Your Progress
Green dots light up as you complete each step.
Step 1 — Paste your sample (minutes)
Reminder: We test the class mean against a target \( \mu_0 \) (default 30). Keep dataset consistent across t, p, and CI.
Step 2 — Summaries (x̄, s, SE, df)
Excel mirror — summaries (with a simple cell map)
Cell map: Put your sample in column A (one value per cell). Then set:
B1 = n, B2 = μ₀, B3 = df, B4 = α; C2 = x̄, C3 = s, C4 = SE.
B1 = n, B2 = μ₀, B3 = df, B4 = α; C2 = x̄, C3 = s, C4 = SE.
B1 (n):
C2 (x̄):
C3 (s):
C4 (SE):
B3 (df):
MIN:
=COUNTA(A:A)
C2 (x̄):
=AVERAGE(A:A)
C3 (s):
=STDEV.S(A:A)
C4 (SE):
=C3/SQRT(B1)
B3 (df):
=B1-1
MIN:
=MIN(A:A)
•
MAX: =MAX(A:A)
Step 3 — Hypotheses & tail
H₀: μ = 30; H₁: μ ≠ 30 (two-sided)
Step 4 — Test statistic t
Excel mirror — t-statistic
Using the cells above:
One-liner from the data column (μ₀ in B2):
=(C2-B2)/(C3/SQRT(B1))
One-liner from the data column (μ₀ in B2):
=(AVERAGE(A:A)-B2)/(STDEV.S(A:A)/SQRT(COUNTA(A:A)))
Step 5 — p-value & decision
Excel mirrors — p-value (copy)
Using t in D2 (or wherever you put it) and df in B3:
Right:
Right:
=T.DIST.RT(D2,B3)
Left: =T.DIST(D2,B3,TRUE)
Two: =T.DIST.2T(ABS(D2),B3)
One-liners (direct from column A; μ₀ in B2):
Right:
Left:
Two:
Right:
=T.DIST.RT((AVERAGE(A:A)-B2)/(STDEV.S(A:A)/SQRT(COUNTA(A:A))), COUNTA(A:A)-1)
Left:
=T.DIST((AVERAGE(A:A)-B2)/(STDEV.S(A:A)/SQRT(COUNTA(A:A))), COUNTA(A:A)-1, TRUE)
Two:
=T.DIST.2T(ABS((AVERAGE(A:A)-B2)/(STDEV.S(A:A)/SQRT(COUNTA(A:A)))), COUNTA(A:A)-1)
Decision templates (compare to α in B4):
Right:
Right:
=IF(T.DIST.RT(D2,B3)<=B4,"Reject H0","Fail to reject H0")
Left: =IF(T.DIST(D2,B3,TRUE)<=B4,"Reject H0","Fail to reject H0")
Two: =IF(T.DIST.2T(ABS(D2),B3)<=B4,"Reject H0","Fail to reject H0")
Step 6 — t* and CI / bound
Excel mirrors — t* & CI/bounds (α in B4, df in B3, x̄ in C2, SE in C4)
Two-sided t*:
=T.INV.2T(B4,B3)
One-sided t* (both cases below use this): =T.INV(1-B4,B3)
Two-sided CI lower:
=C2 - T.INV.2T(B4,B3)*C4
upper: =C2 + T.INV.2T(B4,B3)*C4
Right-tail (H₁: μ > μ₀) — lower bound:
=C2 - T.INV(1-B4,B3)*C4
(upper is +∞)
Left-tail (H₁: μ < μ₀) — upper bound:
=C2 + T.INV(1-B4,B3)*C4
(lower is −∞)
Step 7 — Auto summary
Hypothesis → t → p (cheat-sheet)
A) Hypotheses (H₀ includes “=”).
- Longer than 30 (right tail) ⇒ H₁: μ > 30 (right); H₀: μ ≤ 30 (includes “=”).
- Less than 30 (left tail) ⇒ H₁: μ < 30 (left); H₀: μ ≥ 30 (includes “=”).
- Different from 30 (two tails) ⇒ H₁: μ ≠ 30 (two); H₀: μ = 30.
B) Test statistic t = (x̄ − μ₀)/(s/√n). Excel:
=(C2-B2)/(C3/SQRT(B1))C) p-value Right:
=T.DIST.RT(t,df) • Left: =T.DIST(t,df,TRUE) • Two: =T.DIST.2T(ABS(t),df)\(p=1-F(t)\) (right), \(p=F(t)\) (left), \(p=2\min\{F(t),1-F(t)\}\) (two).
Right-tailed
p = shaded area to the right of t
p = —
Left-tailed
p = shaded area to the left of t
p = —
Two-sided
p = both tails beyond |t|
p = —
Examples — left, right, and two tails for common t (df = 22)
Exact t-CDF. Rows: t ∈ {1.5, 2, 3, 3.8}. Columns: Right, Left, Two.