Estimation and Hypothesis Testing

What you'll learn

This revision guide covers estimation and hypothesis testing for AQA GCSE Statistics. You'll learn how to use samples to make estimates about populations, understand confidence intervals, and conduct hypothesis tests to determine whether results are statistically significant. These skills are essential for interpreting real-world data and are worth significant marks in your exam.

Key terms and definitions

Population — the entire group being studied, from which a sample is taken.

Sample — a subset of the population selected for study, used to make inferences about the whole population.

Bias — systematic error that causes a sample to be unrepresentative of the population, leading to inaccurate estimates.

Hypothesis — a statement or claim about a population parameter that can be tested using sample data.

Null hypothesis (H₀) — the default assumption that there is no effect, no difference, or no change; what we assume to be true unless evidence suggests otherwise.

Alternative hypothesis (H₁) — the claim we're testing for; states that there is an effect, difference, or change.

Significance level — the threshold probability (commonly 5% or 0.05) used to decide whether to reject the null hypothesis; represents the acceptable risk of making a Type I error.

Critical value — the boundary value that defines the rejection region; if the test statistic exceeds this, we reject the null hypothesis.

Core concepts

Sampling methods and estimation

When we cannot study an entire population, we take a sample and use it to estimate population parameters. The quality of our estimate depends on how we select the sample.

Random sampling ensures every member of the population has an equal chance of selection. Methods include:

• Simple random sampling — using random number generators or drawing names from a hat
• Systematic sampling — selecting every nth item (e.g., every 10th person)
• Stratified sampling — dividing the population into groups (strata) and sampling proportionally from each

Non-random sampling methods are quicker but more prone to bias:

• Quota sampling — selecting a set number from each group, but not randomly
• Opportunity sampling — using whoever is available
• Cluster sampling — randomly selecting whole groups, then sampling everyone within those groups

Sample size matters. Larger samples generally provide more reliable estimates because:

• They reduce random variation
• They better represent the population diversity
• They give narrower confidence intervals (more precise estimates)

However, very large samples can be expensive, time-consuming, and may show statistically significant results that aren't practically important.

Understanding confidence intervals

A confidence interval provides a range of values within which we believe the true population parameter lies. At GCSE level, you need to understand what they represent rather than calculate them from raw data.

A 95% confidence interval means:

• If we took 100 different samples and calculated 100 confidence intervals, approximately 95 would contain the true population parameter
• We are 95% confident the true value lies within this range
• There's a 5% chance the interval doesn't contain the true value

Interpreting confidence intervals:

If a 95% confidence interval for mean height is 165 cm to 175 cm:

• We estimate the true population mean height is between 165 cm and 175 cm
• We cannot say "there's a 95% probability the mean is in this range" (the true mean is fixed, not random)
• The interval width indicates precision — narrower intervals show more precise estimates

Factors affecting interval width:

• Sample size — larger samples produce narrower intervals (more precise)
• Variability in data — more spread-out data produces wider intervals
• Confidence level — higher confidence (e.g., 99% vs 95%) produces wider intervals

Setting up hypotheses

Before testing, you must clearly state your hypotheses about the population parameter.

The null hypothesis (H₀) always includes an equals sign and represents no change or difference:

• H₀: p = 0.5 (the probability equals 0.5)
• H₀: μ = 100 (the population mean equals 100)

The alternative hypothesis (H₁) represents what you're testing for and can be:

• Two-tailed: H₁: p ≠ 0.5 (the probability is different from 0.5)
• One-tailed: H₁: p > 0.5 or H₁: p < 0.5 (the probability is greater than or less than 0.5)

Use a two-tailed test when looking for any difference. Use a one-tailed test when you're specifically testing whether a parameter has increased or decreased.

Conducting hypothesis tests with binomial distributions

At GCSE level, you'll conduct hypothesis tests using the binomial distribution when dealing with situations involving success/failure or yes/no outcomes.

