A better interpretation is:
If we repeatedly took random samples and constructed confidence intervals using the same method, approximately 95% of those intervals would contain the true population parameter.
In everyday communication, we often say:
"We are 95% confident that the true population parameter lies within this interval."
🔹 8. Confidence Level and Interval Width
A higher confidence level generally produces a wider confidence interval.
For example:
• 90% CI → [48.5, 51.5]
• 95% CI →[48,52]
• 99% CI →[47,53]
The exact values depend on the data, but the general relationship is:
Higher confidence → Wider interval
Lower confidence → Narrower interval
Why? Because if we want greater confidence that our interval captures the true population parameter, we need to consider a wider range of possible values.
🔹 9. Sample Size and Confidence Interval
Sample size has a major impact on confidence intervals.
For a sample mean:
Standard Error = Standard Deviation / √Sample Size
As sample size increases:
Sample Size ↑ → Standard Error ↓
Therefore: Larger Sample → Smaller Uncertainty → Narrower Confidence Interval
For example:
Suppose Standard Deviation = 20
With n = 100 → SE = 20 / √100 = 20 / 10 = 2
If we increase the sample size to n = 400 → SE = 20 / √400 = 20 / 20 = 1
The standard error has decreased. This means the estimate becomes more precise.
🔹 10. Standard Deviation vs Standard Error
These concepts are often confused.
Standard Deviation
Standard deviation measures how spread out individual observations are.
Example:
How different are individual employee salaries from the average salary?
Standard Error
Standard error measures how much a sample statistic, such as the sample mean, is expected to vary from sample to sample.
For the sample mean: SE = SD / √n
So: SD = 20, n = 100, Then SE = 20 / 10 = 2
Therefore: Standard Deviation = 20, Standard Error = 2
They measure different things.
🔹 11. Example of a Confidence Interval
Suppose we have:
Sample mean = 50
Sample standard deviation = 10
Sample size = 100
Confidence level = 95%
For illustration, let's use a critical value of approximately 1.96.
First calculate the standard error:
SE = 10 / √100 = 10 / 10 = 1
Now calculate the margin of error:
Margin of Error = 1.96 × 1 = 1.96
Therefore: CI = 50 ± 1.96
So: Lower Limit = 48.04, Upper Limit = 51.96
Therefore: 95% CI = [48.04, 51.96]
🔹 12. Confidence Interval Using Python
Python's scipy library can be used to calculate confidence intervals.
import numpy as np
from scipy import stats
data = np.array([48, 51, 49, 52, 50, 47, 53, 51, 49, 50])
mean = np.mean(data)
confidence_level = 0.95
confidence_interval = stats.t.interval(
confidence_level,
df=len(data) - 1,
loc=mean,
scale=stats.sem(data)
)
print("Mean:", mean)
print("95% Confidence Interval:", confidence_interval)