IntroStatsCalc

Paired T-Test

Matched pairs (before/after) test on the differences.

Inputs

Enter

Paste or type numbers separated by commas, spaces, or new lines. A column copied from Excel or Google Sheets works too.

Same order as list 1. Each difference is list 1 − list 2.

Alternative hypothesis Ha
Significance level α

Results update as you type.

Results

Reject H₀: t = −4.4, p-value = 0.0016

The p-value (0.0016) is less than or equal to α = 0.05, so the data give enough evidence to conclude that the mean difference is less than 0.

t-distribution with 7 degrees of freedom curve. The shaded p-value region is the area to the left of t = −4.4, with area 0.0016. Dashed line marks the critical value −1.895.−4−2024t = −4.4critical −1.895

Results

Mean difference d̄−4.625
sd2.9731
n (pairs)8
Test statistic t−4.4
df7
p-value0.0016
Decision at α = 0.05Reject H₀
Differences d = list 1 − list 2
Paird
1−5
2−6
31
4−7
5−6
6−6
7−1
8−7

On the TI-84

Enter list 1 in L1 and list 2 in L2. On the home screen, store the differences: L1 − L2 → L3. Then STAT → TESTS → 2:T-Test, Data, μ0 = 0, List: L3.

L1-L2→L3 then T-Test 0, L3, 1, <

Show steps
  1. Hypotheses. H₀: μd = 0   Ha: μd < 0
  2. Differences. d̄ = −4.625, sd = 2.9731, n = 8.
  3. Test statistic. t = d̄ ÷ (sd ÷ √n) = −4.625 ÷ (2.9731 ÷ √8) = −4.4, df = 7
  4. p-value. Left-tailed test: area to the left of the test statistic: p-value = 0.0016. Since 0.0016 ≤ 0.05, reject H₀.
  5. Critical-value method. At α = 0.05, the rejection region is t ≤ −1.895. The test statistic t = −4.4 is in the rejection region, so we reject H₀. (Same decision as the p-value method.)
  6. Conclusion. The p-value (0.0016) is less than or equal to α = 0.05, so the data give enough evidence to conclude that the mean difference is less than 0.

About this calculator

When data are paired

Paired (matched) data come from measuring the same subjects twice, or from natural pairs such as twins. Instead of comparing two separate means, compute each pair's difference and do a one-sample t-test on those differences with μ0 = 0.

Pay attention to the order of subtraction. If d = after − before, then "the treatment lowered the scores" means μd < 0.

More inference calculators: CI for a Mean · CI for a Proportion · Sample Size · Z-Test (mean) · T-Test (mean) · 1-Prop Z-Test · 2-Sample T-Test · 2-Prop Z-Test