Hypothesis testing, significance testing, confidence intervals and power — Edexcel A-Level Statistics
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Hypothesis testing, significance testing, confidence intervals and power explained
A confidence interval for a population mean μ gives a range of plausible values based on sample data.
Read the full explanation
For a large sample or known population standard deviation σ, use the z-interval: x̄ ± z* × σ/√n. When σ is unknown and the sample is small, use the t-interval: x̄ ± t* × s/√n with n − 1 degrees of freedom. For example, with x̄ = 50, s = 8, n = 25 and 95% confidence, t* ≈ 2.064, giving 50 ± 2.064 × 8/5 = 50 ± 3.302, so the interval is approximately (46.70, 53.30). Interpret this as: we are 95% confident that the interval contains μ. The confidence level describes the long-run capture rate of the method, not the probability that a fixed μ lies in this particular interval. In context, state the interval in the units of the problem and comment on what it suggests about the mean.
Your focus
- Construct a confidence interval for a mean using the appropriate z or t critical value.
- Calculate the standard error correctly from σ or s and the sample size.
- Interpret a confidence interval and confidence level accurately in a practical context.
Hypothesis testing, significance testing, confidence intervals and power exam tips
Marking Points
- Selects the z-interval when σ is known or the sample is large, and the t-interval when σ is unknown and the sample is small.
- Uses the correct formula x̄ ± critical value × standard error, with standard error σ/√n or s/√n as appropriate.
- Obtains the correct critical value from the standard normal or t distribution with n − 1 degrees of freedom for the given confidence level.
- Calculates the interval correctly and states it in the context of the problem, including units where relevant.
- Interprets the confidence level correctly as the long-run proportion of intervals that would contain μ, avoiding the incorrect claim that there is a 95% probability that μ lies in the calculated interval.
Examiner Tips
- 💡Check whether the question gives σ or s and whether n is large; this determines whether to use z or t.
- 💡Write the interval in context, for example 'The mean waiting time is estimated to be between 12.3 and 15.7 minutes.'
- 💡In multiple-choice questions, eliminate options that misinterpret the confidence level as a probability statement about μ.
Common Mistakes
- Using z when the sample is small and σ is unknown. Correction: use the t-distribution with n − 1 degrees of freedom in that situation.
- Interpreting a 95% confidence interval as meaning there is a 95% probability that μ lies in the calculated interval. Correction: the confidence level refers to the long-run performance of the method; the parameter μ is fixed.
- Forgetting to divide s or σ by √n when computing the standard error. Correction: the standard error of x̄ is s/√n or σ/√n, not s or σ.