Math · Introductory statistics · Worked example
Interpret a p-value correctly
The bottle test gave p ≈ 0.028. Which statements are correct? (a) There is a 2.8% chance that the mean is 500 mL. (b) If the mean were 500 mL, samples this extreme would occur about 2.8% of the time. (c) The machine’s error is large.
Statement (a)
Incorrect. The p-value is computed assuming H₀ is true, so it cannot be the probability that H₀ is true.
Statement (b)
Correct. That is the definition: the chance, under H₀, of a t at least 2.33 from 0 in either direction.
Statement (c)
Not supported by the p-value alone. The estimated shortfall is 2.8 mL, and the interval 494.72 to 499.68 mL says it is somewhere between about 0.3 and 5.3 mL. Whether that matters is a practical judgment.
Result
Only (b) is correct.
Your turn
A test gives p = 0.30. Does that show H₀ is true?
Show the answer and explanation
No.
It shows only that the data are not unusual under H₀. A small or noisy sample can miss a real difference.
Keep exploring
In Statistics, 100 bottles with the same mean and standard deviation would give t = −2.8 ÷ 0.6 ≈ −4.67 with 99 degrees of freedom: enter those, and p drops below 0.0001.
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