O: Statistical hypothesis testing — AQA A-Level Mathematics
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O: Statistical hypothesis testing explained
You must use hypothesis testing language correctly.
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For a binomial test, define H0 (the default hypothesis) and H1 (the alternative), choose a significance level, calculate the test statistic (number of successes), and compare with critical values or p-value. Hypothesis tests assess evidence against H0, not test H1. Know 1-tail vs 2-tail, critical region, acceptance region. For correlation, interpret a given r using a given p-value or critical value: if p < significance level, reject H0. For a two-tailed test, reject H0 if |r| > critical value; for a one-tailed test, compare the signed r with a direction-specific critical value. You do not calculate r. For example, testing a coin for bias: H0: p=0.5, H1: p>0.5, n=20, X~B(20,0.5), find critical region for 5% one-tail.
Conduct a statistical hypothesis test for the proportion in the binomial distribution and interpret the results in context. Understand that a sample is being used to make an inference about the population and appreciate that the significance level is the probability of incorrectly rejecting the null hypothesis.
This topic covers testing a claim about a population proportion using a binomial model. Set up a null hypothesis (H0: p = p0) and an alternative (H1: p < p0, p > p0 or p ≠ p0). Calculate the probability of the observed result or something more extreme, assuming H0 is true. For a one-tailed test, compare this probability to the significance level α. For a two-tailed test, compare the observed tail probability to α/2. If the probability is less than or equal to the threshold, reject H0. Interpret the decision in context, noting the sample provides evidence about the unknown population proportion. The significance level α is the probability of incorrectly rejecting a true H0.
Conduct a statistical hypothesis test for the mean of a Normal distribution with known, given or assumed variance and interpret the results in context.
This topic covers testing a claim about the mean μ of a Normal population when the variance σ² is known, given or assumed. You set up H0: μ = μ0 and an alternative (μ < μ0, μ > μ0 or μ ≠ μ0). Under H0, the sample mean X̄ of n independent observations follows N(μ0, σ²/n). Standardise to Z = (X̄ − μ0)/(σ/√n), which has a standard Normal distribution. For a one-tailed test, find P(Z ≥ z) or P(Z ≤ z); for two-tailed, double the smaller tail. Compare with the significance level α. If p ≤ α, reject H0; otherwise do not reject. Interpret the decision in context, linking it to the population mean. The significance level is the probability of incorrectly rejecting a true H0.
Your focus
- Apply the language of hypothesis testing to a binomial model.
- Distinguish between 1-tail and 2-tail tests and use critical regions.
- Interpret a given correlation coefficient using a given p-value or critical value.
Show all 9 objectives
- Formulate null and alternative hypotheses for a population proportion and select an appropriate binomial model.
- Calculate p-values or critical regions for one- and two-tailed binomial tests and compare them with a given significance level.
- Interpret the outcome of a hypothesis test in context, including an explanation of the significance level as the probability of a Type I error.
- Formulate null and alternative hypotheses for a population mean and determine the distribution of the sample mean under H0.
- Calculate p-values for one- and two-tailed z-tests and compare them with a given significance level.
- Interpret the outcome of a hypothesis test in context, including an explanation of the significance level as the probability of a Type I error.
O: Statistical hypothesis testing exam tips
Marking Points
- Define null and alternative hypotheses in context, using correct symbols.
- State the significance level and whether the test is 1-tail or 2-tail.
- Identify the test statistic and its distribution under H0.
- Find critical value(s) and critical region, and state acceptance region.
- Calculate or use p-value to make a decision, comparing with significance level.
- Interpret a given correlation coefficient using a given p-value or critical value, linking to linear relationship; for a two-tailed test use |r| > critical value, for a one-tailed test use the signed r and a direction-specific critical value.
- Correctly state H0 and H1 in terms of the population proportion p, ensuring the alternative hypothesis matches the direction of the investigation.
- Identify the binomial distribution X ~ B(n, p0) under the null hypothesis, where n is the sample size and p0 is the hypothesised proportion.
- Calculate the appropriate tail probability for the observed test statistic, comparing it to α for a one-tailed test or α/2 for a two-tailed test.
- Make a formal decision to reject or not reject H0 and write a conclusion in context, linking back to the original claim about the population proportion.
- Explain that the significance level is the probability of incorrectly rejecting a true null hypothesis (a Type I error).
- State H0 and H1 clearly in terms of the population mean μ, ensuring the alternative matches the direction of the test.
- Identify the distribution of the sample mean under H0: X̄ ~ N(μ0, σ²/n), using the given variance and sample size.
- Standardise the observed sample mean to a z-score using z = (x̄ − μ0)/(σ/√n).
- Find the appropriate tail probability (or probabilities) using the standard Normal distribution, taking care with one- and two-tailed tests.
- Compare the p-value with the significance level and make a formal decision about H0.
- Write a conclusion in context, interpreting the result in terms of the population mean and acknowledging the role of the significance level.
Examiner Tips
- 💡Always write hypotheses in terms of the population parameter, not sample statistic.
- 💡For binomial, state X~B(n,p) under H0 and use tables or calculation for critical region.
- 💡For correlation, compare given p-value with significance level or given r with critical value, and conclude in context; check whether the test is one- or two-tailed before using |r|.
- 💡Always define the population proportion p in words before writing H0 and H1, so your hypotheses are unambiguous.
- 💡Show the distribution you are using under H0, e.g. X ~ B(20, 0.3), and state the observed value of X clearly.
- 💡For two-tailed tests, it is often safer to find the critical regions for each tail using α/2 rather than calculating p-values, especially if the distribution is highly skewed.
- 💡Write down the distribution of X̄ under H0 explicitly, e.g. X̄ ~ N(50, 16/25), before standardising.
- 💡Sketch the standard Normal curve and shade the relevant tail(s) to avoid errors with one- and two-tailed tests.
- 💡State the z-value and the corresponding probability clearly, showing whether you used tables or a calculator.
- 💡In your conclusion, refer back to the original claim about the mean and avoid absolute statements like 'the mean is 50'.
Common Mistakes
- Confusing 1-tail and 2-tail tests; correct by reading the alternative hypothesis direction.
- Misinterpreting p-value as probability H0 is true; correct by stating it is probability of observed or more extreme given H0.
- For correlation, thinking r close to 1 means causation; correct by explaining it measures linear association only.
- Applying |r| > critical value to a one-tailed correlation test; correct by using the signed r and a direction-specific critical value.
- Doubling the tail probability for a two-tailed test: this is inaccurate for asymmetric binomial distributions (when p0 ≠ 0.5). Instead, compare the observed tail probability directly to α/2.
- Using the sample proportion as the null hypothesis proportion: the null hypothesis must use the claimed population proportion p0, not the observed sample proportion.
- Stating the conclusion as 'accept H0' rather than 'do not reject H0': the test does not prove H0, it only fails to find sufficient evidence against it.
- Misinterpreting the significance level as the probability that H0 is true: it is actually the probability of rejecting H0 given that H0 is true.
- Using the sample standard deviation instead of the known population standard deviation: the test requires the population variance to be known or assumed.
- Forgetting to divide the standard deviation by √n when calculating the standard error of the sample mean.
- Mixing up one- and two-tailed tests: for a two-tailed test, double the smaller tail probability; for a one-tailed test, use only the relevant tail.
- Concluding 'accept H0' instead of 'do not reject H0': failing to reject does not prove the null hypothesis.