Step by Step
1
When to use z-procedures
Use z-procedures only when the population standard deviation (σ) is actually known — a fairly rare situation in real-world practice.
2
When to use t-procedures
Use t-procedures when σ is unknown, which is the case in nearly all real-world inference about means — instead, the sample standard deviation (s) is used to estimate σ.
3
Degrees of freedom
For a one-sample t-procedure, degrees of freedom = n − 1, where n is the sample size. Degrees of freedom affects the exact shape of the t-distribution used.
4
How the t-distribution compares to the normal (z) distribution
The t-distribution is bell-shaped and symmetric, like the normal distribution, but has heavier tails — reflecting the extra uncertainty introduced by estimating σ with s. As degrees of freedom increases, the t-distribution approaches the normal (z) distribution.
Applied Walkthrough
1
A researcher wants to construct a confidence interval for a population mean, using a random sample of size 25, but doesn't know the true population standard deviation.
2
Since σ is unknown (the typical real-world situation), a t-procedure is required rather than a z-procedure.
3
Degrees of freedom = n − 1 = 25 − 1 = 24.
4
The t-interval is calculated as: x̄ ± t*×(s/√n), using the critical t-value corresponding to 24 degrees of freedom — slightly wider than a z-interval would be, reflecting the extra uncertainty from estimating σ with the sample standard deviation s.
Exam Application
Exams test whether you can correctly determine when to use a t-procedure versus a z-procedure (based on whether σ is known), and whether you can correctly calculate degrees of freedom and apply the t-interval formula.
⚠ Common Trap
The most common trap is defaulting to z-procedures out of habit, when in reality σ is almost never known in real-world scenarios — the correct default assumption should be to use t-procedures unless σ is explicitly stated as known.
✓ Quick Self-Check
1. When should you use a z-procedure instead of a t-procedure?
Only when the population standard deviation σ is actually known.
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2. What is used instead of σ in a t-procedure?
The sample standard deviation, s.
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3. What is the degrees of freedom formula for a one-sample t-procedure?
n − 1.
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4. How does the t-distribution compare to the normal distribution?
It's bell-shaped and symmetric like the normal distribution, but has heavier tails, and approaches the normal distribution as degrees of freedom increases.
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5. What is the formula for a one-sample t-interval?
x̄ ± t*×(s/√n).
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