One-sample t test
A one-sample t test compares a sample mean with a claimed value when sigma is unknown. The statistic is t = (sample mean - claimed mean) / (s / square root of n), with n - 1 degrees of freedom.
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Analogy
It asks how many standard errors the sample mean sits from the claim.
Example
n = 16, mean = 52, s = 8H0: mean = 48SE = 8 / 4 = 2t = (52 - 48) / 2 = 2.0, df = 15Critical t (5%) = 2.131; p = 0.064Fail to reject H0
In practice
Checks whether a machine's average output matches its specification.
Practice
Flashcards
- Degrees of freedom in a one-sample t test?n minus 1.
Interview questions
- A machine should fill bottles with a mean of 500 ml. A sample of 16 bottles has a mean of 504 and s of 8. Test H0: mean = 500 at the 5% level.
- SE = 8 / 4 = 2
- t = 4 / 2 = 2.0 with 15 df
- Critical value 2.131 (two-sided)
- Fail to reject H0
- State the conclusion in words
Questions people ask
What assumptions does a t test make?
The data should be independent observations, and roughly normal when the sample is small. With larger samples, the central limit theorem makes the test robust to non-normality. Strong outliers or heavy skew in small samples can make the results unreliable.
How does the t statistic relate to the confidence interval?
They give matching answers. If the 95 percent interval for the mean excludes the claimed value, a two-sided test at alpha 0.05 rejects it. The interval has the advantage of also showing the plausible size of the difference, which a bare test result never does.