L-INF-3: Confidence Intervals for a Population Mean (σ Unknown)
Module 4 · Statistical Inference
Section 1: Introduction
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Dublin, 1908. William Sealy Gosset, a chemist at the Guinness brewery, ran experiments on barley with very small samples, sometimes only four measurements at a time. Nobody knew the population standard deviation , so the standard recipe of the day was L-INF-2’s interval with the sample’s own standard deviation dropped in where should be:
Make a bet before reading on. With measurements from a normal population, how often would intervals built this way capture the true mean: about 95% of the time, about 90%, about 85%, or about 75%?
Pause and write. Write your bet and one sentence of reasoning on paper or in your notes. You will check it with a simulation in C1, and it comes back in the Boss Fight.
Gosset worked out the answer, and the fix. The brewery did not let its staff publish under their own names, so he published it under the pen name “Student”. That fix is the t-distribution, and this lesson is about when and how to use it.
After this lesson, you will be able to:
Explain why an interval built with and captures less often than it claims
Find and from a t-table, and construct a t-interval for
Choose between a t-interval, a z-interval, or no mean interval yet, from what the study tells you
Use a t-interval to decide whether a population mean is above or below a threshold
Section 2: Prerequisites
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A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
This lesson changes one ingredient of the mean interval you built in L-INF-2.
From L-INF-2 — interval structure: point estimate ± margin of error. You will keep this structure and replace its critical value and standard-error ingredients.
From L-INF-2 — standard error: a mean varies by a standard deviation divided by . You will use because is unknown.
From L-INF-1 — shape and independence: a probability model needs independent observations and a defensible sampling-distribution shape. These conditions matter more when the sample is small.
Retrieval Warm-up — from L-INF-2
Before continuing, check the foundations you will reuse:
Bridge to the new rule: L-INF-2 supplied the interval blueprint. This lesson changes how uncertainty is measured when is unknown: use , use a t critical value, and look it up with . A large sample can make z a close approximation, but it does not make known.
Section 3: Core Concepts
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Your route through the ideas:
C1–C4: see why t is needed, how its shape works, why , and how t approaches z.
C5–C6: find the critical value and build the interval.
C7–C8: check conditions and choose the defensible procedure.
C9–C10: interpret the interval and explain its width.
C1 — Why the -Distribution Falls Short
In a z-interval, the population standard deviation is treated as known. When is unknown, we replace it with the sample standard deviation . That estimate changes from sample to sample, so the interval must account for more uncertainty than a z critical value does.
Settle your bet. In the simulator below, choose , keep 95%, and draw 40 samples with the z rule. Before you press the button, write how many of the 40 intervals you expect to miss . Then switch to the t rule: the same 40 samples, each interval stretched by the same factor. Finally, sort the samples by and look at where the misses are.
Predict first: of 40 intervals built as x̄ ± z*·s/√n with n = 6 at 95%, how many will miss
μ? Then draw.
Confidence
Rule
μ = 0
One sample's interval (z rule)
Pale band = t* interval, dark core = z* interval
✗ missed μ
Figure 1: Each row is one random sample of size n from a normal population
with μ = 0, and its interval x̄ ± (critical value)·s/√n. Draw once, then switch the rule: the
same samples are re-scored with t* instead of z*, so every interval stretches in place about
its own x̄. Sort by s to see which intervals the z rule misses.
The misses are mostly the short intervals: samples whose happened to come out smaller than . A of 1.96 gives those samples too little room. Under a normal population the z recipe captures only about 85.5% of the time when , and about 89.3% when ; your run of 40 will wobble around those rates. Gosset’s bet comes out at “about 85%”.
Why t replaces z when σ is unknown
When is estimated by , the standardized statistic follows a t-distribution under the one-sample t model. For a finite sample, using with produces an interval that is too narrow for its advertised confidence level.
A sample size of 30 does not make known. With unknown , t is the principled procedure at every sample size; z may become a close approximation as the degrees of freedom grow.
C2 — The -Distribution: Shape and Properties
The -distribution is a family of bell-shaped, symmetric distributions. One member of the family is selected by its degrees of freedom. We write to mean degrees of freedom.
