Difficulty: Intermediate | Prerequisites: Part 1 conceptual foundations (t-test vs z-test, assumptions, degrees of freedom). Familiarity with null/alternative hypotheses and critical values.
Knowing which t-test to use and why is only half the job. The other half is carrying out the calculation: stating hypotheses, computing t-obtained, finding t-critical from the table, and making a decision. This set of notes walks through three complete worked examples, one for each type of t-test (independent samples, one-sample, and dependent/paired samples). These mirror the kind of problems that appear on exams, and the steps are identical every time regardless of the scenario.
Every t-test follows the same five-step pattern: state hypotheses, compute t-obtained, look up t-critical, compare the two, decide whether to reject H₀. The formulas differ slightly by test type, but the logic never changes. One-tailed tests put the entire rejection region on one side; two-tailed tests split it across both.
Null hypothesis (H₀)
The default claim that there is no effect or no difference. In a t-test, this typically states that the population mean (or difference between population means) equals some specific value.
Think of it as: the boring explanation, the one that says nothing interesting is going on.
Alternative hypothesis (H₁ or Hₐ)
The research claim, stating that the population mean (or difference) is different from, greater than, or less than the null value.
In simple terms, this means: the thing you actually believe is true and are trying to find evidence for.
T-obtained (t_obs)
The test statistic you compute from your sample data. It tells you how many estimated standard errors the sample result is from the null hypothesis value.
T-critical (t_crit)
The threshold value from the t-distribution table for your chosen alpha level and degrees of freedom. If t-obtained exceeds t-critical (in the predicted direction), you reject H₀.
One-tailed test
A test where the alternative hypothesis specifies a direction (greater than or less than). The entire rejection region sits in one tail of the distribution.
Two-tailed test
A test where the alternative hypothesis states only that there is a difference (not specifying direction). The rejection region is split between both tails.
Rejection region
The area in the tails of the t-distribution where t-obtained values would lead you to reject H₀. Defined by alpha and the directionality of the test.
State the null and alternative hypotheses.
Compute t-obtained from the sample data using the appropriate formula.
Find t-critical from the t-table using df and alpha.
Compare t-obtained to t-critical.
Decide: if |t-obtained| ≥ |t-critical| in the predicted direction, reject H₀. Otherwise, fail to reject H₀.
Scenario: Dr. X compares short-term memory scores (out of 100) between participants viewing nature pictures vs city pictures. The prediction is that nature scores will be significantly higher.
Given values:
x̄₁ (nature) = 68, x̄₂ (city) = 64
n₁ = 5, n₂ = 5
s₁ = 1.27, s₂ = 1.90
α = .05
Step a: Hypotheses
Because Dr. X predicts a specific direction (nature > city), this is a one-tailed test.
H₀: μ₁ ≤ μ₂ (nature scores are equal to or lower than city scores)
H₁: μ₁ > μ₂ (nature scores are higher than city scores)
Step b: T-obtained
Formula for independent samples:
t = (x̄₁ − x̄₂) / √(s²_p / n₁ + s²_p / n₂)
First, compute the pooled variance:
s²_p = [(n₁ − 1)s₁² + (n₂ − 1)s₂²] / (n₁ + n₂ − 2)
s₁² = 1.27² = 1.6129
s₂² = 1.90² = 3.6100
s²_p = [(4)(1.6129) + (4)(3.6100)] / (5 + 5 − 2)
s²_p = [6.4516 + 14.4400] / 8
s²_p = 20.8916 / 8 = 2.6115
Now compute t:
t = (68 − 64) / √(2.6115/5 + 2.6115/5)
t = 4 / √(0.5223 + 0.5223)
t = 4 / √1.0446
t = 4 / 1.0221
t-obtained ≈ 3.91
Step c: T-critical
df = n₁ + n₂ − 2 = 5 + 5 − 2 = 8
One-tailed test, α = .05, df = 8:
t-critical = 1.860
Step d: Distribution sketch
Imagine a bell-shaped t-distribution centred at 0. The rejection region is the right tail beyond t-critical = 1.860 (shaded). The t-obtained of 3.91 falls well inside the rejection region, far to the right of the critical boundary.
Step e: Decision
t-obtained (3.91) > t-critical (1.860), so we reject H₀. There is sufficient evidence at the .05 level to conclude that viewing nature pictures produces significantly higher short-term memory scores than viewing city pictures.
Scenario: A teacher wants to determine if their AP calculus class comes from a population with a mean higher than the district mean of 3.71 (scale out of 5).
Given values:
x̄ = 3.89, μ = 3.71
s = 0.36, n = 32
α = .05
Step a: Hypotheses
The teacher predicts their class mean is greater than the population mean, so this is one-tailed.
H₀: μ ≤ 3.71
H₁: μ > 3.71
Step b: T-obtained
Formula for one sample:
t = (x̄ − μ) / (s / √n)
t = (3.89 − 3.71) / (0.36 / √32)
t = 0.18 / (0.36 / 5.6569)
t = 0.18 / 0.06364
t-obtained ≈ 2.83
Step c: T-critical
df = n − 1 = 32 − 1 = 31
One-tailed test, α = .05, df = 31:
t-critical ≈ 1.696
Step d: Distribution sketch
A t-distribution centred at 0. The rejection region is the right tail beyond 1.696 (shaded). The t-obtained of 2.83 falls within the rejection region.
Step e: Decision
t-obtained (2.83) > t-critical (1.696), so we reject H₀. There is sufficient evidence at the .05 level to conclude that this AP calculus class comes from a population with a mean higher than the district mean of 3.71.
