Hypothesis testing and contextual conclusions
| English | 中文 | Pinyin |
|---|---|---|
| significance level/sɪɡˈnɪfɪkəns ˈlevl/ | 显著性水平 | xiǎn zhù xìng shuǐ píng |
Could chance explain the result?
- A factory claims that only 10% of items are defective. A sample contains more defects, but chance alone may explain some difference.
- This lesson studies significance level 显著性水平: The chosen probability threshold for rejecting a null hypothesis.
Choose the mathematical structure
- State H₀ and H₁ in population parameters before inspecting the outcome. Calculate the appropriate tail probability under H₀. Reject H₀ when the evidence meets the specified significance rule; otherwise say there is insufficient evidence.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines significance level?
The chosen probability threshold for rejecting a null hypothesis.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For H₀:p=0.1 and H₁:p>0.1 with n=10, observing at least 3 defects has probability 1-(0.9^10+10×0.1×0.9^9+45×0.1²×0.9^8)≈0.070191. At 5%, this is insufficient evidence that the defect rate exceeds 10%.
Hypothesis testing and contextual conclusions
State H₀ and H₁ in population parameters before inspecting the outcome
Compare the model with the worked case and explain one change.
Find the significance threshold written as a decimal for 5%.
5% means 5/100=0.05.
Test a tempting shortcut
- Failing to reject H₀ is not proof that H₀ is true. Choose the tail from H₁, not from whichever tail gives a small result. Statistical significance does not measure the practical size of an effect.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Failing to reject a null hypothesis proves it is true. This claim is false. Explain which definition or assumption it violates.
A correct one-tailed p-value is 0.03 at a 5% threshold. Enter 1 for reject or 0 for do not reject.
The correct tail probability 0.03 is below 0.05, so reject H₀. This is evidence, not proof.
Failing to reject a null hypothesis proves it is true.
Failing to reject H₀ is not proof that H₀ is true. Choose the tail from H₁, not from whichever tail gives a small result. Statistical significance does not measure the practical size of an effect.
Interpret a new situation
- Finish with a sentence about the population and the original claim. State the model's assumptions and consider whether the sampling procedure supports them.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find P(X≥3) for X binomial(10,0.1).
Sum the binomial probabilities for k=3 through 10; equivalently use 1-P(X≤2)=0.0701908264.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- edexcel IAL mathematics; official unit S2. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The chosen probability threshold for rejecting a null hypothesis. Choose the relationship, show the method, check its assumptions and interpret the result.