LinRegTTest Calculator (Slope t-Test)

Test whether there is a real linear relationship between two variables — that is, whether the population slope β is zero. Enter paired X and Y data; the output matches the TI-84 LinRegTTest screen: t, p, df, a, b, s, r², and r.

Just need the line and predictions? Use the regression calculator. Part of the TI-84 statistics functions guide.

The Slope t-Test

t = b / SE(b), SE(b) = s / √Σ(x − x̄)², df = n − 2

s = √(Σ(y − ŷ)² / (n − 2))

H₀: β = 0 (no linear relationship) vs. Hₐ: β ≠ 0, β < 0, or β > 0

Testing β = 0 is equivalent to testing that the population correlation ρ = 0, which is why the TI-84 labels the alternative "β & ρ". Conditions: a linear pattern, independent observations, residuals roughly normal with constant spread (check a residual plot).

How to Run LinRegTTest on a TI-84

  1. Enter X in L1 and Y in L2 (STAT → 1:Edit).
  2. Press STAT → TESTS and choose F:LinRegTTest (the letter can vary by model).
  3. Set Xlist: L1, Ylist: L2, Freq: 1, choose the alternative, and select Calculate.

If r² and r do not appear, turn on DiagnosticOn (TI-84 Plus) or set Stat Diagnostics to On in MODE (TI-84 Plus CE).

Worked Example: Study Hours and Scores

X = hours studied: 1, 2, 3, 4, 5, 6, 7, 8. Y = test score: 52, 55, 61, 60, 66, 70, 71, 77. Click Load example to reproduce.

  1. Line: ŷ = 48.5714 + 3.4286x — each extra hour adds about 3.4 points.
  2. s = 1.5430 and SE(b) = 0.2381, so t = 3.4286 / 0.2381 = 14.4 with df = 6.
  3. Two-tailed p-value ≈ 7.0242 × 10⁻⁶; r² = 0.9719, r = 0.9858.

The slope is overwhelmingly significant: study time explains about 97% of the variation in scores in this sample. For the correlation alone, see the correlation calculator.

Frequently Asked Questions

What does LinRegTTest test?

Whether the population slope β of the regression line is zero. A small p-value means the sample slope is too far from zero to be explained by chance, so there is evidence of a linear relationship between X and Y.

What is s in the LinRegTTest output?

s is the standard deviation of the residuals — the typical distance between the observed y values and the fitted line, in the units of y. It is also called the residual standard error.

Why are the degrees of freedom n − 2?

Two parameters, the intercept a and the slope b, are estimated from the data before the residual spread can be measured, which uses up two degrees of freedom.

Does a significant slope mean X causes Y?

No. The test shows a linear association that is unlikely to be due to chance. Causation requires a controlled experiment or other design that rules out confounding variables.

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