Step-by-step process:

1. State the hypotheses clearly

   • Define H₀ (always contains the assumed probability)
   • Define H₁ (one-tailed or two-tailed)

2. State the significance level

   • Usually 5% (0.05) unless told otherwise
   • For two-tailed tests, split between both tails (2.5% each)

3. Define the random variable

   • State what X represents (e.g., "Let X = number of successes in the sample")
   • Identify n (sample size) and p (probability under H₀)

4. Find the critical region

   • Calculate probabilities using binomial tables or formula
   • Find values where P(X ≤ x) ≤ 0.05 (or appropriate level)
   • The critical region contains the most extreme values that would lead to rejection

5. Compare the observed value with the critical region

   • If the result falls in the critical region, reject H₀
   • If not, do not reject H₀ (never say "accept H₀")

6. Write a conclusion in context

   • State whether there is sufficient evidence to reject H₀
   • Relate this to the original claim in plain English

Making conclusions and interpreting results

When the test statistic falls in the critical region:

• "There is sufficient evidence at the 5% significance level to reject H₀"
• "The result is statistically significant"
• State what this means in context (e.g., "There is evidence the coin is biased")

When the test statistic does not fall in the critical region:

• "There is insufficient evidence to reject H₀"
• "The result is not statistically significant"
• "We do not reject H₀" (never say we "accept" H₀ — we simply lack evidence against it)

Important distinctions:

• Statistical significance doesn't always mean practical importance
• Not rejecting H₀ doesn't prove it's true; we simply lack evidence against it
• The 5% significance level means we accept a 5% risk of incorrectly rejecting H₀ when it's actually true (Type I error)

Understanding Type I and Type II errors

Type I error — rejecting H₀ when it's actually true

• The probability of this is equal to the significance level (α)
• Example: Concluding a coin is biased when it's actually fair

Type II error — not rejecting H₀ when it's actually false

• Accepting a false null hypothesis
• Example: Concluding a coin is fair when it's actually biased

The significance level helps control the Type I error rate. Lowering the significance level (e.g., from 5% to 1%) reduces Type I errors but increases Type II errors.

Worked examples

Example 1: Setting up and interpreting a hypothesis test

Question: A teacher claims that 40% of students in her school walk to school. A student thinks the percentage is higher. She samples 30 students and finds that 16 walk to school. Test at the 5% significance level whether there is evidence the percentage is higher than 40%.

Solution:

Step 1: State hypotheses

• H₀: p = 0.4 (40% walk to school)
• H₁: p > 0.4 (more than 40% walk to school)
• This is a one-tailed test

Step 2: Significance level = 5% = 0.05

Step 3: Define the random variable

• Let X = number of students who walk to school in a sample of 30
• Under H₀, X ~ B(30, 0.4)

Step 4: Find the critical region

• We need P(X ≥ x) ≤ 0.05
• Using binomial tables or calculation:
  • P(X ≥ 17) = 1 - P(X ≤ 16) = 1 - 0.8329 = 0.1671 (too high)
  • P(X ≥ 18) = 1 - P(X ≤ 17) = 1 - 0.9155 = 0.0845 (too high)
  • P(X ≥ 19) = 1 - P(X ≤ 18) = 1 - 0.9597 = 0.0403 (less than 0.05)
• Critical region is X ≥ 19

Step 5: Compare observed value

• Observed value is 16
• 16 < 19, so 16 is not in the critical region

Step 6: Conclusion There is insufficient evidence at the 5% significance level to reject H₀. We conclude there is no significant evidence that the percentage of students who walk to school is higher than 40%.

(2 marks for hypotheses, 1 mark for significance level, 2 marks for critical region, 1 mark for comparison, 2 marks for conclusion = 8 marks total)

Example 2: Interpreting a confidence interval

Question: A survey of 200 adults found a 95% confidence interval for the proportion who support a new policy to be 0.52 to 0.64.

(a) Explain what this confidence interval tells us. (2 marks)

(b) The government claims that exactly 50% of adults support the policy. Comment on this claim using the confidence interval. (2 marks)

Solution:

(a) We are 95% confident that the true proportion of all adults who support the policy lies between 0.52 (52%) and 0.64 (64%). If we repeated this survey many times, approximately 95% of the confidence intervals produced would contain the true population proportion.

(1 mark for correct interpretation of the interval, 1 mark for explaining the 95% confidence)

(b) The value 0.50 (50%) does not lie within the confidence interval of 0.52 to 0.64. This provides evidence against the government's claim, suggesting the true proportion is likely higher than 50%.

