🧪 t-Test

Last Updated: Jan 2026

A t-Test is a hypothesis test used to determine whether a sample mean (or means) is significantly different when:

  • Sample size is small
  • Population standard deviation is unknown

It uses the t-distribution, which is wider than the normal distribution.

🗣 Hinglish Tip: t-Test = jab sample chhota ho aur σ pata na ho

When to Use t-Test?

Use t-Test when:

  • Sample size: n < 30
  • Population standard deviation unknown
  • Data is approximately normal
  • Samples are independent (except paired test)

Types of t-Test

  1. One-Sample t-Test
  2. Two-Sample (Independent) t-Test
  3. Paired t-Test

👉 In this tutorial, we cover One-Sample t-Test

t-Test Notation (Math Standard)

  • Population mean → μ
  • Sample mean → x̄
  • Sample standard deviation → s
  • Sample size → n
  • Degrees of freedom → df = n − 1
  • Significance level → α
  • t statistic → t

t-Test Formula (One-Sample)

t = (x̄ − μ) / (s / √n)

Example

A coaching institute claims that the average score of students is 70.A sample of 10 students is taken and their scores are:

65, 68, 70, 72, 74, 69, 71, 73, 67, 66

Test the claim at 5% significance level.

Step 1: State the Hypotheses

Two-tailed test (checking difference):

H₀: μ = 70
H₁: μ ≠ 70

Step 2: Identify Test Type

  • n = 10 (< 30) ✅
  • σ unknown ✅

👉 Use One-Sample t-Test

Step 3: Calculate Sample Mean (x̄)

Σx = 695
x̄ = 695 / 10 = 69.5

Step 4: Calculate Sample Standard Deviation (s)

xx − x̄(x − x̄)²
65-4.520.25
68-1.52.25
700.50.25
722.56.25
744.520.25
69-0.50.25
711.52.25
733.512.25
67-2.56.25
66-3.512.25
Σ(x − x̄)² = 82.5
s² = 82.5 / (10 − 1) = 9.17
s = √9.17 ≈ 3.03

Step 5: Calculate t-Statistic

t = (69.5 − 70) / (3.03 / √10)
t = -0.5 / 0.958
t ≈ -0.52

Step 6: Find Critical t-Value

  • α = 0.05
  • df = n − 1 = 9

From t-table:

t₀.₀₂₅,₉ = ±2.262

Step 7: Decision

  • Calculated t = −0.52
  • Critical t = ±2.262

Since:

|t| = 0.52 < 2.262

👉 Fail to Reject H₀

Step 8: Conclusion

There is no sufficient evidence to reject the institute’s claim. The average score can be considered 70.

🗣 Hinglish Tip: t limit ke andar hai → claim acceptable

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