AP Syllabus focus:
‘Null Hypothesis (H0): There is no difference between the two population means (H0: μ1 - μ2 = 0 or H0: μ1 = μ2). - Alternative Hypothesis (Ha): Indicates the nature of the difference; it could be less than (<), greater than (>), or not equal to (≠) depending on the research question (Ha: μ1 - μ2 < 0, Ha: μ1 - μ2 > 0, or Ha: μ1 ≠ 0).’
Formulating hypotheses establishes the foundation of a significance test, clarifying the competing claims about population means and guiding the statistical reasoning used to evaluate observed sample data.
Understanding the Role of Hypotheses in Two-Sample Mean Tests
In two-sample inference settings, formulating hypotheses provides a structured comparison between two population means. Because inference relies on sample data that inherently vary, clearly stating expectations about population behavior is essential. The hypotheses determine how evidence is evaluated and what kinds of conclusions can be drawn from the test statistic and p-value. For AP Statistics, students must distinguish between the null hypothesis, which represents no difference, and the alternative hypothesis, which represents the claim or idea the study seeks to support.
The Null Hypothesis (H0)
The null hypothesis asserts that there is no true difference between population means. This establishes a baseline assumption that any observed difference in the sample is due to random variation rather than a real effect.
Null Hypothesis (H0): A statement proposing no difference between population means, represented as .
Under the null hypothesis, researchers assume equality between means unless sufficient evidence suggests otherwise. This hypothesis is the reference point for calculating the test statistic and p-value.
The Alternative Hypothesis (Ha)
The alternative hypothesis expresses the direction or form of the suspected difference between population means. It reflects the research question and dictates whether the test is one-sided or two-sided.
Alternative Hypothesis (Ha): A statement indicating the presence and direction of a difference between population means, expressed using <, >, or ≠.
A well-written alternative hypothesis should align precisely with how researchers expect the means to differ. It is important that students choose the correct form before analyzing data, since choosing afterward would bias the interpretation.
Forms of Hypotheses in Two-Sample Mean Tests
Three forms of alternative hypotheses may be used, depending on the research goal.
Two-Sided Hypothesis (μ1 − μ2 ≠ 0)
A two-sided hypothesis tests for any difference between population means, without specifying direction. This form is appropriate when researchers suspect the means differ but lack a reason to believe one is consistently larger or smaller.
This figure depicts null and alternative hypotheses for a two-sample t test, emphasizing how researchers choose between two-sided and directional claims when comparing population means. Source.
One-Sided Hypothesis (μ1 − μ2 < 0 or μ1 − μ2 > 0)
A one-sided hypothesis tests for a directional difference.
• Less than (<) indicates that the first population mean is expected to be smaller than the second.
• Greater than (>) indicates that the first population mean is expected to be larger than the second.
Selecting a one-sided hypothesis requires strong contextual justification because it limits the kinds of differences the test can detect.
This illustration compares two-sided and one-sided hypothesis tests, showing how rejection regions differ depending on the form of the alternative hypothesis and its directional claim. Source.
Writing Hypotheses in Symbolic and Contextual Forms
In AP Statistics, hypotheses must be stated using both symbols and contextual language. While symbolic notation provides mathematical clarity, the contextual explanation shows understanding of how the test applies to the real situation.
EQUATION
= Mean of population 1
= Mean of population 2
A correct contextual statement might say that the two population means are equal with no genuine difference, expressed relative to the situation being studied. Students should avoid referencing sample statistics in hypotheses, since hypotheses always refer to population parameters.
After presenting hypotheses, the test statistic and p-value (not covered in this subsubtopic) will later determine whether sample data provide convincing evidence against .
Aligning Hypotheses With Research Questions
The hypothesis pair must reflect the research objective. If the question asks whether one treatment improves outcomes relative to another, a one-sided alternative is appropriate. If the question simply asks whether the treatments differ, a two-sided alternative should be used. The decision affects the test’s sensitivity, so students must articulate hypotheses carefully and intentionally.
