Difficulty: Intermediate | Prerequisites: Sampling distributions, t-distribution basics, standard deviation vs. standard error, inference for proportions (helpful but not required).
Big picture: This topic covers how to estimate and test claims about population means using sample data. Means come up whenever your variable is quantitative (measured on a number line) rather than categorical. Because you almost never know the population standard deviation σ, you use the t-distribution instead of the z-distribution. The t-distribution is wider and has heavier tails than the normal, which accounts for the extra uncertainty from estimating σ with s. Everything else, the logic of confidence intervals and hypothesis tests, works the same way it does with proportions.
When your data is quantitative (measurements, scores, times, amounts), you use t-based procedures. A confidence interval gives a range of plausible values for the true population mean. A hypothesis test checks whether the data provide evidence against a claimed value for the mean. One-sample procedures deal with a single group; two-sample procedures compare two groups.
Population mean (μ)
The true average value in the entire population. This is the parameter you are trying to estimate or test.
In simple terms, it is the answer you would get if you could measure every single individual.
Sample mean (x̄)
The average of the values in your sample. It is a statistic that serves as your point estimate of μ.
Think of it as your best guess for the population mean, based on the data you have.
Sample standard deviation (s)
A measure of how spread out the values in your sample are. It estimates the population standard deviation σ.
Standard error of the mean (SE)
SE = s / √n. It measures how much x̄ is expected to vary from sample to sample. Larger samples and smaller standard deviations both reduce the standard error.
In simple terms, this is the "wobble" in your estimate.
t-distribution
A family of bell-shaped distributions, each defined by its degrees of freedom (df). With small samples it is wider than the normal distribution. As df increases, the t-distribution approaches the standard normal.
Think of it as the normal distribution's cautious sibling: it gives you wider intervals to compensate for the fact that you are estimating σ with s.
Degrees of freedom (df)
For a one-sample t-procedure: df = n − 1. For a two-sample t-procedure, the calculator computes df using a formula (Welch's approximation) that is not worth memorising. The degrees of freedom determine which t-distribution you use.
t-statistic
The number of standard errors your sample mean sits away from the hypothesised mean: t = (x̄ − μ₀) / (s / √n).
Purpose: Estimate the true population mean μ with a confidence interval.
When to use: You have one sample from one population and want a range of plausible values for μ.
Conditions to check:
Random sample (or random assignment)
Independence: n ≤ 10% of the population
Normal/large sample: either the population is roughly normal, or n ≥ 30 (Central Limit Theorem), or the sample data show no strong skewness or outliers
Calculator function: TInterval
Inputs required: x̄ (sample mean), s (sample standard deviation), n (sample size), C-Level. Alternatively, you can enter raw data into a list and select "Data" mode.
Purpose: Test a claim about the value of a single population mean.
When to use: You want to know whether the data provide sufficient evidence that μ differs from (or is greater/less than) a specific value μ₀.
Conditions to check: Same as the T Interval.
Calculator function: T-Test
Inputs required: μ₀ (hypothesised mean), x̄, s, n, direction of Hₐ (≠, <, or >)
Purpose: Estimate the difference between two population means (μ₁ − μ₂) with a confidence interval.
When to use: You have independent samples from two populations and want a range for how much their means differ.
Conditions to check:
Both samples are random and independent of each other
Independence within each sample: n₁ ≤ 10% of population 1, n₂ ≤ 10% of population 2
Normal/large sample condition for each group separately
Calculator function: 2-SampTInt
Inputs required: x̄₁, s₁, n₁, x̄₂, s₂, n₂, C-Level. The "Pooled" option should typically be set to "No" unless you have strong reason to assume equal variances.
Purpose: Test whether two population means are equal (or whether one is greater/less than the other).
When to use: You want to know whether there is a statistically significant difference between the means of two independent groups.
Conditions to check: Same as the two-sample T interval.
Calculator function: 2-SampTTest
Inputs required: x̄₁, s₁, n₁, x̄₂, s₂, n₂, direction of Hₐ (≠, <, or >), Pooled (usually "No")
One-sample t confidence interval:
x̄ ± t* × (s / √n)
where t* is the critical value from the t-distribution with df = n − 1.
One-sample t-test statistic:
t = (x̄ − μ₀) / (s / √n)
Two-sample t confidence interval:
(x̄₁ − x̄₂) ± t* × √(s₁²/n₁ + s₂²/n₂)
Two-sample t-test statistic:
t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂)
Degrees of freedom for the two-sample case are computed by Welch's formula (let the calculator handle this).