(1 mark for noting 0.50 is outside the interval, 1 mark for appropriate conclusion)

Example 3: Two-tailed hypothesis test

Question: A manufacturer claims that a machine produces components with a 10% defect rate. A quality inspector suspects the rate may have changed. She examines 40 components and finds 1 defective. Test at the 5% significance level whether the defect rate has changed.

Solution:

Step 1: State hypotheses

• H₀: p = 0.1
• H₁: p ≠ 0.1 (two-tailed test)

Step 2: Significance level = 5%, so 2.5% in each tail

Step 3: Let X = number of defective components in a sample of 40

• Under H₀, X ~ B(40, 0.1)

Step 4: Find critical regions (both tails)

• Lower tail: P(X ≤ x) ≤ 0.025
  • P(X ≤ 0) = 0.0148 < 0.025
  • P(X ≤ 1) = 0.0805 > 0.025
  • Lower critical region: X ≤ 0
• Upper tail: P(X ≥ x) ≤ 0.025
  • P(X ≥ 9) = 1 - P(X ≤ 8) = 1 - 0.9576 = 0.0424 (too high)
  • P(X ≥ 10) = 1 - P(X ≤ 9) = 1 - 0.9829 = 0.0171 < 0.025
  • Upper critical region: X ≥ 10

Step 5: Observed value is 1

• 1 is not in either critical region (X ≤ 0 or X ≥ 10)

Step 6: Conclusion There is insufficient evidence at the 5% significance level to reject H₀. The data does not provide significant evidence that the defect rate has changed from 10%.

(8-9 marks total for complete solution)

Common mistakes and how to avoid them

• Confusing "not rejecting H₀" with "accepting H₀" — You can only ever reject or fail to reject the null hypothesis. Not having evidence against H₀ doesn't prove it's true; you simply lack sufficient evidence to conclude otherwise.

• Using the wrong tail for the test — Read the question carefully. "Has increased" requires a one-tailed test (H₁: p > value), while "has changed" requires a two-tailed test (H₁: p ≠ value). Two-tailed tests split the significance level between both tails.

• Forgetting to state conclusions in context — Don't just say "reject H₀." Explain what this means for the actual situation: "There is evidence the coin is biased" or "The percentage of students has increased."

• Misinterpreting confidence intervals — A 95% confidence interval doesn't mean "95% probability the parameter is in this range." The parameter is fixed (not random); the interval either contains it or doesn't. The 95% refers to the long-run success rate of the method.

• Including the boundary value in the wrong region — When the critical region is X ≥ 19, this means 19 IS in the critical region. When P(X ≥ 19) = 0.0403, the probability includes 19.

• Rounding errors in probability calculations — Keep at least 4 decimal places during calculations. Only round your final answer. Check whether your comparison should be < or ≤ the significance level.

Exam technique for "Estimation and Hypothesis Testing"

• Identify command words precisely — "Test whether" requires a full hypothesis test with all six steps. "State" means write it down clearly without explanation. "Interpret" means explain what it means in context, not just repeat numbers.

• Show all working systematically — Even if you can use a calculator for binomial probabilities, write down what you're calculating (e.g., "P(X ≤ 16) = 0.8329"). Method marks are awarded for correct process even if your final answer is wrong.

• For hypothesis tests, follow the structure — State both hypotheses clearly (1-2 marks), define your variable and distribution (1 mark), find the critical region with supporting calculations (2-3 marks), compare with observed value (1 mark), write conclusion in context (2 marks). This structure typically earns 7-9 marks.

• Pay attention to mark allocation — A 2-mark question needs two distinct points. If asked to "explain" for 3 marks, give three separate reasons or develop your answer with detail. Single-word answers rarely earn full marks on explanations.

Quick revision summary

Hypothesis testing uses sample data to test claims about populations. State null (H₀) and alternative (H₁) hypotheses, then find the critical region using the significance level (usually 5%). Reject H₀ if your observed value falls in the critical region; otherwise, don't reject it. Confidence intervals estimate population parameters with a level of certainty. Larger samples produce more precise estimates. Always interpret results in context and understand that statistical significance doesn't prove absolute truth—it indicates evidence strength. Master the systematic approach and show all working for maximum marks.