Structuring Hypotheses for Matched Contexts
In two-sample tests, hypotheses always refer to population means from two independent groups. They do not refer to sample differences or observational patterns. The correctness of the hypotheses is foundational for interpreting results logically and responsibly. Once hypotheses are set, the subsequent steps of checking conditions, calculating the test statistic, and making decisions rely on this initial formulation.
Importance of Consistency and Precision
Hypotheses must be:
• Mutually exclusive, meaning only one can be true.
• Exhaustive, representing all possible outcomes.
• Clear, with no ambiguity about direction.
• Parameter-based, never referencing sample values.
Clear formulation supports accurate interpretation of the p-value and ensures valid reasoning throughout the inference process.
FAQ
There is no statistical rule that determines which group must be labelled first; the choice will not affect the numerical results of the test.
However, choosing labels consistently helps avoid sign errors when interpreting the difference between means.
A good practice is:
• Assign population 1 to the group expected to have the smaller or larger mean if the research question suggests a direction.
• Keep labels consistent throughout the analysis, especially if the alternative hypothesis is one-sided.
Hypotheses describe claims about the underlying populations, not the particular sample observed.
Sample statistics vary from study to study, so using them in hypotheses would make the hypotheses dependent on the data and therefore meaningless.
Using parameters ensures the hypotheses remain fixed statements that can be tested using sample evidence.
Practice Questions
Question 1 (1–3 marks)
A researcher wants to determine whether the mean reaction time of Group A differs from that of Group B. Write the appropriate null hypothesis and an appropriate two-sided alternative hypothesis for this investigation.
Question 1
• 1 mark for correctly stating the null hypothesis: H0: mean of Group A minus mean of Group B equals 0, or wording indicating no difference in population means.
• 1 mark for correctly stating a two-sided alternative hypothesis: Ha: mean of Group A minus mean of Group B is not equal to 0, or wording indicating a difference in population means.
• 1 mark for hypotheses clearly referring to population means rather than sample means.
Question 2 (4–6 marks)
A sports scientist is studying whether a new training programme results in faster average sprint times compared with the standard programme. Let population 1 be athletes using the new programme and population 2 be athletes using the standard programme.
(a) State the null hypothesis and a suitable one-sided alternative hypothesis for this context.
(b) Explain why the direction of the alternative hypothesis must be chosen before looking at the sample data.
(c) Briefly describe how the wording of the research question determines whether a one-sided or two-sided alternative is used.
Question 2
(a)
• 1 mark for correct null hypothesis: H0: mean of population 1 minus mean of population 2 equals 0, or the two population means are equal.
• 1 mark for correct one-sided alternative hypothesis: Ha: mean of population 1 minus mean of population 2 is less than 0, indicating faster sprint times (smaller values) for the new programme.
(b)
• 1 mark for explaining that choosing the direction after seeing the data would bias the test or inflate the Type I error rate.
• 1 mark for stating that hypotheses must be set before data analysis to ensure a valid and unbiased inference procedure.
(c)
• 1 mark for explaining that a one-sided alternative is used only when the research question predicts a specific direction of difference.
• 1 mark for explaining that if the question merely asks whether the programmes differ without specifying direction, a two-sided alternative is appropriate.
Can the alternative hypothesis ever be written using an equality sign?
No. The equality sign is reserved exclusively for the null hypothesis, which specifies a single parameter value representing no difference.
The alternative hypothesis must express a departure from that value.
Using equality in the alternative would make it impossible to identify which outcomes contradict the null, undermining the logic of significance testing.
What happens if the research question seems ambiguous about the direction of the expected difference?
When the research question does not clearly predict one group to have a larger or smaller mean, a two-sided alternative is safer.
Consider whether:
• Previous research strongly suggests a direction.
• There is a plausible justification that one group should outperform the other.
If neither is present, examiners expect a two-sided hypothesis because it avoids unjustified directional claims.
Why is the difference form (mu1 − mu2) preferred over writing hypotheses separately for each mean?
Expressing hypotheses as a single difference emphasises that the test concerns the relationship between two means rather than the means individually.
Advantages include:
• Clear identification of the parameter of interest.
• Direct alignment with the test statistic formula, which is based on the difference between sample means.
• Reduced likelihood of contradictory or incomplete hypothesis statements.
