Confidence Intervals

A confidence interval gives a range of plausible values for a population parameter rather than a single estimate.

Formula for a Mean

CI = x̄ ± z × (σ / √n)

• x̄: the sample mean, the center of the interval.
• z: the critical value for the chosen confidence level. It is 1.645 for 90%, 1.96 for 95% and 2.576 for 99%.
• σ / √n: the standard error of the mean.
• z × (σ / √n): the margin of error, how far the interval extends on each side.

When σ is unknown and the sample is small, we use the sample standard deviation s and replace z with a t value, which gives slightly wider intervals.

How to Read It

A 95% confidence level describes the method. If you repeated the sampling many times and built an interval each time, about 95% of those intervals would contain the true value.

It does not mean there is a 95% chance that the true value sits inside one particular interval. Higher confidence or a smaller sample both widen the interval.

Example: A sample of 100 orders has a mean value of $52, and the population standard deviation is $15. Build a 95% confidence interval.

1. SE = 15 / √100 = 1.5
2. Margin of error = 1.96 × 1.5 = 2.94
3. CI = 52 ± 2.94, which gives (49.06, 55.94)

The true average order value is plausibly between $49.06 and $55.94.

from scipy import stats

stats.norm.interval(0.95, loc=52, scale=1.5)

Code Explanation:

• 0.95 sets the confidence level.
• loc=52 is the center of the interval, the sample mean.
• scale=1.5 is the standard error.
• The function returns the lower and upper limits, about (49.06, 55.94).