Section 9.2

Estimating a Population Mean

Use the t-distribution to construct confidence intervals for population means when σ is unknown.

1

Point Estimate

The sample mean is the best point estimate for the population mean .

Key Difference from Proportions: When estimating μ, we usually don't know the population standard deviation σ, so we use the sample standard deviation s instead.

2

Student's t-Distribution

Used when σ is unknown and is estimated using the sample standard deviation s.

Properties

Centered at 0 and symmetric

Area under the curve equals 1

Has heavier tails than the standard normal

Approaches Z as df → ∞

Degrees of Freedom:

Interactive t-Distribution Explorer

See how the t-distribution changes with degrees of freedom

Why Not Just Use the Normal Distribution?

When we don't know the population standard deviation , we estimate it using the sample standard deviation . This adds extra uncertainty to our calculations. The t-distribution accounts for this by having heavier tails, making our confidence intervals wider (more conservative).

1 (very wide tails)100 (nearly normal)
-4-2024t (df = 5)Standard Normal (Z)
5
Degrees of Freedom
Sample size n = 6
+20.6%
Tail Density at t=2
vs. standard normal
Wider Tails
Distribution Shape
Noticeably wider
Key Insight: As degrees of freedom increase (larger sample size), the t-distribution approaches the standard normal distribution. This is because with more data, our estimate becomes a better approximation of , reducing the extra uncertainty. When df ≥ 30, the t-distribution is nearly identical to Z.
Confidence Level (Normal) (df = 5)Difference
90%1.6452.015+22.5%
95%1.9602.571+31.2%
99%2.5764.032+56.5%
3

Confidence Interval for μ

Model Requirements

Random sample or randomized experiment
(sample ≤ 5% of population)
Population is normal OR
4

Determining Sample Size

To estimate μ within a margin of error E:

Note: Always round n up to the next whole number.

5

Try It Yourself

t-Interval Calculator for μ

Large Sample (n ≥ 30)

df = n - 1 = 34,

Calculations

Standard Error:1.4030
Margin of Error:2.8354
95% Confidence Interval
(49.565, 55.235)
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