PSYC 354 | Statistics for the Behavioral Sciences Module 6 | Distribution of Sample Means | 2026 Update
PSYC 354 MODULE 6 QUIZ
The Distribution of Sample Means | Questions and Answers | 2026 Update | Score A+
Course PSYC 354 — Statistics for the Behavioral Sciences (Liberty University)
Module 6 Quiz — The Distribution of Sample Means, CLT, Standard Error &
Assessment
Probability
Total Questions 30 (Multiple Choice, Single Best Answer)
Total Points 100 points (3.33 points per question; Q30 = 3.40 pts)
Cognitive Distribution 30% Recall | 50% Application | 20% Analysis
Format Mix 75% Scenario-Based Calculation & Interpretation | 25% Direct Conceptual
Alignment APA 7th Edition Reporting | 2026–2027 Academic Standards
Distribution of Sample Means & CLT; Standard Error Calculation & Influencing
Domains Covered Factors; Z-Scores for Sample Means, Probability & Behavioral Applications
(SPSS/JASP, APA 7 CIs, AI-Assisted Probability)
Examiner Directive: This Module 6 examination assesses mastery-level competency in the
distribution of sample means, the Central Limit Theorem (CLT), standard error (σ_M = σ/√n),
z-scores for sample means (z = (M − µ)/σ_M), and probability applications in behavioral
research, consistent with the Liberty University PSYC 354 curriculum. Each item requires
absolute mathematical precision, proper statistical notation (µ for population mean, σ for
population standard deviation, M for sample mean, σ_M for standard error), and foolproof
step-by-step rationales. Distractors are engineered to represent the most common student
calculation errors — including forgetting to take the square root of n in the standard error
formula (σ_M = σ/n instead of σ/√n), using the population z-score formula z = (X−µ)/σ instead of
the sample-mean z-score formula z = (M−µ)/σ_M, confusing the population standard deviation
σ with the standard error σ_M, and misreading the unit normal table (e.g., body vs. tail
probabilities). The 2026 update integrates contemporary SPSS/JASP sampling-distribution
output interpretation, AI-assisted probability estimation, and APA 7th-edition reporting
standards for standard error and confidence intervals.
SECTION 1: Fundamentals of the Distribution of Sample Means
& Central Limit Theorem (Q1–Q10)
Q1: Which of the following best defines the distribution of sample means?
A. The distribution of all individual scores in a population
B. The collection of sample means (M values) from all possible random samples of a
particular size (n) drawn from a single population, along with their associated
probabilities [CORRECT]
C. The distribution of population parameters estimated from a single sample
D. The distribution of standard deviations computed across multiple populations
Correct Answer: B — B. The collection of sample means (M values) from all possible
random samples of a particular size (n) drawn from a single population, along with their
Page 1 | Complete Solution Key
, PSYC 354 | Statistics for the Behavioral Sciences Module 6 | Distribution of Sample Means | 2026 Update
associated probabilities
Rationale: The distribution of sample means is the set of sample means (M values) obtained from
all possible random samples of a fixed size n drawn from a single population, together with the
probability of each M. It is a sampling distribution, not a distribution of individual scores. Option A
describes the population distribution. Option C describes parameter estimation, not a sampling
distribution. Option D confuses standard deviations with sample means. The defining feature is that
the values in the distribution are means (M), not individual scores.
Q2: According to the Central Limit Theorem, under what condition does the distribution
of sample means approach a normal distribution even when the population is NOT
normally distributed?
A. When the population standard deviation is at least 10
B. When the sample mean equals the population mean
C. When the sample size is sufficiently large — conventionally n ≥ 30 [CORRECT]
D. When the number of samples drawn equals the population size
Correct Answer: C — C. When the sample size is sufficiently large — conventionally n ≥
30
Rationale: The Central Limit Theorem states that, regardless of the population's shape, the
distribution of sample means approaches a normal distribution as the sample size n increases. The
conventional rule of thumb is n ≥ 30 for non-normal populations. Option A is irrelevant — population
SD magnitude does not determine shape. Option B is always true (µ_M = µ) but does not address
shape. Option D confuses the number of samples with sample size. For non-normal populations, n
≥ 30 is the threshold that permits the CLT to ensure approximate normality of the sampling
distribution.
Q3: If a population is normally distributed, what can be concluded about the shape of
the distribution of sample means for ANY sample size n (including small samples)?
A. The distribution of sample means is exactly normal for any n, including n = 1
[CORRECT]
B. The distribution of sample means is normal only when n ≥ 30
C. The distribution of sample means is skewed for small n
D. The distribution of sample means is uniform for small n
Correct Answer: A — A. The distribution of sample means is exactly normal for any n,
including n = 1
Rationale: When the population itself is normally distributed, the distribution of sample means is
exactly normal for every sample size n — there is no minimum-n requirement. The n ≥ 30 rule is
needed only when the population is non-normal, to invoke the CLT. Option B incorrectly applies the
n ≥ 30 threshold to a normal population. Option C is wrong — a normal population produces a
normal sampling distribution at any n. Option D is wrong — uniformity is unrelated to sampling from
a normal population. For normally distributed populations, normality of M holds at all sample sizes.
Q4: A population has a mean of µ = 80. What is the mean of the distribution of sample
means (µ_M) for samples of size n = 25?