The t-Distribution
The t-distribution with degrees of freedom is symmetric and centered at 0. Compared with the standard normal :
It has heavier tails — more probability in the extremes
Its peak is slightly lower than the normal
As , it converges to
The heavier tails produce larger critical values. For example, at 95% confidence and ,, compared with .
Figure 2: The t-distribution (dashed) against the standard normal (solid). Slide df to
watch the t curve's heavier tails shrink toward the normal; at the last stop, df = ∞, the two
curves are the same and t* = z*. With critical values shown, t* sits farther out than z* for the
same confidence level, and the shaded tails each hold the stated tail area.
Quick Check — Small-Sample Critical Values
For a 95% confidence interval with and unknown , how does the correct compare with ?
C3 — Degrees of Freedom: Why ?
Once the sample mean is fixed, the deviations from it must sum to zero. If deviations are known, the last is forced. Only pieces can vary freely.
Figure 3: Five data values with x̄ = 10 (already computed). Each row shows one value's
deviation from x̄. Drag any green point: the locked point x₅ moves automatically to keep the mean
at 10, because the five deviations must add to zero. Only 4 of the 5 values are free; the last
is determined. That is why df = n − 1 = 4.
Degrees of Freedom
For a one-sample t-interval, the degrees of freedom are:
where n is the sample size. This is the row you look up in the t-table.
The single most common arithmetic error in this lesson is using instead of . For :, not 12. Always subtract 1 before looking up . Getting this wrong gives you the wrong critical value and therefore the wrong CI.
Quick Check — Degrees of Freedom
A researcher with observations reads the t-table in the row labelled 17. What went wrong?
C4 — How t Approaches z
As grows, becomes a more stable estimate of . The extra uncertainty shrinks, so the t-distribution approaches the standard normal.
Convergence of t to z
For any fixed cumulative probability, the t critical value approaches the corresponding z critical value as . At every finite , the positive two-sided t critical value remains larger than .
At 95% confidence, for , for , and for . The values move toward from above.
The convergence is gradual, not a switch at one magic sample size. That is why t remains the principled procedure whenever is estimated.
C5 — Reading the -Table
We write to mean the positive critical value that leaves the chosen central confidence area under a t curve. It depends on both and the confidence level.
Choosing t* from a Table
First compute . Then use that row and the desired two-sided confidence-level column. Their intersection is .
n = 6→
df = n − 1 = 5
Confidence level
Figure 4: Pick a sample size and confidence level. The lookup always goes
n → df = n − 1 → confidence column → t* — you read the table at row df, never row n. The highlighted cell and the readout above update together so the
off-by-one step stays visible.
The critical values in these tables are rounded to three decimal places. Treat a displayed critical value—and any decimal calculation that uses it—as an approximation. Keep unrounded calculator values through intermediate arithmetic, then round the requested final answer. The full t-table is in Section 10.
The table row is , not . With , use the row; the critical value belongs to a sample of 13.
Quick Check — Reading the t-Table
A sample of observations has unknown. You are constructing a 95% CI. What is the correct critical value ?
C6 — Confidence Interval Formula
The interval keeps the L-INF-2 structure: estimate ± margin of error. We write for the population mean, for its sample estimate, for the sample standard deviation, and for sample size.
Confidence Interval for μ (σ Unknown)
which gives the interval:
where is the critical value from the -table with and the desired confidence level.
Figure 5: Adjust n and s to watch every piece of the formula update live.
Notice that df = n − 1 (not n) controls which row of the t-table you look up, and
that t* is always larger than z*, so the t-interval (solid) is always wider than the z-approximation
(dashed) built from the same data.
If ,,, and , then , so the margin is and the interval is approximately .
C7 — When Can You Use the -Interval?
Conditions for a Mean Interval
A confidence interval for needs observations from a random/independent process and a sampling distribution of that is plausibly close to normal. For a small sample, that second condition requires population-shape evidence with no severe skew or extreme outliers. Once those conditions license inference, use the t procedure when is unknown and estimated by .
What does “approximately normal” look like in practice? When you inspect a histogram or dot plot, look for:
Single peak — the data should mound in one place, not two or three
Rough symmetry — neither tail should be dramatically longer than the other
No extreme outliers — one or two points far beyond the rest can distort the t-interval, especially when is very small
There is no universal sample-size cutoff that turns a visibly problematic distribution into a safe one. Smaller samples demand stronger shape evidence; larger samples are more robust to moderate departures.
