Degrees of freedom and interpretation MCQs With Answer

Degrees of freedom and interpretation MCQs With Answer — Introduction

This question set is designed for M.Pharm students to deepen understanding of degrees of freedom (df) and their interpretation across common statistical tests used in pharmaceutical research. The items cover conceptual definitions, calculation rules for t-tests, ANOVA, chi-square, regression and more advanced topics such as Welch’s approximation and residual df implications. Each question focuses on practical application and interpretation — for example how df affect test critical values, estimation precision, and unbiased variance estimation. Working through these MCQs will strengthen your ability to choose correct formulas, interpret output from statistical software, and critically appraise results in experimental and clinical research contexts.

Q1. In a one-sample t-test for the mean using n observations, what is the degrees of freedom used to reference the t-distribution?

• n
• n − 1
• n + 1
• n − 2

Correct Answer: n − 1

Q2. For an independent two-sample t-test assuming equal variances with sample sizes n1 and n2, what is the appropriate degrees of freedom?

• n1 + n2
• n1 + n2 − 1
• n1 + n2 − 2
• min(n1, n2) − 1

Correct Answer: n1 + n2 − 2

Q3. Which formula gives the approximate degrees of freedom used in Welch’s t-test when variances are unequal?

• (n1 + n2 − 2)
• A weighted approximation using sample variances and sizes (Satterthwaite’s formula)
• min(n1 − 1, n2 − 1)
• n1 + n2 − 1

Correct Answer: A weighted approximation using sample variances and sizes (Satterthwaite’s formula)

Q4. In a one-way ANOVA comparing k groups with total sample size N, what is the degrees of freedom for the between-groups (treatment) mean square?

• N − k
• k − 1
• N − 1
• k

Correct Answer: k − 1

Q5. In the same one-way ANOVA, what is the degrees of freedom for the within-groups (error) mean square?

• N − k
• k − 1
• N − 1
• k

Correct Answer: N − k

Q6. For a chi-square test of independence using an r × c contingency table, how are the degrees of freedom calculated?

• r + c − 2
• (r − 1)(c − 1)
• r × c − 1
• min(r, c) − 1

Correct Answer: (r − 1)(c − 1)

Q7. In a simple linear regression with one predictor plus an intercept and sample size n, what is the residual degrees of freedom used to estimate residual variance?

• n − 2
• n − 1
• n
• n − 3

Correct Answer: n − 2

Q8. For multiple linear regression with p parameters including the intercept, what is the residual degrees of freedom?

• n − p
• p − 1
• n − 1
• n − p − 1

Correct Answer: n − p

Q9. In a paired t-test with n pairs, what degrees of freedom are used?

• n
• n − 1
• 2n − 1
• n − 2

Correct Answer: n − 1

Q10. When estimating a population variance from a sample, why do we divide the sum of squared deviations by n − 1 rather than n?

• To make the estimate biased downward
• Because one observation is always lost
• To obtain an unbiased estimator of the population variance using sample mean as estimate
• To reflect the larger population

Correct Answer: To obtain an unbiased estimator of the population variance using sample mean as estimate

Q11. Which statement about degrees of freedom and the t-distribution is correct?

• As degrees of freedom increase, the t-distribution becomes more spread out than normal
• As degrees of freedom increase, the t-distribution approaches the standard normal distribution
• Degrees of freedom do not affect the shape of the t-distribution
• The t-distribution is identical to the chi-square distribution for high df

Correct Answer: As degrees of freedom increase, the t-distribution approaches the standard normal distribution

Q12. In a goodness-of-fit chi-square test where you estimate m parameters from the data, what is the correct degrees of freedom for the test with k categories?

• k − 1
• k − m
• k − 1 − m
• k − m − 2

Correct Answer: k − 1 − m

Q13. You perform a two-way ANOVA without interaction on factors A (a levels) and B (b levels) with n observations per cell. What is the residual degrees of freedom?

• abn − 1
• ab(n − 1)
• a + b − 2
• n − 1

Correct Answer: ab(n − 1)

Q14. Which of the following best describes “effective degrees of freedom” in the context of smoothing or mixed models?

• The raw sample size used in the model
• An integer equal to the number of estimated parameters only
• A continuous quantity reflecting the model’s flexibility, often non-integer
• The number of levels of a categorical variable

Correct Answer: A continuous quantity reflecting the model’s flexibility, often non-integer

Q15. In an F-test comparing two nested linear models, how are the numerator and denominator degrees of freedom defined?

• Numerator = residual df of full model, Denominator = residual df of reduced model
• Numerator = difference in number of parameters between models, Denominator = residual df of full model
• Numerator = sample size, Denominator = number of predictors
• Numerator = number of predictors in full model, Denominator = number of predictors in reduced model

Correct Answer: Numerator = difference in number of parameters between models, Denominator = residual df of full model

Q16. If you have N independent observations and you estimate both the mean and variance from the same data, what is the total degrees of freedom available for partitioning variation?

• N
• N − 1
• N − 2
• 2N

Correct Answer: N − 1

Q17. In survival analysis using the Cox proportional hazards model with p covariates and n events, what degrees of freedom are typically used when testing a single regression coefficient?

• n − p
• 1
• p
• n

Correct Answer: 1

Q18. Which effect does increasing degrees of freedom (holding sample variance constant) have on the standard error of the mean estimate?

• Standard error increases
• Standard error decreases
• Standard error remains unchanged
• Standard error becomes undefined

Correct Answer: Standard error decreases

Q19. For a 3×4 contingency table with one parameter estimated from data, what is the appropriate chi-square degrees of freedom?

• 3 × 4 − 1 − 1 = 10
• (3 − 1)(4 − 1) = 6
• (3 − 1)(4 − 1) − 1 = 5
• (3 − 1)(4 − 1) + 1 = 7

Correct Answer: (3 − 1)(4 − 1) − 1 = 5

Q20. In repeated measures ANOVA with subject as a blocking factor (s subjects) and t time points, what is the degrees of freedom for the subject × time interaction error term if each subject is observed at all times?

• (s − 1)(t − 1)
• s(t − 1)
• (s − 1)t
• s − t

Correct Answer: (s − 1)(t − 1)

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