Scenario: Dr. Y examines how different marijuana dosages (1g vs 2g per hour) affect short-term memory. Each participant experiences both conditions (within-subjects design).
Given values:
D̄ (mean difference) = 7.67
S_D (standard deviation of difference scores) = 15.64
n = 40
α = .01
Step a: Hypotheses
The research question is whether the two dosages produce different scores, with no specified direction. This is a two-tailed test.
H₀: μ_D = 0 (no difference between conditions)
H₁: μ_D ≠ 0 (there is a difference between conditions)
Step b: T-obtained
Formula for dependent samples:
t = D̄ / (S_D / √n)
t = 7.67 / (15.64 / √40)
t = 7.67 / (15.64 / 6.3246)
t = 7.67 / 2.4729
t-obtained ≈ 3.10
Step c: T-critical
df = n − 1 = 40 − 1 = 39
Two-tailed test, α = .01, df = 39:
t-critical ≈ ±2.708
Step d: Distribution sketch
A t-distribution centred at 0 with two rejection regions: one in the left tail beyond −2.708 and one in the right tail beyond +2.708 (both shaded). The t-obtained of 3.10 falls in the right rejection region.
Step e: Decision
|t-obtained| (3.10) > t-critical (2.708), so we reject H₀. There is sufficient evidence at the .01 level to conclude that the two marijuana dosages produce significantly different short-term memory scores.
One-sample t-test:
t = (x̄ − μ) / (s / √n), df = n − 1
Independent-samples t-test:
s²_p = [(n₁ − 1)s₁² + (n₂ − 1)s₂²] / (n₁ + n₂ − 2)
t = (x̄₁ − x̄₂) / √(s²_p/n₁ + s²_p/n₂), df = n₁ + n₂ − 2
Dependent-samples t-test:
t = D̄ / (S_D / √n), df = n − 1
The independent-samples t-test is the workhorse of experimental research in psychology and medicine: any time you randomly assign people to two groups and measure an outcome, this is the test. The paired t-test shows up in before/after designs (weight loss programmes, training interventions, drug trials with crossover designs). The one-sample test is used in quality control, such as checking whether a production batch's mean weight matches the specification.
Students often forget to square the standard deviations when computing pooled variance. The formula uses s², not s.
Mixing up one-tailed and two-tailed tests is common. If the research hypothesis specifies a direction (greater than, less than), it is one-tailed. If it just says "different," it is two-tailed.
Students sometimes use the wrong degrees of freedom formula. One-sample and paired tests use n − 1; independent samples use n₁ + n₂ − 2.
A common error is comparing t-obtained to the wrong t-critical (using a two-tailed critical value for a one-tailed test, or vice versa). Always match the critical value lookup to the type of test.
⚠️ You will very likely need to carry out a full five-step hypothesis test on the exam. Practise the arithmetic.
⚠️ Know which formula goes with which test type. The scenario description tells you whether it is one-sample, independent, or paired.
⚠️ The direction of the hypothesis (one-tailed vs two-tailed) changes the t-critical value. Read the problem carefully for directional language.
⚠️ Be able to sketch the distribution with t-critical, t-obtained, and the rejection region labelled. This is a common exam sub-question.
⚠️ State your decision in context ("reject H₀" or "fail to reject H₀") and tie it back to the research question.
True or False: In a one-tailed test at α = .05, the entire .05 rejection area is in one tail.
Fill in the blank: The pooled variance formula uses __________ (s or s²) values.
True or False: For a dependent-samples t-test with n = 40 participants, df = 39.
Fill in the blank: If t-obtained does not exceed t-critical, we __________ H₀.
True or False: A two-tailed test at α = .01 places .005 in each tail.
Answers: 1. True. 2. s² (squared). 3. True. 4. fail to reject. 5. True.
Q: In the independent-samples t-test example (nature vs city), why is the test one-tailed?
A: Because Dr. X made a directional prediction: nature scores would be higher. The alternative hypothesis specifies μ₁ > μ₂, so the rejection region is only in the right tail.
Q: Calculate the pooled variance if n₁ = 5, n₂ = 5, s₁ = 1.27, and s₂ = 1.90.
A: s²_p = [(4)(1.6129) + (4)(3.61)] / 8 = 20.8916 / 8 ≈ 2.61.
Q: For a one-sample t-test with n = 32 and α = .05 (one-tailed), what are the degrees of freedom and approximate t-critical?
A: df = 31, t-critical ≈ 1.696.
Q: Why does the dependent-samples t-test in the marijuana example use a two-tailed test?
A: Because the research question asks whether the dosages produce different scores without predicting which dosage would yield higher scores.
Q: What is the general decision rule for rejecting H₀?
A: If the absolute value of t-obtained is greater than or equal to the absolute value of t-critical (in the direction specified by H₁), reject H₀. Otherwise, fail to reject H₀.
These calculation skills transfer directly to ANOVA, which extends the logic of comparing means to three or more groups. The concept of pooled variance reappears in ANOVA's mean square within. Effect size measures (Cohen's d) build on the same t-test components. Understanding p-values and critical regions here also lays the groundwork for non-parametric tests (Mann-Whitney, Wilcoxon) when t-test assumptions are not met.
t-test calculation, t-obtained, t-critical, pooled variance, independent-samples t-test, one-sample t-test, paired t-test, dependent-samples t-test, null hypothesis, alternative hypothesis, one-tailed test, two-tailed test, rejection region, alpha level, degrees of freedom, hypothesis testing steps, fail to reject, statistically significant, UCF mathematical statistics, Quiz 6