Statistical Inference and Interpretation
What you must understand
Inference means drawing conclusions about a population from sample data. Know how to construct and interpret a confidence interval for a mean: x̄ ± z × (σ ÷ √n), where z = 1.96 for 95% confidence and 1.645 for 90%. Interpret it correctly: if we repeated the sampling many times, about 95% of such intervals would contain the true mean - not that the true mean has a 95% chance of lying in this particular interval.
Understand hypothesis testing language: null and alternative hypotheses, significance level, test statistic, critical value, and the meaning of a p-value. A small p-value suggests the observed result is unlikely under the null hypothesis. Know the difference between one-tailed and two-tailed tests and the risk of Type I (rejecting a true null) and Type II (failing to reject a false null) errors.
Interpretation skills matter: comment on sample size, representativeness, bias, causation versus association, and whether a conclusion is justified by the data shown.
Traps that appear in questions
- Saying "there is a 95% probability the mean lies in this interval"; the interval either contains it or not.
- Confusing statistical significance with practical importance; a tiny effect can be significant with a huge sample.
- Claiming proof of a hypothesis rather than "evidence to reject the null".
- Ignoring bias when interpreting a confidence interval - intervals say nothing about biased samples.
- Confusing Type I and Type II errors.
Worked example
A sample of 100 students has mean study time 12 hours; σ = 4. Construct a 95% confidence interval.
Answer: Standard error = 4 ÷ √100 = 0.4. Margin = 1.96 × 0.4 = 0.784. Interval ≈ 11.2 to 12.8 hours. State: "We are 95% confident the population mean lies between 11.2 and 12.8 hours, assuming a random, unbiased sample."
Rule of thumb: always report the estimate, the margin and the assumption of random sampling.