Robustness guide for a t-interval after sampling and independence conditions hold.
Shape of population
n = 6
n = 12
n = 20
ShapeApproximately normalSymmetric, single peak
n = 6✓Strong shape support
n = 12✓Strong shape support
n = 20✓Strong shape support
ShapeModerate skewOne tail longer, no outliers
n = 6✗Weak support — seek more evidence
n = 12⚠Case-dependent — inspect influence
n = 20✓Often robust — still inspect influence
ShapeStrong skew / outliersHeavy tail or extreme values
n = 6✗Poor support
n = 12✗Poor support
n = 20✗Poor support — size does not repair outliers
✓ Strong shape support⚠ Case-dependent — inspect skew and influential values✗ Poor shape support — seek another method or more evidence
Figure 6: Assuming a representative, independent sample, shape evidence matters most for very
small samples. Larger samples are often more robust to moderate skew, but no sample-size cutoff
guarantees a trustworthy interval, and influential outliers remain a serious concern.
The t-distribution accounts for estimating ; it does not repair dependence, biased sampling, severe skew, or extreme outliers.
Quick Check — Checking the Conditions
A researcher collects an independent random sample of reaction-time measurements from the target population. is unknown, and a histogram of the sample looks roughly bell-shaped with no extreme outliers. Which statement best describes whether a t-interval is valid?
You will judge real dot plots in Guided Practice Problem 3, where some samples pass and some do not.
C8 — When to Use vs. : The Practical Decision Rule
First check whether inference is valid. Then choose the critical-value family from what is known about the population standard deviation.
t-versus-z Decision Rule
Is known? → Use
Is unknown and estimated by ? → Use
For large , z may be a close approximation to t, but t remains the principled choice.
Figure 7: Choosing a one-sample mean interval. First ask whether the sampling
process and the population-shape evidence support a mean interval at all; if not, neither
procedure rescues the data. If they do, use z only when σ is known from outside this
sample, and t when σ is estimated by s.
A sample of with only reported still calls for t. Its large makes the t and z answers close; it does not change which standard deviation is known.
C9 — Interpreting the -Interval
Confidence-Interval Interpretation
“We are confident that the population mean lies between and ” means the interval came from a procedure that captures in about of repeated samples under the model conditions.
What it does NOT mean: “There is a probability that lies in this interval.” The mean is a fixed constant — it either is or isn’t in your interval. Everything you learned about reading an interval in L-INF-2 carries over unchanged; only the critical value is new.
A confidence interval for is not a prediction interval for individual observations, and it does not place a post-data probability on the fixed parameter.
C10 — Why the -Interval is Wider (and That’s Correct)
Width Comparison
For the same center, standard error, and confidence level, a finite-df t-interval is wider than the corresponding z-interval because . The difference shrinks as grows.
At 95% confidence with , while . With the same standard error, the t margin is about 13.7% larger.
The wider interval is not a penalty or a mistake. It is the visible cost of estimating instead of knowing it. A narrower z-interval can even reach a firmer-looking conclusion than the data support; the Boss Fight is about exactly that.
Section 4: Worked Examples
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Example 1 — Computing a t-Interval (Fully Worked)
A clinic records the change in pain score for an independent random sample of patients following one treatment protocol. A dot plot of the changes is roughly symmetric with no extreme outliers. The sample gives points and points; is unknown. Construct a 95% confidence interval for the population mean change .
Step 1 — Notice the evidence and choose the model.
I notice an independent random sample, unknown , and small-sample shape evidence with no extreme outlier. That supports a t-interval.
Step 2 — Find and .
The 95% t-table entry for is:
Step 3 — Compute the standard error and margin of error.
Using the unrounded standard error,
Step 4 — Build and interpret the interval.
We are 95% confident that the population mean change under this protocol is between 2.91 and 4.69 points. Because this is a one-sample interval with no comparison group or assignment design described, it does not by itself isolate a treatment effect.
If we incorrectly used , the margin would be:
That interval would be approximately —narrower because it fails to account for estimating .
