Explain Student's t test with its types and steps. Add a worked example.

1. 1 · Define the t test. A test of significance that compares means in small samples (n < 30) when the population SD is unknown. Given by W.S. Gosset, who wrote as 'Student'.
2. 2 · Assumptions.
   • Data are roughly normal.
   • Samples are drawn at random.
   • For two groups, the variances are similar.
3. 3 · Types with formulas.
   • One-sample: t = (x̄ − μ) ÷ (s ÷ √n), df = n − 1.
   • Independent: t = (x̄1 − x̄2) ÷ [S × √(1/n1 + 1/n2)], df = n1 + n2 − 2.
   • Paired: t = d̄ ÷ (s_d ÷ √n), df = n − 1.
4. 4 · Steps. Hypotheses → 5% level → calculate t → degrees of freedom → table value → compare → conclude.
5. 5 · Worked example. [Table: before, after, d, d − d̄, (d − d̄)², with totals]
6. 6 · Decision. State the calculated t, df and table t (e.g. 5.20 > 2.571 at df 5), then reject or accept H0.
7. 7 · Nutrition link. Before-and-after nutrition education, two diet groups, or a group's intake against the RDA.
8. 8 · Close in one line. The t test tells us whether a difference is real or due to chance.

Scoring tip: Always write H0, df and the table value you used. Many students lose marks by giving only the t value.