Determining the probability that observed results could have occurred by chance is accomplished through
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______.

Question

Determining the probability that observed results could have occurred by chance is accomplished through 
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______.

BlackTom AI · Accepted Answer

Statistical analysis determines how likely observed results are due to chance by using probability models, test statistics, and p-values.

1. Relevant knowledge points and concepts
- Statistics: collection of methods for analyzing data, including hypothesis testing and estimation.
- Null hypothesis (H0): a statement of no effect or no difference; used as the baseline assumption.
- Test statistic: a numerical summary (e.g., z, t, χ²) that measures how far the data deviate from H0.
- Sampling distribution and p-value: the probability, under H0, of observing a test statistic as extreme as (or more extreme than) the observed value. The p-value quantifies chance.
- Significance level (α): a threshold for deciding whether the observed result is unlikely under H0.

2. Step-by-step reasoning process
- State H0 (e.g., population mean μ = μ0) and an alternative hypothesis H1.
- Choose an appropriate test (z-test, t-test, χ², etc.) based on data type and sample size.
- Compute the test statistic from the sample data.
- Use the sampling distribution under H0 to find the p-value.
- Compare p-value to α to decide whether the result is unlikely due to chance.

3. Example calculation with intermediate steps (symbolic)
- Suppose testing a mean: H0: μ = μ0. Sample mean x̄, sample standard deviation s, sample size n.
- Compute t = (x̄ − μ0) / (s/√n).
- Determine p-value = P(T(df = n−1) ≥ |t|) (two-sided) from t-distribution tables or software.
- If p-value < α, reject H0; otherwise do not reject H0.

4. Why the correct answer is correct
The phrase “Determining the probability that observed results could have occurred by chance” directly refers to computing p-values and performing hypothesis tests—core procedures within statistics. Statistics provides the formal mathematical framework and tools to make that probability assessment.

5. Why the other options are incorrect
- Theories: Broad explanatory frameworks; they describe and predict phenomena but do not compute probabilities for observed sample results.
- Hypotheses: Specific testable statements; they are the targets of statistical tests but are not the method used to determine the probability that results arose by chance.

6. Conclusion
The correct choice is statistics, which matches the provided answer.

Determining the probability that observed results could have occurred by chance is accomplished through Blank ______.单项选择题

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