Hypothesis Testing
Understand the framework of hypothesis testing, including null and alternative hypotheses, p-values, significance levels, and Type I/II errors in HSC Advanced Mathematics.
Null and Alternative Hypotheses
Hypothesis testing is a statistical method for making decisions about a population based on sample data. Every test begins with two competing statements:
H0 (Null Hypothesis): The default assumption — no effect, no difference, or no change. For example, "the population mean is 50."
H1 (Alternative Hypothesis): What we are trying to find evidence for — there is an effect, difference, or change. For example, "the population mean is not 50."
The alternative hypothesis can be one-tailed (testing for a specific direction, e.g., μ > 50 or μ < 50) or two-tailed (testing for any difference, e.g., μ ≠ 50).
Example: A factory claims its widgets weigh 100g on average. A quality inspector suspects they are lighter.
H0: μ = 100 (the mean weight is 100g)
H1: μ < 100 (the mean weight is less than 100g) — this is a one-tailed test.
P-Values and Significance Level
The p-value is the probability of observing data as extreme as (or more extreme than) the sample result, assuming the null hypothesis is true. A small p-value suggests the observed result is unlikely under H0.
If p-value ≤ α, reject H0. If p-value > α, do not reject H0.
The significance level α (commonly 0.05 or 5%) is the threshold for deciding when evidence is strong enough.
The significance level α is chosen before the test is conducted. Common values are α = 0.05 (5%), α = 0.01 (1%), and α = 0.10 (10%).
Example: A test yields a p-value of 0.032 with α = 0.05.
Since 0.032 < 0.05, we reject H0. There is sufficient evidence at the 5% significance level to support the alternative hypothesis.
Type I and Type II Errors
Because we make decisions based on sample data, there is always a chance of error:
Type I Error (False Positive): Rejecting H0 when it is actually true. The probability of this error equals the significance level α.
Type II Error (False Negative): Failing to reject H0 when it is actually false. The probability of this error is denoted β.
There is a trade-off: decreasing α (making it harder to reject H0) reduces Type I errors but increases Type II errors, and vice versa. The power of a test is 1 − β, representing the probability of correctly rejecting a false H0.
Real-world analogy: In a court trial, H0 is "the defendant is innocent."
Type I Error: Convicting an innocent person (rejecting a true H0).
Type II Error: Acquitting a guilty person (failing to reject a false H0).
Key Vocabulary
Null Hypothesis (H0)
The default claim about a population parameter, assumed true until evidence suggests otherwise.
P-value
The probability of obtaining a result at least as extreme as the observed data, assuming H0 is true.
Significance Level (α)
The pre-set threshold for the p-value below which H0 is rejected, typically 0.05.
Type I Error
Incorrectly rejecting a true H0 (false positive), with probability equal to α.
Worked Examples
A coin is flipped 100 times and lands heads 62 times. Test whether the coin is biased at the 5% significance level.
Step 1: H0: p = 0.5 (coin is fair). H1: p ≠ 0.5 (coin is biased). Two-tailed test, α = 0.05.
Step 2: Under H0, the sample proportion &hat;p = 0.62. The test statistic z = (0.62 − 0.5) / √(0.5 × 0.5/100) = 0.12/0.05 = 2.4.
Step 3: For a two-tailed test, p-value ≈ 2 × P(Z > 2.4) ≈ 2 × 0.0082 = 0.0164.
Answer: Since 0.0164 < 0.05, reject H0. There is sufficient evidence the coin is biased.
A test gives a p-value of 0.08. What conclusion do you draw at α = 0.05 and at α = 0.10?
At α = 0.05: p-value (0.08) > 0.05, so do not reject H0. Insufficient evidence.
At α = 0.10: p-value (0.08) < 0.10, so reject H0. Sufficient evidence at the 10% level.
A medical test has α = 0.01. Describe the Type I and Type II errors in context.
Context: H0: The patient does not have the disease. H1: The patient has the disease.
Type I Error: Concluding the patient has the disease when they don't (false positive). Probability = 0.01.
Type II Error: Concluding the patient doesn't have the disease when they do (false negative). This could be very dangerous in medical contexts.
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
If a hypothesis test has α = 0.05 and the p-value is 0.03, what is the correct decision?
Question 2
What is a Type I error?
Question 3
A manufacturer claims 95% of products pass quality control. A sample finds 90% pass. Which hypotheses are correct for testing the claim?
Question 4
If α = 0.05, what is the probability of making a Type I error?
Question 5
A p-value of 0.72 for a hypothesis test at α = 0.05 suggests:
Key Concepts Summary
- ● H0 is the null (default) hypothesis; H1 is the alternative hypothesis we seek evidence for.
- ● The p-value measures the strength of evidence against H0.
- ● Reject H0 when the p-value ≤ α (the significance level).
- ● Type I error = rejecting a true H0 (probability = α). Type II error = failing to reject a false H0.
- ● We never "accept" H0 — we either reject it or fail to reject it.