A hypothesis test does not prove a claim directly. It begins with a baseline assumption and asks how unusual the data are.
A battery manufacturer claims that a new process increases mean battery life beyond 10 hours. A sample of 40 batteries has a mean life of 11.2 hours.
Assumption
Mean life = 10 hr
Observed
Sample mean = 11.2 hr
Question
How unusual is this?
A sample mean above 10 hours does not by itself prove improvement. Sampling variation could also produce a higher mean.
Key question
If the true mean were still 10 hours, how unusual would a sample mean of 11.2 hours be?
Begin with no effect
A hypothesis test begins by assuming no change or no difference. This baseline statement is the null hypothesis, H₀.
The claim that the new process increases mean life is the alternative hypothesis, H₁.
| Hypothesis | Statement |
|---|---|
| H₀ | Mean battery life is 10 hours |
| H₁ | Mean battery life is greater than 10 hours |
Why reason backward?
A hypothesis test does not directly calculate the probability that H₀ is false. It asks how unusual the observed data would be if H₀ were true.
If the result would be very unusual under H₀, the data are considered inconsistent with that assumption.
The four steps
| Step | Action |
|---|---|
| 1 | State H₀ and H₁ |
| 2 | Calculate a test statistic |
| 3 | Evaluate extremeness under H₀ |
| 4 | Compare with a preselected decision rule |
The p-value and significance level implement the final two steps. They are covered in the next lesson.
Failure to reject is not proof
Weak evidence leads us not to reject H₀, but that does not prove the null hypothesis true.
Say it carefully
Incorrect: The new process has no effect.
Better: The current data do not provide sufficient evidence that mean battery life increased.
Change the null value
The curve is the world where the average is still 10 hours. The shaded tail is everything at least as high as what was measured.
Standard error
0.474
Distance ÷ SE
2.53
How rare
0.0057
If the average really were still 10 hours, the sample mean would wobble by about 0.47 hours. Measuring 11.2 sits 2.53 standard errors above the claim, which happens 0.57% of the time.
The same 1.2-hour gap is ordinary or remarkable depending on the noise and the sample size.
Hypothesis testing checks whether the data are compatible with a baseline assumption; it does not directly prove a claim.