Pharmaceutical companies use two-sample t-tests to compare average blood pressure reductions between a drug group and a placebo group. Quality control engineers use one-sample t-intervals to estimate the average weight or length of manufactured parts and check whether production is meeting specifications.
"If the population is not normal, I cannot use a t-procedure." You can, as long as your sample size is large enough (n ≥ 30 is a common guideline) for the Central Limit Theorem to kick in. For smaller samples, check for severe skewness or outliers.
"The degrees of freedom for a two-sample t-test is n₁ + n₂ − 2." That formula applies only when you assume equal variances (pooled procedure). The default, unpooled (Welch's) procedure uses a more complicated formula. Let the calculator compute it.
"I should always set Pooled to Yes on my calculator." In most intro stats courses, you should set Pooled to No unless the problem explicitly tells you to assume equal population standard deviations.
"A wider confidence interval is worse." It is less precise, but it has a higher confidence level. The width is a trade-off, not a flaw. The only way to get a narrower interval at the same confidence level is to increase the sample size.
⚠️ Means always use t-procedures in intro stats, even when the sample is large. The z-procedures for means assume σ is known, which is rare in practice.
⚠️ Checking conditions is required for full marks. State all three (random, independent, normal/large sample) and show your work.
⚠️ Know the difference between independent samples and paired/matched samples. If the two groups are naturally linked (before/after on the same subjects, husband/wife pairs), you take the differences and run a one-sample t-procedure on those differences, not a two-sample test.
⚠️ On the calculator, "Stats" mode means you enter summary statistics (x̄, s, n). "Data" mode means your raw data is stored in a list.
True or False: A t-distribution with 100 degrees of freedom is practically identical to the standard normal distribution.
Fill in the blank: The calculator function for testing whether two population means are different is __________.
True or False: For a one-sample t-test, the degrees of freedom are n.
Fill in the blank: If subjects are measured before and after a treatment, you should use a __________ t-test on the differences, not a two-sample t-test.
True or False: The standard error of the mean increases as the sample size increases.
Answers: 1. True. 2. 2-SampTTest. 3. False (df = n − 1). 4. One-sample (or "paired"). 5. False (it decreases).
Q: A sample of 36 students has a mean test score of 78 and a standard deviation of 12. Construct a 95% confidence interval for the population mean score. What calculator function do you use?
A: Use TInterval. Enter x̄ = 78, s = 12, n = 36, C-Level = 0.95. The interval is 78 ± t* × (12/√36) = 78 ± t* × 2. With df = 35, t* ≈ 2.030, giving approximately (73.94, 82.06).
Q: A manufacturer claims their bolts have a mean length of 5.00 cm. A sample of 25 bolts has x̄ = 5.08 cm and s = 0.15 cm. At α = 0.01, is there evidence the mean length differs from 5.00 cm?
A: Use T-Test. H₀: μ = 5.00, Hₐ: μ ≠ 5.00. Enter μ₀ = 5.00, x̄ = 5.08, s = 0.15, n = 25. The test statistic is t = (5.08 − 5.00) / (0.15/√25) = 0.08/0.03 = 2.667, with df = 24. The two-sided p-value is approximately 0.013. Since 0.013 > 0.01, you fail to reject H₀ at the 1% level. There is not sufficient evidence at α = 0.01.
Q: Group A (n = 40, x̄ = 85, s = 10) and Group B (n = 50, x̄ = 80, s = 12). Is there a significant difference at α = 0.05?
A: Use 2-SampTTest. H₀: μ₁ = μ₂, Hₐ: μ₁ ≠ μ₂. Enter the summary statistics for both groups. Set Pooled to No. The calculator gives the t-statistic and p-value. Compare the p-value to 0.05 to decide.
Q: When should you use a paired t-test instead of a two-sample t-test?
A: Use a paired (one-sample) t-test when the two sets of observations are dependent, meaning each value in one group has a natural partner in the other. Common examples include before/after measurements on the same subjects, or measurements on matched pairs (twins, left eye vs. right eye). Compute the difference for each pair and run a one-sample t-test on those differences.
Inference for means is the t-distribution counterpart to inference for proportions (which uses the z-distribution). The same logic, confidence intervals estimate parameters, hypothesis tests evaluate claims, carries through to chi-squared tests (for categorical data with more than two categories) and linear regression inference (for testing slopes). Paired t-tests connect to the concept of blocking in experimental design.
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