Which of the following is most accurate regarding the null hypothesis when comparing two samples?

Question

BlackTom AI · Accepted Answer

The null hypothesis for comparing two samples is that the populations they come from are the same (i.e., there is no difference in the population parameter of interest).

1. Relevant concepts
- Null hypothesis (H0): a statement of no effect or no difference about population parameters.
- Alternative hypothesis (H1): statement that there is an effect or difference.
- For means: H0: μ1 = μ2 (equivalently μ1 − μ2 = 0).
- For proportions: H0: p1 = p2.
- Test statistic and standard error quantify how far the observed sample difference is from 0 given sampling variability.

2. Step-by-step reasoning
- When comparing two samples, we care whether observed sample differences could plausibly arise if both samples were drawn from the same population distribution (same mean/proportion).
- Formulate H0 as equality of the population parameter: e.g., H0: μ1 = μ2.
- Compute the observed difference d = x̄1 − x̄2. Under H0 the expected difference = 0.
- Compute the test statistic by standardizing d with its standard error.

3. Intermediate (symbolic) calculations
- Difference: d = x̄1 − x̄2.
- Standard error (for independent samples, means): SE = sqrt(s1^2/n1 + s2^2/n2).
- Test statistic (t or z): T = d / SE.
- Compare T to the appropriate distribution (t with df or z) to get a p-value. If p is small, reject H0; if p is large, fail to reject H0.

4. Why the correct answer is correct
- Stating that "the two samples were drawn from the same population" is another way of asserting no difference in the underlying population parameter(s). That is the standard formulation of the null in two-sample tests (no difference).

5. Why other common phrasings are incorrect
- Saying the null is “the samples are independent” is about design/assumption, not the hypothesis being tested.
- Saying the null is “the sample means are equal” confuses sample statistics with population parameters; the null concerns population means.
- Saying the null is “there is an effect” is the opposite of the null; that is the alternative.

Conclusion: The most accurate statement is: It is the hypothesis that the two samples were drawn from the same population — which matches the provided answer.

Which of the following is most accurate regarding the null hypothesis when comparing two samples?单项选择题

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