STAT 2020

CONFIDENCE INTERVAL FOR POPULATION MEAN µ

NORMAL NON-NORMALPARENT PARENT

σ2 known x̄± zα2

σ√n

n small: § [NO!]

n > 30: x̄± zα2

σ√n

[CLT helps you]

σ2 unknown x̄± tα2,n−1

s√n

n small: §

n > 60: x̄± zα2

s√n

NOTE

1. Point Estimate of µ

µ̂ = x̄, and it is the mid-point of the 2-sided confidence interval for µ.

2. Standard Error of Sample Mean:

s.e.(X̄) = σ√n

3. E or Error or Margin of Error (or Precision):

E = zα2

σ√n, and it is half the width of the confidence interval.

4. As n increases, the width of confidence interval decreases. (More sample givesbetter precision.)

As confidence level increases, the width of confidence interval increases. (Inorder to be more confident to catch the fish, you need a bigger net.)

5. Commonly used z scores: z0.05 z0.025 z0.005

1.645 1.96 2.575

6. Interpretations of Confidence Interval:

• Probabilistic Interpretation:

With repeated sampling, 100(1−α) percent of all confidence intervals willinclude the true population mean µ in the long run.

• Practical Interpretation:

For a single sampling, we are 100(1 − α) percent confident that the truepopulation mean µ falls in the confidence interval.

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