Question 121 markMCQSection A

For the purpose of t-test of significance, if a random sample of size 34 is drawn from a normal population, then the degree of freedom (ν)(\nu) is :

Inferential Statistics
Degrees of Freedom in t-test

Options

(A)3232
(B)3333
(C)3434
(D)3535

Official answer

The correct option is (B) 33.


Reason

  • For a single-sample t-test, degrees of freedom ν=n−1\nu = n - 1.
  • Here n=34n = 34, so ν=34−1=33\nu = 34 - 1 = 33.
Answer keywords:degrees of freedomt-testn minus 1sample size 34single sample testsignificance testinferential statisticsnu

Marking scheme

  1. 1.1 mark: correct option (B) 33, obtained from ν=n−1=34−1\nu = n - 1 = 34 - 1.
  2. 2.No marks for 32, 34 or 35.

Hint

For a single sample t-test, degrees of freedom = n−1n - 1.

Quick oral answer

For a single-sample t-test the degrees of freedom equal the sample size minus one, so a sample of 34 gives thirty-three degrees of freedom.

Common mistakes

  1. 1.Using n itself (34) and forgetting to subtract 1.
  2. 2.Subtracting 2 (giving 32), which is the rule for a two-sample t-test, not a single sample.
  3. 3.Adding 1 to the sample size instead of subtracting.

Analysis and explanation

Degrees of freedom for a one-sample t-test equal the sample size reduced by one.


Concept

  • When the sample mean is used to estimate the population mean, one constraint is imposed on the data.
  • This costs one degree of freedom, so ν=n−1\nu = n - 1.

Calculation

  • n=34n = 34, therefore ν=34−1=33\nu = 34 - 1 = 33.

Why the key is right

  • (B) 33=34−133 = 34 - 1, matching the formula.

Why the distractors are wrong

  • (A) 32 wrongly subtracts 2 (that rule applies to a two-sample test, n1+n2−2n_1 + n_2 - 2).
  • (C) 34 uses n itself, forgetting the -1 correction.
  • (D) 35 adds 1 instead of subtracting.

Interesting facts

The t-distribution and its degrees of freedom were introduced by William Sealy Gosset in 1908, who published under the pen name 'Student' because his employer, the Guinness brewery, forbade staff from publishing.

As degrees of freedom grow large, the t-distribution approaches the standard normal curve; by ν=33\nu = 33 the two are already very close.

Question details

Q12 of 44 questions

Board
CBSE
Class
Class 12
Subject
Applied Mathematics
Year
2026
Section
Section A
Marks
1
Type
MCQ
Chapter
Inferential Statistics
Topic
Degrees of Freedom in t-test
Bloom's level
understand
Syllabus topic
Unit: Inferential Statistics — t-test of Significance: Degrees of Freedom
Author
CBSE
Reviewed by
Padhantu Academic Team

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Frequently asked questions

When is degrees of freedom n−2n - 2 instead of n−1n - 1?

For a two-sample t-test comparing two independent means, the degrees of freedom are n1+n2−2n_1 + n_2 - 2. For a single sample, as here, it is simply n−1n - 1.

Why do we subtract 1 from the sample size?

Because the sample mean is estimated from the data, it imposes one linear constraint, so only n−1n - 1 of the deviations are free to vary. This lost information is the one degree of freedom that is subtracted.