Key takeaway: A t-test is a significance test about a population mean, or a difference of means, that uses the sample standard deviation and a t-distribution to judge whether the observed result is unusual.
Grade 12 – AP StatisticsAP Statistics Unit 4Statistics and Data
Definition of t-Test
A t-test answers questions such as: is the average fill of these bags really 15 ounces? The one-sample version compares a sample mean with a claimed value. The two-sample version compares the means of two independent groups, and the paired version works with differences.
The test statistic measures how many standard errors the sample mean lies from the claim. Because the population spread is estimated from the sample, the tail probability comes from a t curve with the right degrees of freedom. Conditions: random sampling and an approximately normal population or a large sample.
Formula and notation
For a one-sample test of \(H_0:\mu=\mu_0\):
\[ t=\frac{\,\overline{x}-\mu_0}{s/\sqrt{n}},\qquad df=n-1. \]
The p-value is the area in the tail or tails of the t curve beyond the observed \(t\).
Key facts
- The sign of t shows the direction of the difference from the claimed mean.
- A larger absolute t gives a smaller p-value.
- Check normality with a graph if the sample is small; heavy skew or outliers cause trouble.
- A paired design runs a one-sample t-test on the differences within each pair.
t-Test example
A cereal maker claims boxes hold 15 ounces on average. A random sample of 16 boxes has mean 14.2 ounces and standard deviation 1.6 ounces. Test \(H_a:\mu\neq 15\) at the 5% level.
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Standard error: \(\frac{1.6}{\sqrt{16}}=0.4\).
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Statistic: \(t=\frac{14.2-15}{0.4}=-2.0\), with \(df=15\).
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Two-sided p-value: about 0.064, and the critical value is \(t^{*}=2.131\).
Since \(0.064\gt 0.05\), fail to reject \(H_0\): the sample does not give convincing evidence that the mean differs from 15 ounces.
Common mistake
Watch out
A student looks up \(z=-2.0\) on the normal table, gets a two-sided area of 0.0455, and concludes that the mean differs from 15.
Why it is wrong
The standard deviation was estimated from only 16 boxes, so the reference curve is the t-distribution with 15 degrees of freedom, whose tails are heavier. Its two-sided area at 2.0 is about 0.064, which is above 0.05, so the decision changes.
Related terms and lessons
Browse the glossary: A–Z list · AP Statistics vocabulary · Statistics and Data