Example 2 — Constructing a 90% t-Interval (Partially Scaffolded)
An environmental scientist takes 8 independent soil samples from randomly selected locations near an industrial site. The measurements are roughly symmetric with no extreme outliers and give ppm and ppm. Construct and interpret a 90% confidence interval for the population mean lead concentration. Before revealing the solution, predict the degrees of freedom and the t-table row you will use.
Pause and write. Record your answer and reasoning on paper or in your notes before continuing. Even if you are unsure, committing your thinking helps you see what you understood and what you need to revisit.
Show Solution
Step 1 — Choose the procedure.
The sample is independent, the small-sample shape is defensible, and is unknown, so use t.
Step 2 — Find and .
Step 3 — Compute the standard error and margin.
Step 4 — Form and interpret the interval.
We are 90% confident that the population mean lead concentration for the sampled area is between 19.69 and 25.31 ppm.
Example 3 — 99% Confidence Interval (Minimally Scaffolded)
A nutritionist takes an independent random sample of students. A plot of daily caloric intake shows no extreme outliers and only mild asymmetry. The sample gives calories and calories; is unknown. Construct and interpret a 99% confidence interval. Determine the method, ,, margin, and bounds without a supplied setup.
Pause and write. Complete the requested analysis in your notes before opening the solution.
Show Solution
Step 1 — Choose t.
The independent random sample, acceptable shape evidence, and unknown support a t-interval.
Step 2 — Find and .
Step 3 — Compute the standard error and margin.
Step 4 — Form and interpret the interval.
We are 99% confident that the population mean daily caloric intake is between about 1952 and 2348 calories.
Example 4 — Choosing the Right Distribution (Application Twist)
A quality-control engineer randomly samples independent machined parts. A process plot is approximately symmetric with no extreme outliers. The sample gives mm and mm; is unknown.
A colleague’s analysis: “Because 25 is close to 30, use . The 95% interval is approximately mm.”
Diagnose the method error, construct the correct interval, and explain why the two answers are close without treating 30 as a magic boundary.
Pause and write. Complete the requested diagnosis and interval in your notes before opening the solution.
Show Solution
Step 1 — Diagnose the method.
Because is unknown and estimated by , use t. The value 30 is not a method switch.
Step 2 — Find the critical value.
Step 3 — Compute the t interval.
Step 4 — Compare with the colleague’s interval.
The answers are close because already makes fairly close to . The t interval is still the principled result because was estimated.
Section 5: Guided Practice
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Work through each problem step by step. Immediate feedback identifies the decision or calculation that needs repair, and generator-backed problems provide fresh practice.
Problem 1 — Build a t-Interval, Step by Step (C3 + C5 + C6)
Every step is guided. The t-table excerpt in the problem has three rows; choosing the right one is part of the work.
Problem 2 — What Does the Table Header Mean? (C5)
Answer on the Graph
Which area is the 0.05?
In the t-table, the 95% column is headed “0.025 (1)” and “0.05 (2)”: one tail of 0.025, or two tails together of 0.05. On this t curve with 11 degrees of freedom, select every region that together make up the two-tail area 0.05.
Select each region you want to shade. Select a shaded region again to clear it. Keyboard: arrow keys move through the figure; Enter or Space selects.
Nothing shaded yet.
Problem 3 — z, t, or No Interval Yet? (C7 + C8)
Sort into groups
Which interval would you report?
Each card describes a study of a population mean. Sort it by the procedure its information supports. Read the dot plots: shape is part of the decision.
Drag an item into its group, or select it and then choose the group. On a keyboard: select an item and press the group’s number (1–3); 0 sends it back to Unsorted. Each item also has its own “move” buttons.
Unsorted Drop here
Every item is sorted.
Problem 4 — What Changes the Method? (C7 + C8)
These studies are deliberately almost identical. Choose the interval procedure for each one, then use the feedback to name the single fact that changes the method.
Structural Difference
Known population spread or estimated sample spread?
Identify what key word or condition separates Case A from Case B, then select the method for each.
Scenario A
A normal production process is studied with a random sample of n = 16. The population standard deviation is known: σ = 5.8.
Scenario B
A normal production process is studied with a random sample of n = 16. The population standard deviation is unknown; the sample standard deviation is s = 5.8.
