Chapter 8 CI Review

🧠 Concept Review

A confidence interval gives a range of plausible values for a population parameter (like the mean μ).

CI formula (σ known): CI = X̄ ± z_α/2 × (σ / √n) CI formula (σ unknown, small n): CI = X̄ ± t_{α/2, df} × (s / √n)

The critical value (z or t) depends on your confidence level. For 95%, z ≈ 1.96.

🔢 Try a Sample Problem

Suppose a sample of n = 10 has a sample mean of X̄ = 64.46 and population SD σ = 1.

CI = 64.46 ± 1.96 × (1 / √10) = 64.46 ± 0.62 = [63.84, 65.08]

✅ What Does This Tell Us?

We are 95% confident that the true population mean μ lies between 63.84 and 65.08. This means that if we repeated this sampling process many times, 95% of the calculated intervals would contain the true μ.

📏 FYI: CI for Variance or Standard Deviation (σ²)

This type of CI is based on the chi-squared distribution and is rarely used in business or finance but can appear in engineering quality control.

CI for Variance: Lower = (n−1)s² / χ²_1−α/2 Upper = (n−1)s² / χ²_α/2 CI for Std Dev = Square root of variance bounds.

Example: A sample of n = 10 has s = 3. Compute a 95% CI for variance.

df = 9, χ²_0.975 ≈ 19.02, χ²_0.025 ≈ 2.70 CI = [ (9)(3²)/19.02, (9)(3²)/2.70 ] = [4.25, 30.00] CI for σ = [√4.25, √30.00] ≈ [2.06, 5.48]

This tells us the true standard deviation is likely between 2.06 and 5.48.

🧮 CI Calculator

Sample Mean X̄: Standard Deviation σ: Sample Size n: Confidence Level %:

📝 Quick Quiz

1. If n = 100 and σ = 4, what is the margin of error at 95% confidence for X̄ = 50?

z = 1.96, MOE = 1.96 × (4 / √100) = 1.96 × 0.4 = 0.784
Margin of Error (MOE) is the distance from the sample mean to either CI boundary.

2. What happens to the CI length when sample size increases?