MATH 110 Module 6 Exam: Confidence Intervals
and Estimation – Introduction to Statistics
2026/2027 Edition – Verified Questions &
Answers – Portage Learning
SECTION 1: CONCEPTUAL FOUNDATIONS (Questions 1-5)
Q1: [Conceptual] What is a confidence interval?
A. A range of values that definitely contains the population parameter
B. A range of values likely to contain the population parameter based on sample data
C. A single value that estimates the population parameter
D. The standard deviation of the sampling distribution
Correct Answer: B
Explanation: A confidence interval is a range of values, calculated from sample data,
that is likely to contain the true population parameter. It provides an interval estimate
rather than a single point estimate. Option A is incorrect because confidence intervals
do not guarantee the parameter is within the range; they provide a level of confidence
(typically 90%, 95%, or 99%). Option C describes a point estimate (like the sample
mean). Option D describes the standard error or standard deviation, not a confidence
interval.
Formula Reference: Confidence Interval = Point Estimate ± Margin of Error
,Q2: [Conceptual] What does a 95% confidence level indicate?
A. The population parameter has a 95% chance of being in the interval
B. 95% of sample means will fall within the interval
C. If we took many samples and constructed intervals, about 95% of them would
contain the true population parameter
D. The interval will be correct 95% of the time for any given sample
Correct Answer: C
Explanation: The confidence level refers to the long-run success rate of the method. If
we repeated the sampling process many times and constructed confidence intervals
each time, approximately 95% of those intervals would contain the true population
parameter. This is the frequentist interpretation of confidence. Option A is a common
misinterpretation; the parameter is fixed (not random), so it either is or is not in the
interval. Option B confuses the sampling distribution of means with confidence
intervals. Option D is incorrect because for any single interval, we cannot say it has a
95% probability of being correct—the parameter is either in it or not.
Memory Aid: "95% confidence means 95% of intervals from repeated sampling would
capture the true parameter."
Q3: [Conceptual] Which factor increases the width of a confidence interval?
A. Larger sample size
B. Smaller confidence level
C. Larger standard error
, D. Smaller population standard deviation
Correct Answer: C
Explanation: The width of a confidence interval is determined by: Width = 2 × (Critical
Value × Standard Error). A larger standard error directly increases the margin of error
and thus the interval width. Option A (larger sample size) decreases standard error (SE
= σ/√n) and narrows the interval. Option B (smaller confidence level) decreases the
critical value (z or t) and narrows the interval. Option D (smaller population standard
deviation) decreases standard error and narrows the interval.
Key Relationship: Width ∝ (Critical Value × Standard Deviation) / √n
Q4: [Conceptual] If the sample size increases, what happens to the confidence interval
(assuming all else remains constant)?
A. The interval widens
B. The interval narrows
C. The interval stays the same
D. The interval becomes invalid
Correct Answer: B
Explanation: Increasing sample size (n) decreases the standard error (since SE = σ/√n
or s/√n), which reduces the margin of error and narrows the confidence interval. This
reflects the increased precision that comes with larger samples—more data gives us a
more precise estimate of the population parameter. The interval does not become
invalid (D); rather, it becomes more precise.
and Estimation – Introduction to Statistics
2026/2027 Edition – Verified Questions &
Answers – Portage Learning
SECTION 1: CONCEPTUAL FOUNDATIONS (Questions 1-5)
Q1: [Conceptual] What is a confidence interval?
A. A range of values that definitely contains the population parameter
B. A range of values likely to contain the population parameter based on sample data
C. A single value that estimates the population parameter
D. The standard deviation of the sampling distribution
Correct Answer: B
Explanation: A confidence interval is a range of values, calculated from sample data,
that is likely to contain the true population parameter. It provides an interval estimate
rather than a single point estimate. Option A is incorrect because confidence intervals
do not guarantee the parameter is within the range; they provide a level of confidence
(typically 90%, 95%, or 99%). Option C describes a point estimate (like the sample
mean). Option D describes the standard error or standard deviation, not a confidence
interval.
Formula Reference: Confidence Interval = Point Estimate ± Margin of Error
,Q2: [Conceptual] What does a 95% confidence level indicate?
A. The population parameter has a 95% chance of being in the interval
B. 95% of sample means will fall within the interval
C. If we took many samples and constructed intervals, about 95% of them would
contain the true population parameter
D. The interval will be correct 95% of the time for any given sample
Correct Answer: C
Explanation: The confidence level refers to the long-run success rate of the method. If
we repeated the sampling process many times and constructed confidence intervals
each time, approximately 95% of those intervals would contain the true population
parameter. This is the frequentist interpretation of confidence. Option A is a common
misinterpretation; the parameter is fixed (not random), so it either is or is not in the
interval. Option B confuses the sampling distribution of means with confidence
intervals. Option D is incorrect because for any single interval, we cannot say it has a
95% probability of being correct—the parameter is either in it or not.
Memory Aid: "95% confidence means 95% of intervals from repeated sampling would
capture the true parameter."
Q3: [Conceptual] Which factor increases the width of a confidence interval?
A. Larger sample size
B. Smaller confidence level
C. Larger standard error
, D. Smaller population standard deviation
Correct Answer: C
Explanation: The width of a confidence interval is determined by: Width = 2 × (Critical
Value × Standard Error). A larger standard error directly increases the margin of error
and thus the interval width. Option A (larger sample size) decreases standard error (SE
= σ/√n) and narrows the interval. Option B (smaller confidence level) decreases the
critical value (z or t) and narrows the interval. Option D (smaller population standard
deviation) decreases standard error and narrows the interval.
Key Relationship: Width ∝ (Critical Value × Standard Deviation) / √n
Q4: [Conceptual] If the sample size increases, what happens to the confidence interval
(assuming all else remains constant)?
A. The interval widens
B. The interval narrows
C. The interval stays the same
D. The interval becomes invalid
Correct Answer: B
Explanation: Increasing sample size (n) decreases the standard error (since SE = σ/√n
or s/√n), which reduces the margin of error and narrows the confidence interval. This
reflects the increased precision that comes with larger samples—more data gives us a
more precise estimate of the population parameter. The interval does not become
invalid (D); rather, it becomes more precise.