A. 20 — dividing µ by n
B. 80 — the mean of the distribution of sample means equals the population mean: µ_M =
µ = 80 [CORRECT]
C. 16 — multiplying µ by √n / n
D. Cannot be determined without the population standard deviation
Correct Answer: B — B. 80 — the mean of the distribution of sample means equals the
population mean: µ_M = µ = 80
Page 2 | Complete Solution Key
PSYC 354 MODULE 6 QUIZ
The Distribution of Sample Means | Questions and Answers | 2026 Update | Score A+
Course PSYC 354 — Statistics for the Behavioral Sciences (Liberty University)
Module 6 Quiz — The Distribution of Sample Means, CLT, Standard Error &
Assessment
Probability
Total Questions 30 (Multiple Choice, Single Best Answer)
Total Points 100 points (3.33 points per question; Q30 = 3.40 pts)
Cognitive Distribution 30% Recall | 50% Application | 20% Analysis
Format Mix 75% Scenario-Based Calculation & Interpretation | 25% Direct Conceptual
Alignment APA 7th Edition Reporting | 2026–2027 Academic Standards
Distribution of Sample Means & CLT; Standard Error Calculation & Influencing
Domains Covered Factors; Z-Scores for Sample Means, Probability & Behavioral Applications
(SPSS/JASP, APA 7 CIs, AI-Assisted Probability)
Examiner Directive: This Module 6 examination assesses mastery-level competency in the
distribution of sample means, the Central Limit Theorem (CLT), standard error (σ_M = σ/√n),
z-scores for sample means (z = (M − µ)/σ_M), and probability applications in behavioral
research, consistent with the Liberty University PSYC 354 curriculum. Each item requires
absolute mathematical precision, proper statistical notation (µ for population mean, σ for
population standard deviation, M for sample mean, σ_M for standard error), and foolproof
step-by-step rationales. Distractors are engineered to represent the most common student
calculation errors — including forgetting to take the square root of n in the standard error
formula (σ_M = σ/n instead of σ/√n), using the population z-score formula z = (X−µ)/σ instead of
the sample-mean z-score formula z = (M−µ)/σ_M, confusing the population standard deviation
σ with the standard error σ_M, and misreading the unit normal table (e.g., body vs. tail
probabilities). The 2026 update integrates contemporary SPSS/JASP sampling-distribution
output interpretation, AI-assisted probability estimation, and APA 7th-edition reporting
standards for standard error and confidence intervals.
SECTION 1: Fundamentals of the Distribution of Sample Means
& Central Limit Theorem (Q1–Q10)
Q1: Which of the following best defines the distribution of sample means?
A. The distribution of all individual scores in a population
B. The collection of sample means (M values) from all possible random samples of a
particular size (n) drawn from a single population, along with their associated
probabilities [CORRECT]
C. The distribution of population parameters estimated from a single sample
D. The distribution of standard deviations computed across multiple populations
Correct Answer: B — B. The collection of sample means (M values) from all possible
random samples of a particular size (n) drawn from a single population, along with their
Page 1 | Complete Solution Key
, PSYC 354 | Statistics for the Behavioral Sciences Module 6 | Distribution of Sample Means | 2026 Update
associated probabilities
Rationale: The distribution of sample means is the set of sample means (M values) obtained from
all possible random samples of a fixed size n drawn from a single population, together with the
probability of each M. It is a sampling distribution, not a distribution of individual scores. Option A
describes the population distribution. Option C describes parameter estimation, not a sampling
distribution. Option D confuses standard deviations with sample means. The defining feature is that
the values in the distribution are means (M), not individual scores.
Q2: According to the Central Limit Theorem, under what condition does the distribution
of sample means approach a normal distribution even when the population is NOT
normally distributed?
A. When the population standard deviation is at least 10
B. When the sample mean equals the population mean
C. When the sample size is sufficiently large — conventionally n ≥ 30 [CORRECT]
D. When the number of samples drawn equals the population size
Correct Answer: C — C. When the sample size is sufficiently large — conventionally n ≥
30
Rationale: The Central Limit Theorem states that, regardless of the population's shape, the
distribution of sample means approaches a normal distribution as the sample size n increases. The
conventional rule of thumb is n ≥ 30 for non-normal populations. Option A is irrelevant — population
SD magnitude does not determine shape. Option B is always true (µ_M = µ) but does not address
shape. Option D confuses the number of samples with sample size. For non-normal populations, n
≥ 30 is the threshold that permits the CLT to ensure approximate normality of the sampling
distribution.
Q3: If a population is normally distributed, what can be concluded about the shape of
the distribution of sample means for ANY sample size n (including small samples)?
A. The distribution of sample means is exactly normal for any n, including n = 1
[CORRECT]
B. The distribution of sample means is normal only when n ≥ 30
C. The distribution of sample means is skewed for small n
D. The distribution of sample means is uniform for small n
Correct Answer: A — A. The distribution of sample means is exactly normal for any n,
including n = 1
Rationale: When the population itself is normally distributed, the distribution of sample means is
exactly normal for every sample size n — there is no minimum-n requirement. The n ≥ 30 rule is
needed only when the population is non-normal, to invoke the CLT. Option B incorrectly applies the
n ≥ 30 threshold to a normal population. Option C is wrong — a normal population produces a
normal sampling distribution at any n. Option D is wrong — uniformity is unrelated to sampling from
a normal population. For normally distributed populations, normality of M holds at all sample sizes.
Q4: A population has a mean of µ = 80. What is the mean of the distribution of sample
means (µ_M) for samples of size n = 25?
A. 20 — dividing µ by n
B. 80 — the mean of the distribution of sample means equals the population mean: µ_M =
µ = 80 [CORRECT]
C. 16 — multiplying µ by √n / n
D. Cannot be determined without the population standard deviation
Correct Answer: B — B. 80 — the mean of the distribution of sample means equals the
population mean: µ_M = µ = 80
Page 2 | Complete Solution Key
