Q12

1 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 $(\nu)$ is :

Q: 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 $(\nu)$ is :

The correct option is **(B) 33**. **Reason** - For a single-sample t-test, degrees of freedom $\nu = n - 1$. - Here $n = 34$, so $\nu = 34 - 1 = 33$.

Inferential Statistics

Degrees of Freedom in t-test

Options

(A)

32

(B)

33

(C)

34

(D)

35

Official Answer

The correct option is (B) 33.

Reason

For a single-sample t-test, degrees of freedom $\nu = n - 1$ .
Here $n = 34$ , so $\nu = 34 - 1 = 33$ .

degrees of freedomt-testn minus 1sample size 34single sample testsignificance testinferential statisticsnu

Marking Scheme

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

Hint

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

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.

Analysis & 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 $\nu = n - 1$ .

Calculation

$n = 34$ , therefore $\nu = 34 - 1 = 33$ .

Why the key is right

(B) $33 = 34 - 1$ , matching the formula.

Why the distractors are wrong

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

Common Mistakes

1Using n itself (34) and forgetting to subtract 1.
2Subtracting 2 (giving 32), which is the rule for a two-sample t-test, not a single sample.
3Adding 1 to the sample size 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 $\nu = 33$ the two are already very close.

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Frequently Asked Questions

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

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

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 $n - 1$ of the deviations are free to vary. This lost information is the one degree of freedom that is subtracted.