Hypothesis testing for 1 and 2 samples — Edexcel A-Level Statistics
Test yourself on Hypothesis testing for 1 and 2 samples with PEARSON EDEXCEL A-Level practice questions.
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Hypothesis testing for 1 and 2 samples explained
When a single sample is modelled as coming from a normal distribution whose variance is unknown, the sample variance s² replaces σ² and the test statistic follows a t-distribution with n − 1 degrees of freedom.
Read the full explanation
You first state H₀ and H₁ about the population mean μ, choose a significance level, then compute t = (x̄ − μ₀)/(s/√n), where x̄ is the sample mean, μ₀ the hypothesised mean, s the sample standard deviation and n the sample size. Compare t with the critical value from tₙ₋₁ for the correct tail, or use the p-value. For a two-tailed test at 5% with n = 16, df = 15 and the critical values are approximately ±2.131. Reject H₀ when the statistic lies beyond the critical value or the p-value is below the significance level, and write the conclusion in context.
Your focus
- Set up a one-sample t-test for a normal mean with unknown variance, including H₀, H₁ and significance level.
- Calculate the t-statistic using the sample mean, hypothesised mean, sample standard deviation and sample size.
- Use t-tables or technology to find critical values or p-values with n − 1 degrees of freedom and reach a justified conclusion.
Hypothesis testing for 1 and 2 samples exam tips
Marking Points
- State H₀ and H₁ in terms of the population mean μ, choosing a one-tailed or two-tailed alternative appropriate to the question.
- Recognise that σ is unknown, so the sample standard deviation s is used and the test statistic has a t-distribution with n − 1 degrees of freedom.
- Calculate the test statistic as t = (x̄ − μ₀)/(s/√n), keeping the correct order in the numerator and dividing by the standard error s/√n.
- Obtain the critical value or p-value from the t-distribution with n − 1 degrees of freedom and the chosen significance level, matching the tail(s) of H₁.
- Compare the test statistic with the critical value, or the p-value with the significance level, and state the decision to reject or not reject H₀.
- Write the conclusion in the context of the original variable, making clear what the evidence does or does not support about the mean.
Examiner Tips
- 💡Write down H₀, H₁, the significance level and the degrees of freedom before calculating, so the method is clear and easy to follow.
- 💡Show the substitution into t = (x̄ − μ₀)/(s/√n) with the values you use, then give the test statistic to an appropriate accuracy.
- 💡State the critical value or p-value with its source (t-distribution, n − 1 df) and finish with a conclusion in context, not just 'reject H₀'.
Common Mistakes
- Using the normal distribution with σ known when the variance is unknown: the correct approach is to use s and the t-distribution with n − 1 degrees of freedom.
- Dividing by s instead of s/√n in the denominator: the standard error of the sample mean is s/√n, so the statistic is t = (x̄ − μ₀)/(s/√n).
- Using n degrees of freedom instead of n − 1: for a one-sample t-test the degrees of freedom are n − 1.
- Choosing a one-tailed critical value when H₁ is two-tailed, or vice versa: the tail(s) of the test must match the alternative hypothesis.