Download A Level Maths Statistics Hypothesis Testing and more Study notes Mathematics in PDF only on Docsity!
Hypothesis Testing
Edexcel A Level Mathematics – Statistics Revision Notes
The Big Idea
Hypothesis testing is about using data to decide whether something unusual is really happening, or whether it’s just random chance.
The logic: You assume nothing special is going on (the null hypothesis). You collect data. If the data would be extremely unlikely under that assumption, you reject it and conclude something IS different.
Key Terms
Null hypothesis H 0 – your starting assumption. The “nothing has changed” posi- tion. Always stated as an equality, e.g. H 0 : p = 0.3. Alternative hypothesis H 1 – what you’re trying to find evidence for. States a direction or just “not equal.” Test statistic – the value you calculate from your sample data. Significance level α – how much evidence you demand before rejecting H 0. Common values: 1%, 5%, 10%. Critical region – the range of values of the test statistic that would lead you to reject H 0. Critical value – the boundary of the critical region. p-value – the probability of getting a result at least as extreme as yours, assuming H 0 is true.
One-Tailed vs Two-Tailed Tests
One-tailed test: You’re only looking for a change in one direction. H 1 : p < 0 .3 (testing if the probability has decreased) H 1 : p > 0 .3 (testing if the probability has increased) Two-tailed test: You’re looking for any change, up or down. H 1 : p ̸= 0. 3
Two-Tailed Significance Level
In a two-tailed test at 5% significance, you split the 5% equally between both tails – 2.5% in each tail. So your critical region is in both ends of the distribution.
Hypothesis Testing for a Binomial Proportion
This is the Year 1 content. The test statistic is the number of “successes” in your sample.
Step-by-Step Method
- State H 0 and H 1 , defining p in context.
- State the significance level α.
- Assume H 0 is true: let X ∼ B(n, p 0 ) where p 0 is the value in H 0.
- Find P (X ≤ observed value) or P (X ≥ observed value) depending on H 1.
- Compare this p-value with α: If p-value ≤ α: reject H 0. Sufficient evidence to support H 1. If p-value > α: do not reject H 0. Insufficient evidence.
- Write a conclusion in context.
Worked Example – Binomial Test
A bag is supposed to contain 30% red sweets (p = 0.3). Someone suspects there are fewer red sweets. They pick 20 sweets and find 3 are red. Test at the 5% significance level.
Step 1: H 0 : p = 0.3. H 1 : p < 0 .3 (one-tailed, lower).
Step 2: Significance level: 5%.
Step 3: Under H 0 : X ∼ B(20, 0 .3). Observed value: X = 3.
Step 4: P (X ≤ 3) = binomcdf(20, 0. 3 , 3) = 0.1071.
Step 5: 0. 1071 > 0 .05, so we do not reject H 0.
Conclusion: There is insufficient evidence at the 5% significance level to conclude that the proportion of red sweets is less than 0.3.
Finding the Critical Region
Instead of comparing a p-value, you can find the critical region – the set of values that would lead to rejection.
For a lower-tailed test at 5%: find the largest c such that P (X ≤ c) ≤ 0 .05.
For an upper-tailed test at 5%: find the smallest c such that P (X ≥ c) ≤ 0 .05, i.e., 1−P (X ≤ c − 1) ≤ 0 .05.
Critical Region Example
X ∼ B(20, 0 .3), upper-tailed test at 5%. P (X ≥ 10) = 1 − P (X ≤ 9) = 1 − 0 .9520 = 0. 0480 ≤ 0. 05 P (X ≥ 9) = 1 − P (X ≤ 8) = 1 − 0 .8867 = 0. 1133 > 0. 05 So the critical region is X ≥ 10. The actual significance level is 4.80% (not exactly 5% because the binomial is discrete).
Hypothesis Testing for a Normal Distribution Mean (Year
In Year 2, you test whether a population mean μ has changed, when you know the variance σ^2.
evidence to reject it. Two-tailed tests: forgetting to double the tail probability, or using 5% instead of 2.5% per tail. Not mentioning the significance level in the conclusion. For normal tests: using σ^2 instead of σ in the denominator.
