Confidence intervals, chi-square and regression
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| confidence interval/ˈkɒnfɪdəns ˈɪntəvl/ | 置信区间 | zhì xìn qū jiān |
How uncertain is the sample mean?
- Two samples give different means. We need a statement about sampling uncertainty, not a promise that one sample equals the population.
- This lesson studies confidence interval 置信区间: An interval calculated by a procedure designed to cover the population parameter at a stated long-run rate.
Choose the mathematical structure
- For a normal mean with known σ, a 95% interval is sample mean±1.96σ/√n. For chi-square, sum (observed-expected)²/expected with the correct degrees of freedom. Choose a test that fits the data and assumptions.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines confidence interval?
An interval calculated by a procedure designed to cover the population parameter at a stated long-run rate.
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.
With mean=50,σ=10,n=100, standard error=1 and the 95% interval is [48.04,51.96]. For observed counts 30 and 20 against expected 25 and 25, chi-square=25/25+25/25=2, before deciding degrees of freedom and the rejection threshold.
Confidence intervals, chi-square and regression
For a normal mean with known σ, a 95% interval is sample mean±1
Compare the model with the worked case and explain one change.
Find the standard error when σ=10 and n=100.
Standard error=σ/√n=10/10=1.
Test a tempting shortcut
- A 95% confidence procedure does not give a 95% probability that a fixed population mean moves within this one interval. Expected counts and estimated parameters affect the validity and degrees of freedom of chi-square tests.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A strong correlation alone establishes a causal relationship. This claim is false. Explain which definition or assumption it violates.
Find the lower 95% confidence bound when mean=50,σ=10,n=100.
Lower endpoint=50-1.96×1=48.04 under the stated known-σ normal model.
A strong correlation alone establishes a causal relationship.
A 95% confidence procedure does not give a 95% probability that a fixed population mean moves within this one interval. Expected counts and estimated parameters affect the validity and degrees of freedom of chi-square tests.
Interpret a new situation
- Report uncertainty in the context. In regression, inspect residuals, avoid unjustified extrapolation, and distinguish a fitted relationship from a causal explanation. S3 and IB AI HL require different verified scope boundaries.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find chi-square for observed 30,20 and expected 25,25.
Chi-square=(30-25)²/25+(20-25)²/25=1+1=2.
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 further mathematics; official unit S3. 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.
An interval calculated by a procedure designed to cover the population parameter at a stated long-run rate. Choose the relationship, show the method, check its assumptions and interpret the result.