Problem 5 — Same Spread, Three Sample Sizes (C4 + C5 + C10)
Complete the table
Same spread, three sample sizes
Fill in every empty cell, then check the whole table once. The cells depend on one another.
A colleague looks at a table like this one and says: “Report the z-based margins. They are smaller, so they are more informative.” Which response is most defensible?
Problem 6 — On Your Own, with Steps on Request (C3 + C5 + C6)
A new problem is generated each time. Try the whole interval first; open the steps only if you need them.
Section 6: Independent Practice
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Each problem begins with a cold commitment. Numeric work is typed; longer procedures offer steps only after you choose that support. From here on, sample sizes are no longer perfect squares, so keep the unrounded standard error through the calculation.
Problem 1 — Margin of Error from Sample Data (C5 + C6)
Problem 2 — Choose the Distribution, Then Compute (C5 + C6 + C8)
Problem 3 — Read the Interval Against a Threshold (C6 + C8)
Build the Conclusion
Read the t-interval against the threshold
Choose a phrase for each blank so that the sentence states the correct conclusion.
Problem 4 — Diagnose a Confidence-Interval Report (C3 + C6 + C7 + C8 + C9)
Problem 5 — Estuary Salinity Study (C5 + C6 + C8 + C10)
Part (a): Compute the t margin of error for the salinity study, then check the full interval in the feedback.
Part (b): Compare it with a matched known- analysis of the same readings.
Part (c): After checking your work, explain why Part (a) uses a t-interval while the matched analysis in Part (b) uses a z-interval.
Pause and write. Record your answer and reasoning on paper or in your notes before continuing. Even if you are unsure, committing your thinking helps you see what you understood and what you need to revisit.
Compare your explanation
Part (a) uses a t-interval because is unknown and must be estimated by . That extra estimation uncertainty makes larger than at finite , so the t margin is wider. In the matched analysis in Part (b), an external model supplies known , so a z-interval is justified.
Part (d): Hold ,, and the confidence level fixed. If the marine biologist increased the sample size from to , how would the two ingredients in the margin change?
Mixed Review — Retrieval from Earlier Lessons
Section 7: Mastery Check
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Question 1 — Feynman Test
Explain in plain language why estimating with changes a z-interval into a t-interval, and connect the heavier tails to the interval’s width. Aim for three or four sentences that you could use to teach a classmate.
Pause and write. Record your answer and reasoning on paper or in your notes before continuing. Even if you are unsure, committing your thinking helps you see what you understood and what you need to revisit.
Compare with a model answer
When is unknown, we estimate it with , and that estimate changes from sample to sample. Samples whose comes out too small would give intervals that are too short. The t-distribution accounts for this extra uncertainty with heavier tails and a larger critical value at finite . Multiplying the same standard error by that larger produces a wider interval. The extra width is honest uncertainty, not a defect.
Question 2 — Cold Interval
Question 3 — Which Procedure?
Question 4 — Why the Shortcut Falls Short
Question 5 — Decide With the Interval
Self-Assessment
Still confusedReady for the Boss Fight
Section 8: Boss Fight
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Choose one of two equally demanding missions. Neither tells you which method to use: choosing the tool is part of the fight. Each starts with your own complete attempt; open the steps only if you need to find where it went wrong. Lose, and you get a fresh rematch with new numbers.
⚖️ Two Analysts, One Sample
Back to Gosset’s problem: two analysts read the same small sample and reach different verdicts. Whose report do you sign?
🏗️ The Report Architect
A lab’s draft reports look polished. Find the first error in one, then rebuild another before it is approved.
⚖️ Two Analysts, One Sample
Then, see it happen. The figure starts from one worked case. Run more trials and watch how often the two analysts disagree.
Barley trial (worked case). Five plot yields (t/ha): 4.51, 5.07, 3.93, 5.24, 4.25. n = 5, x̄ =
4.60, s = 0.55.
Figure 8: The same five plot yields, two analysts. Analyst Z uses z* = 1.96 as if the
sample SD s were the population σ; Analyst T uses t* from the df = n − 1 row. A verdict is "met"
when the whole interval is above the contract value, "not met" when it is entirely below, and
"can't tell" when it contains it. Simulation assumption for "Run another trial": each trial
draws five plot yields from a normal field with true mean 4.3 t/ha and σ = 0.55 t/ha (in
a real trial the true mean is unknown).
