AP Syllabus focus:
‘Independence: Each sample should be drawn independently using simple random sampling or through a randomized experiment. Check that the sample sizes are less than 10% of their populations if sampling without replacement. Normality: The sampling distribution of the difference between sample means should be approximately normal. If sample sizes are less than 30, ensure the distribution of each sample is not strongly skewed or outlier-ridden.’
Understanding when a two-sample t-test is valid requires verifying specific conditions that justify using a t-distribution to model the test statistic. These conditions ensure reliable conclusions.
Verifying Conditions for a Two-Sample Significance Test
The appropriateness of the two-sample t-test depends on meeting requirements related to independence, sampling design, and normality of the underlying distributions. These conditions protect against misleading inferences caused by biased sampling or non-normal data behavior, especially with small samples.
Independence Condition
The independence requirement ensures that observations within and between samples do not influence one another. Without independence, the variability of the sampling distribution may be underestimated, producing unreliable p-values. Before using a two-sample t-test, we must verify that its key conditions are met: independence within and between samples, and approximate normality of the sampling distribution.
Diagram showing the independent-samples t-test comparing two separate groups on a quantitative outcome, highlighting independence and normality as key assumptions. The figure also includes homogeneity of variance, which is optional enrichment beyond AP requirements. This supports understanding of why assumptions must be checked before conducting the test. Source.
Independence: When the value of any observation does not affect or predict the value of another observation, either within the same sample or across samples.
To meet this condition, data must arise from simple random sampling or a randomized experiment, both of which help guarantee that each observational unit has an equal chance of selection. Randomization also reduces systematic influences that could bias group comparisons.
10% Condition for Sampling Without Replacement
When sampling from a finite population without replacement, selected observations may become slightly dependent. The 10% condition controls this effect by limiting the sample size relative to the population size.
10% Condition: A rule stating that when sampling without replacement, the sample size must be no more than 10% of the population to maintain approximate independence.
If two samples are drawn, each must separately satisfy
Meeting these requirements preserves the validity of the estimated standard error used in the t-statistic.
Normality Condition
For the two-sample t-test, the sampling distribution of the difference in sample means should be approximately normal. This requirement ensures that the test statistic follows a t-distribution closely enough for accurate inference.
Small sample sizes require extra scrutiny because the Central Limit Theorem provides less protection when n < 30. Thus, the shapes of the sample distributions must be evaluated for deviations from normality.
Normality (for inference): A condition stating that the population distributions or sample data must be approximately symmetric and free of strong skewness or outliers to justify using a t-distribution for inference.
Between definition blocks, students should note that the normality condition emphasizes analyzing data patterns rather than assuming perfection.
Assessing Normality in Practice
Because AP Statistics emphasizes practical judgment over rigid rules, students should review graphical and numerical features of each sample when evaluating normality. The sampling distribution of the difference between sample means is approximately normal when each sample either comes from a roughly normal population or has a sufficiently large sample size (usually at least 30).
Grid of histograms illustrating how the sampling distribution of the mean becomes more nearly normal as sample size increases, even for skewed populations. The dashed red curves mark the corresponding normal approximations. The figure compares several population shapes, extending slightly beyond AP requirements but strengthening intuition about normality conditions. Source.
Key indicators include:
No strong skewness: Moderate skew is acceptable for larger samples but problematic for small ones.
No outliers: Extreme values distort estimates of the standard error and affect the t-statistic.
Sample size thresholds:
If , the sampling distribution is approximately normal regardless of population shape.
If n < 30, each sample must exhibit near-normal behavior to justify inference.
Why Conditions Matter for Inference
Verifying the independence and normality conditions ensures that the t-distribution appropriately models the sampling distribution of the test statistic. When any condition is violated, the resulting inference may have incorrect Type I or Type II error rates, leading to flawed conclusions about differences between population means.
Relationship to the Two-Sample t-Test Structure
A correctly executed two-sample t-test depends on a reliable estimate of the standard error of the difference in sample means. This estimate assumes independent observations and stable variation within each group.
EQUATION
= Sample standard deviations
= Sample sizes
When independence or normality is compromised, this formula may no longer accurately estimate variability, rendering the test statistic and p-value unreliable.
A single sentence following the equation helps reinforce that verifying assumptions directly supports the mathematical validity of the test.
Summary of Condition-Checking Steps
Students should systematically check conditions before conducting a two-sample t-test. A structured approach enhances clarity and prevents overlooking critical assumptions.
Step 1: Verify independence within and between samples.
Confirm random sampling or random assignment and ensure no design element links the two groups.
Step 2: Apply the 10% condition if sampling without replacement.
