_interval = stats.t.interval(
    confidence=0.95,
    df=n-1,
    loc=sample_mean,
    scale=sample_std/np.sqrt(n)
)

print(f"95% Confidence Interval: {confidence_interval}")
                        

The output will show the lower and upper bounds of the confidence interval. For our example data, this would be approximately (22.5, 27.5).

Worked Example

Let's calculate a 95% confidence interval for the following test scores: 82, 85, 78, 90, 88, 84, 79, 86, 81, 83.

Step 1: Calculate Sample Statistics

• Sample mean (x̄) = 83.5
• Sample standard deviation (s) = 3.5
• Sample size (n) = 10

Step 2: Find Critical t-value

For n=10, degrees of freedom (df) = 9. The critical t-value for 95% CI is approximately 2.262.

Step 3: Calculate Standard Error

SE = s / √n = 3.5 / √10 ≈ 1.17

Step 4: Compute Confidence Interval

CI = x̄ ± t × SE = 83.5 ± 2.262 × 1.17 ≈ (80.9, 86.1)

This means we're 95% confident the true population mean test score is between 80.9 and 86.1.

Interpreting Results

When you calculate a 95% confidence interval, you're making a probabilistic statement about the population parameter. Key points to remember:

• The interval is not the probability that the true value is within the interval
• If you took many samples and calculated 95% CIs, about 95% would contain the true value
• A narrower interval indicates more precise estimation
• Wider intervals occur with smaller sample sizes or higher variability

Common Mistakes to Avoid

1. Using the wrong distribution: For small samples (n < 30), always use the t-distribution. For larger samples, you can use the normal distribution.
2. Incorrect degrees of freedom: Remember df = n - 1 for sample standard deviation calculations.
3. Misinterpreting the confidence level: A 95% CI doesn't mean there's a 95% chance the true value is in the interval.
4. Ignoring sample size: Smaller samples will naturally produce wider confidence intervals.