A second dispute. Both analysts’ reports are on the table this time. Complete the review.
Build the Conclusion
Whose report do you sign?
Choose a phrase for each blank so that the sentence states the correct conclusion.
Then, on paper: go back to the bet you wrote in the Introduction. In two or three sentences, explain why Gosset’s small-sample intervals captured less often than “95%”, and why the narrower of the two analysts’ intervals is not the safer one to sign.
Compare with a model answer
With , the recipe captures only about 85.5% of the time under a normal population, because is sometimes much smaller than and the interval comes out too short. The t critical value stretches every interval by the same factor, which restores the advertised 95%. Analyst Z’s interval is narrower only because it pretends is ; when the contract value falls between the two intervals’ ends, that pretence turns “can’t tell” into a confident “met” that the data do not support.
🏗️ The Report Architect
You chair a review panel for a lab’s small-sample mean reports. The drafts look orderly, but the arithmetic can be clean and the method still wrong.
Task 1. Find the first statistical error in this draft.
Task 2. Rebuild the interval in a second draft before you approve it.
Then, on paper: write a two-sentence revision note for the second draft: name the choice the draft got wrong, and say how the corrected interval changes what the report can claim.
Compare with a model revision note
“The draft uses the wrong critical value for what is known about the spread: with only the sample’s , the interval needs with ; with known from calibration records, is justified and only widens the interval for no reason. The corrected interval states exactly as much uncertainty as the data carry, so its conclusion is one the panel can sign.”
Section 9: Challenge Problems
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Ready for more? These cold-transfer problems go beyond the lesson objectives.
Challenge 1 — Why df = n − 1
After solving the generated case, explain on paper or in your notes why fixing the mean constrains one value and leads to .
Pause and write. Record your answer and reasoning on paper or in your notes before continuing. Even if you are unsure, committing your thinking helps you see what you understood and what you need to revisit.
Compare your explanation
The deviations from the sample mean must sum to zero. Once any deviations are known, the last is determined, so only can vary freely. Estimating spread around the sample mean therefore leaves .
Challenge 2 — How Quickly Does t-star Converge to z-star?
Use the 95% t-table to find the first integer for which . Then explain why that numerical landmark is not a universal rule for replacing t with z.
Pause and write. Record your answer and reasoning on paper or in your notes before continuing. Even if you are unsure, committing your thinking helps you see what you understood and what you need to revisit.
Show the convergence check
At 95% confidence, for and for . Therefore, 28 is the first integer whose displayed table value is below 2.05. The value 2.05 is only a chosen closeness target; it does not make known or override the study’s sampling and shape conditions.
Challenge 3 — Two Studies, One Evidence Question
Show the comparison principle
Separate one-sample intervals are an indirect comparison. Non-overlap is strong but conservative evidence of a difference; overlap does not establish equality and can occur even when a direct comparison detects a difference. Use a confidence interval or test for to quantify the comparison directly.
Section 10: Solutions Reference
▾
A tag such as (C2 + C3) after a title names the Core Concepts parts the problem draws on, so
you know where to look back.
Complete, step-by-step solutions for all problems in Sections 5–9 are available on the solutions page. Solutions include worked arithmetic, common mistakes to watch for, and interpretation guidance.
If you’re stuck: Re-read the relevant Core Concept in Section 3, then find the Worked Example that uses it. If you’re unsure whether to use t or z, review the decision rule in C8. The solutions page shows the reasoning behind every step, not just the final answer.
Quick-Reference Formulas
Confidence Interval for ( Unknown):
Degrees of Freedom ():
Inference license
What is known about spread?
Distribution to use
Random/independent process; has an approximately normal sampling distribution
known from outside the sample
-distribution
Random/independent process; has an approximately normal sampling distribution
unknown, estimated by
-distribution
Sample size affects how plausible the sampling-distribution condition is and how close is to ; it does not by itself choose the procedure. When is estimated by , t is the principled choice even when a large makes z a close approximation.
t-Distribution Reference Table
Student's t-Distribution Table
Critical values (t*) for given degrees of freedom (df) and tail area.