Ensure each sample size is at most 10% of its population.
Step 3: Assess normality for each sample.
Examine data for strong skewness or outliers, especially when n < 30. When sample sizes are small, we rely on plots of the sample data, such as histograms or dotplots, to make sure there is no strong skewness or extreme outliers.
Three histograms showing symmetric, right-skewed, and left-skewed distributions, each with mean, median, and mode marked. This helps students identify skewness when checking normality for two-sample t-tests. The center-measure annotations exceed AP requirements but strengthen conceptual interpretation of distribution shape. Source.
Step 4: Determine whether conditions justify using the t-distribution.
Proceed with a two-sample t-test only when all criteria are sufficiently met.
These structured verification steps provide the conceptual foundation necessary for valid and defensible statistical inference using the two-sample t-test.
FAQ
For sample sizes in the low twenties, the t-procedures are moderately robust, meaning slight skewness is usually acceptable.
However, robustness depends on both samples being similarly well-behaved. If one sample shows clear asymmetry or a few mild outliers, the normality condition may still hold, but strong skew or influential outliers make the t-test unreliable.
When in doubt, students should inspect graphical displays and consider whether the skewness meaningfully distorts the centre and spread.
Not necessarily. Both samples must meet the condition to justify using the t-distribution.
The test statistic and standard error rely on each group's variability being estimated accurately. Severe skewness or outliers in one sample destabilise this estimate and can bias inference.
When the violation occurs in only one group, consider whether transformation, non-parametric methods, or collecting more data is more appropriate.
Independence ensures the difference between sample means reflects only natural sampling variability rather than a structural link between groups.
If samples influence one another—for example, if individuals appear in both groups—variation is artificially reduced. This increases the chance of detecting a false difference.
Independence protects against misleadingly small standard errors and inflated Type I error rates.
Randomised controlled experiments provide the strongest guarantee because treatment assignment is random, ensuring unrelated groups.
Other designs that support independence include:
• Two simple random samples drawn from distinct populations
• Well-designed field studies where sampling units cannot interact
If the study involves clusters, repeated measures, or paired observations, independence is compromised, and a paired t-procedure may be more appropriate.
An outlier threatens the t-test when it substantially shifts the sample mean or enlarges the sample standard deviation.
Useful checks include:
• Observing whether removing the point changes the centre or spread noticeably
• Assessing whether the point appears inconsistent with the rest of the distribution shape
• Considering whether measurement or recording errors may have occurred
If an outlier meaningfully distorts the distribution, the normality condition is not met for small sample sizes.
Practice Questions
Question 1 (1–3 marks)
A researcher compares the mean exam scores of two independent classes using a two-sample t-test. The sample sizes are n1 = 18 and n2 = 20. The researcher notes that the data for Class 1 contain several extreme outliers and are strongly right-skewed, while Class 2's data appear roughly symmetric with no outliers.
Explain whether the normality condition for a two-sample t-test is satisfied, and justify whether the test should or should not be carried out.
Question 1
• 1 mark: States that normality is violated for Class 1 because of strong skewness and outliers.
• 1 mark: States that Class 2’s distribution is acceptable (roughly symmetric, no outliers).
• 1 mark: Concludes the two-sample t-test should not be carried out because the normality condition is not satisfied for small sample size (n < 30) in Class 1.
Question 2 (4–6 marks)
A study investigates whether two different fertilisers lead to different average plant heights. Two independent random samples are collected: 15 plants using Fertiliser A and 12 plants using Fertiliser B. The populations are very large, and the study is conducted without replacement.
(a) State the independence conditions that must be checked before carrying out a two-sample t-test.
(b) Discuss whether the 10% condition is relevant in this context.
(c) The researcher examines the distributions of plant heights. Fertiliser A shows mild right skew, while Fertiliser B shows strong left skew with two outliers. Evaluate whether the normality condition is met, and explain the implications for performing a two-sample t-test.
Question 2
(a)
• 1 mark: States that observations must be independent within each sample (random sampling or random assignment).
• 1 mark: States that the two samples must be independent of one another.
(b)
• 1 mark: Notes that the 10% condition must be checked when sampling without replacement.
• 1 mark: Concludes that the condition is satisfied because the populations are very large, so the samples represent far less than 10%.
(c)
• 1 mark: Describes that Fertiliser A’s mild skew is acceptable for a small sample only with caution.
• 1 mark: States that Fertiliser B’s strong skew and outliers violate the normality condition for small samples.
• 1 mark: Concludes that the two-sample t-test is not appropriate because the normality condition is not adequately met for Fertiliser B.
