Hypothesis Testing
In short: Hypothesis testing is a decision routine that weighs sample data against a stated claim about a population, judging whether the data would be too surprising to expect if the claim held.
AP StatisticsStatisticsAP Statistics Unit 3
What is Hypothesis Testing?
Hypothesis testing follows courtroom logic. The claim under challenge, the null hypothesis, stays presumed true until the evidence makes it hard to believe. The issue is never whether the claim is proved, only whether this sample is too unusual to be a fluke.
The routine has four parts: write the hypotheses; check conditions and pick the test; get the test statistic and its p-value; then conclude in context by comparing that p-value with the significance level \(\alpha\). A tiny p-value is strong evidence against the null; a large one says the data fit it comfortably.
How to spot it
- A company, school or agency makes a claim and a sample is collected to check it with hypothesis testing.
- The question asks “is there convincing evidence that …” or “is the proportion different from …”.
- A p-value and a significance level are given and a decision is required.
- A study compares an observed result with what chance alone would produce.
The method
- Begin hypothesis testing by stating \(H_0\) and \(H_a\) in terms of a population parameter, using the same symbol for both.
- Check conditions (random, independent, large enough) and name the test.
- Calculate the test statistic and its p-value: how often a result so far from \(H_0\) would arise by luck alone.
- Compare with \(\alpha\); conclude in context: reject \(H_0\) (convincing evidence for \(H_a\)) or fail to reject it (not enough evidence).
Hypothesis Testing: solved example
Problem
A café owner believes that 40% of her customers pay by phone. After a loyalty app launches, she watches 200 randomly chosen transactions and sees 94 phone payments. At \(\alpha = 0.05\), does this show the proportion has risen?
Solution
Hypotheses: \(H_0: p = 0.40\), \(H_a: p > 0.40\). Conditions: \(200(0.40) = 80\) and \(200(0.60) = 120\) exceed 10.
Statistic: \(\hat{p} = \frac{94}{200} = 0.47\); \(z = \frac{0.47 – 0.40}{\sqrt{0.40 \times 0.60/200}} = \frac{0.07}{0.03464} \approx 2.02\). P-value \(\approx 0.0217\).
Conclusion: since \(0.0217 < 0.05\), reject \(H_0\). There is convincing evidence that more than 40% of customers now pay by phone.
Try it
Question 1
A test gives a p-value of 0.08 with \(\alpha = 0.05\). State the conclusion.
Show the answer
Since 0.08 is above \(\alpha\), the data do not clear the bar: keep \(H_0\) and report no convincing support for the alternative.
Question 2
Why is “accept \(H_0\)” the wrong phrase after a large p-value?
Show the answer
The data are merely compatible with \(H_0\); nothing is proved, and a different sample might point the other way.
Hypothesis Testing vs. Confidence Interval
Hypothesis testing poses a pass-or-fail question about one specific value of the parameter. A confidence interval lists every plausible value instead. The two agree in the two-sided case: a test at \(\alpha = 0.05\) rejects exactly the values that fall outside the 95% interval.
Keep going
Lessons
Worksheets and quizzes
Related words
- Null and Alternative Hypotheses
- P-Value
- Significance Level (Alpha)
- Type I and Type II Errors
- Confidence Interval
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