,Liberty University PSYC 354 Quiz 8 | Statistics for the
Social Sciences (2026) Q&A
1. Which of the following best describes the primary purpose of an independent-samples t-
test?
A) To compare a sample mean to a known population mean
B) To compare the means of two independent groups to determine if they differ significantly
C) To compare the means of three or more groups
D) To examine the relationship between two continuous variables
Correct Answer: To compare the means of two independent groups to determine if they
differ significantly
Rationale: The independent-samples t-test is specifically designed to compare the means of
two separate, unrelated groups. It determines whether the difference between the two sample
means is statistically significant. This test is used when participants are randomly assigned to
different conditions or when comparing two distinct populations.
2. What is the null hypothesis for an independent-samples t-test?
A) The two population means are equal (μ₁ = μ₂)
B) The two population means are different (μ₁ ≠ μ₂)
C) The sample means are equal (M₁ = M₂)
D) The sample means are different (M₁ ≠ M₂)
Correct Answer: The two population means are equal (μ₁ = μ₂)
Rationale: The null hypothesis for an independent-samples t-test states that there is no
difference between the two population means. The test evaluates whether the observed
difference between sample means is large enough to reject this null hypothesis. The
alternative hypothesis states that the population means are not equal.
,3. Which of the following is a key assumption of the independent-samples t-test?
A) The two samples must be dependent on each other
B) The populations from which samples are drawn must have equal variances
C) The sample sizes must be exactly equal
D) The dependent variable must be measured on a nominal scale
Correct Answer: The populations from which samples are drawn must have equal variances
Rationale: The independent-samples t-test assumes homogeneity of variance, meaning the
two populations have approximately equal variances. It also assumes that the data are
normally distributed and that observations are independent. Equal sample sizes are not
required, though they are desirable for robustness.
4. What is the formula for the independent-samples t-test?
A) t = (M₁ – M₂) / s²
B) t = (M₁ – M₂) / SEM
C) t = (M₁ – M₂) / (s₁ + s₂)
D) t = (M₁ – M₂) / s_diff
Correct Answer: t = (M₁ – M₂) / SEM
Rationale: The independent-samples t-test formula is t = (M₁ – M₂) / SEM, where M₁ and M₂
are the sample means, and SEM is the estimated standard error of the difference between
means. The denominator represents the average distance between sample mean differences
and the population mean difference.
5. How is the estimated standard error for an independent-samples t-test calculated?
A) s / √n
B) √(s₁²/n₁ + s₂²/n₂)
C) (s₁ + s₂) / 2
D) s_diff / √n
, Correct Answer: √(s₁²/n₁ + s₂²/n₂)
Rationale: The estimated standard error for an independent-samples t-test is calculated as the
square root of the sum of the variances divided by their respective sample sizes. This
accounts for the variability in both samples. It reflects the precision of the difference between
the two sample means.
6. What are the degrees of freedom for an independent-samples t-test?
A) n₁ + n₂
B) n₁ + n₂ – 2
C) n₁ + n₂ – 1
D) (n₁ – 1) + (n₂ – 1)
Correct Answer: (n₁ – 1) + (n₂ – 1)
Rationale: Degrees of freedom for an independent-samples t-test are calculated by adding
the degrees of freedom from each sample. For each sample, df = n – 1. The total df is (n₁ – 1)
+ (n₂ – 1). This value is used to determine the critical t-value from the t-distribution table.
7. A researcher conducts an independent-samples t-test with n₁ = 12 and n₂ = 15. What are the
degrees of freedom?
A) 27
B) 26
C) 25
D) 24
Correct Answer: 25
Rationale: Degrees of freedom for an independent-samples t-test are calculated as (n₁ – 1) +
(n₂ – 1) = (12 – 1) + (15 – 1) = 11 + 14 = 25. This value is used to find the critical t-value.
Social Sciences (2026) Q&A
1. Which of the following best describes the primary purpose of an independent-samples t-
test?
A) To compare a sample mean to a known population mean
B) To compare the means of two independent groups to determine if they differ significantly
C) To compare the means of three or more groups
D) To examine the relationship between two continuous variables
Correct Answer: To compare the means of two independent groups to determine if they
differ significantly
Rationale: The independent-samples t-test is specifically designed to compare the means of
two separate, unrelated groups. It determines whether the difference between the two sample
means is statistically significant. This test is used when participants are randomly assigned to
different conditions or when comparing two distinct populations.
2. What is the null hypothesis for an independent-samples t-test?
A) The two population means are equal (μ₁ = μ₂)
B) The two population means are different (μ₁ ≠ μ₂)
C) The sample means are equal (M₁ = M₂)
D) The sample means are different (M₁ ≠ M₂)
Correct Answer: The two population means are equal (μ₁ = μ₂)
Rationale: The null hypothesis for an independent-samples t-test states that there is no
difference between the two population means. The test evaluates whether the observed
difference between sample means is large enough to reject this null hypothesis. The
alternative hypothesis states that the population means are not equal.
,3. Which of the following is a key assumption of the independent-samples t-test?
A) The two samples must be dependent on each other
B) The populations from which samples are drawn must have equal variances
C) The sample sizes must be exactly equal
D) The dependent variable must be measured on a nominal scale
Correct Answer: The populations from which samples are drawn must have equal variances
Rationale: The independent-samples t-test assumes homogeneity of variance, meaning the
two populations have approximately equal variances. It also assumes that the data are
normally distributed and that observations are independent. Equal sample sizes are not
required, though they are desirable for robustness.
4. What is the formula for the independent-samples t-test?
A) t = (M₁ – M₂) / s²
B) t = (M₁ – M₂) / SEM
C) t = (M₁ – M₂) / (s₁ + s₂)
D) t = (M₁ – M₂) / s_diff
Correct Answer: t = (M₁ – M₂) / SEM
Rationale: The independent-samples t-test formula is t = (M₁ – M₂) / SEM, where M₁ and M₂
are the sample means, and SEM is the estimated standard error of the difference between
means. The denominator represents the average distance between sample mean differences
and the population mean difference.
5. How is the estimated standard error for an independent-samples t-test calculated?
A) s / √n
B) √(s₁²/n₁ + s₂²/n₂)
C) (s₁ + s₂) / 2
D) s_diff / √n
, Correct Answer: √(s₁²/n₁ + s₂²/n₂)
Rationale: The estimated standard error for an independent-samples t-test is calculated as the
square root of the sum of the variances divided by their respective sample sizes. This
accounts for the variability in both samples. It reflects the precision of the difference between
the two sample means.
6. What are the degrees of freedom for an independent-samples t-test?
A) n₁ + n₂
B) n₁ + n₂ – 2
C) n₁ + n₂ – 1
D) (n₁ – 1) + (n₂ – 1)
Correct Answer: (n₁ – 1) + (n₂ – 1)
Rationale: Degrees of freedom for an independent-samples t-test are calculated by adding
the degrees of freedom from each sample. For each sample, df = n – 1. The total df is (n₁ – 1)
+ (n₂ – 1). This value is used to determine the critical t-value from the t-distribution table.
7. A researcher conducts an independent-samples t-test with n₁ = 12 and n₂ = 15. What are the
degrees of freedom?
A) 27
B) 26
C) 25
D) 24
Correct Answer: 25
Rationale: Degrees of freedom for an independent-samples t-test are calculated as (n₁ – 1) +
(n₂ – 1) = (12 – 1) + (15 – 1) = 11 + 14 = 25. This value is used to find the critical t-value.
