Statistics Module 6 Exam Actual 2026/2027 –
100% Verified | Detailed Rationales – Pass
Guaranteed – A+ Graded
Content Area Overview: This actual examination reflects the fundamental statistical knowledge and
inferential reasoning required for success in Statistics Module 6. It is designed to evaluate the student's
ability to construct and interpret confidence intervals, perform hypothesis tests, and apply inferential
statistical methods to real-world data. Questions are structured to assess recall of key concepts,
application of statistical procedures, and analysis of data-driven scenarios. This authentic question bank
represents the real exams used in the course and serves as a comprehensive resource for students
demonstrating mastery of inferential statistics content.
SECTION 1: CONFIDENCE INTERVALS (9 Questions)
Q1. A researcher wants to estimate the mean height of adult males in a city. She takes a random sample
of 100 men and finds a sample mean of 70 inches with a standard deviation of 3 inches. Which
distribution should she use to construct a 95% confidence interval?
A. The z-distribution because the population standard deviation is known.
B. The t-distribution because the population standard deviation is unknown [CORRECT]
C. The chi-square distribution because she is estimating a mean.
D. The F-distribution because she has a large sample.
Rationale: The best answer is B. When the population standard deviation is unknown—which is almost
always the case in real research—we use the sample standard deviation as an estimate and construct
the confidence interval using the t-distribution. With a sample size of 100, the t-distribution is very close
to the normal distribution, but the principle remains: unknown population standard deviation means t-
distribution. The chi-square and F-distributions are used for variance and ANOVA applications, not for
mean estimation.
Correct Answer: B
,Q2. A 95% confidence interval for a population mean is calculated as (45.2, 52.8). Which interpretation
is most accurate?
A. There is a 95% probability that the population mean falls between 45.2 and 52.8.
B. We are 95% confident that the true population mean lies between 45.2 and 52.8 [CORRECT]
C. 95% of the sample data falls between 45.2 and 52.8.
D. The sample mean is exactly 49.0.
Rationale: The best answer is B. A confidence interval gives us a range of plausible values for the
population parameter, and the confidence level tells us how often this method would produce an
interval that captures the true parameter if we repeated the sampling process many times. The
population mean is a fixed value, not a random variable, so it doesn't have a probability of falling in the
interval. The sample mean is the midpoint (49.0), but the interval is about the unknown population
mean, not the sample data distribution.
Correct Answer: B
Q3. A researcher wants to decrease the width of a confidence interval without decreasing the
confidence level. Which strategy is most effective?
A. Increase the sample size [CORRECT]
B. Decrease the sample size
C. Increase the confidence level to 99%
D. Use the z-distribution instead of the t-distribution
Rationale: The best answer is A. The margin of error is inversely related to the square root of the sample
size, so increasing n reduces the width of the confidence interval while keeping the confidence level the
same. Decreasing the sample size would widen the interval. Increasing the confidence level to 99%
would actually widen the interval because you'd need a larger critical value. Switching from t to z might
narrow it slightly, but the effect is minimal and doesn't address the fundamental relationship between
sample size and precision.
Correct Answer: A
Q4. A pollster wants to estimate the proportion of voters who support a new tax. She wants a 95%
confidence interval with a margin of error of no more than 3 percentage points. Using the conservative
estimate p = 0.5, what is the minimum required sample size?
A. 385
B. 1,068 [CORRECT]
, C. 2,401
D. 10,000
Rationale: The best answer is B. The formula for sample size for a proportion is n = (z*)² × p(1-p) / (ME)².
Using z* = 1.96 for 95% confidence, p = 0.5, and ME = 0.03: n = (1.96)² × 0.25 / (0.03)² = 3.8416 × 0.25 /
0.0009 = 0..0009 ≈ 1,067.1, which rounds up to 1,068. Using p = 0.5 gives the most conservative
(largest) sample size requirement because p(1-p) is maximized at 0.25 when p = 0.5.
Correct Answer: B
Q5. A 99% confidence interval for a population mean is wider than a 95% confidence interval based on
the same sample. Which explanation is most accurate?
A. The higher confidence level requires a larger critical value, which increases the margin of error
[CORRECT]
B. The sample size increased.
C. The standard deviation decreased.
D. The sample mean changed.
Rationale: The best answer is A. To be more confident that our interval captures the true parameter, we
need to cast a wider net. The critical value for 99% confidence (z* ≈ 2.576 or t* with appropriate df) is
larger than for 95% confidence (z* ≈ 1.96), which directly increases the margin of error and thus the
interval width. The sample size, standard deviation, and sample mean haven't changed—only our
demand for confidence has increased, and that comes at the cost of precision.
Correct Answer: A
Q6. A researcher constructs a 90% confidence interval for a population mean and obtains (18.5, 24.3).
What is the margin of error?
A. 2.9 [CORRECT]
B. 5.8
C. 21.4
D. 42.8
Rationale: The best answer is A. The margin of error is half the width of the confidence interval. The
width is 24.3 - 18.5 = 5.8, so the margin of error is 5. = 2.9. Alternatively, the point estimate (sample
mean) is the midpoint: (18.5 + 24.3) / 2 = 21.4, and the margin of error is the distance from the midpoint
to either endpoint: 24.3 - 21.4 = 2.9. Understanding how margin of error relates to interval width is
fundamental for interpreting confidence intervals.
Correct Answer: A
100% Verified | Detailed Rationales – Pass
Guaranteed – A+ Graded
Content Area Overview: This actual examination reflects the fundamental statistical knowledge and
inferential reasoning required for success in Statistics Module 6. It is designed to evaluate the student's
ability to construct and interpret confidence intervals, perform hypothesis tests, and apply inferential
statistical methods to real-world data. Questions are structured to assess recall of key concepts,
application of statistical procedures, and analysis of data-driven scenarios. This authentic question bank
represents the real exams used in the course and serves as a comprehensive resource for students
demonstrating mastery of inferential statistics content.
SECTION 1: CONFIDENCE INTERVALS (9 Questions)
Q1. A researcher wants to estimate the mean height of adult males in a city. She takes a random sample
of 100 men and finds a sample mean of 70 inches with a standard deviation of 3 inches. Which
distribution should she use to construct a 95% confidence interval?
A. The z-distribution because the population standard deviation is known.
B. The t-distribution because the population standard deviation is unknown [CORRECT]
C. The chi-square distribution because she is estimating a mean.
D. The F-distribution because she has a large sample.
Rationale: The best answer is B. When the population standard deviation is unknown—which is almost
always the case in real research—we use the sample standard deviation as an estimate and construct
the confidence interval using the t-distribution. With a sample size of 100, the t-distribution is very close
to the normal distribution, but the principle remains: unknown population standard deviation means t-
distribution. The chi-square and F-distributions are used for variance and ANOVA applications, not for
mean estimation.
Correct Answer: B
,Q2. A 95% confidence interval for a population mean is calculated as (45.2, 52.8). Which interpretation
is most accurate?
A. There is a 95% probability that the population mean falls between 45.2 and 52.8.
B. We are 95% confident that the true population mean lies between 45.2 and 52.8 [CORRECT]
C. 95% of the sample data falls between 45.2 and 52.8.
D. The sample mean is exactly 49.0.
Rationale: The best answer is B. A confidence interval gives us a range of plausible values for the
population parameter, and the confidence level tells us how often this method would produce an
interval that captures the true parameter if we repeated the sampling process many times. The
population mean is a fixed value, not a random variable, so it doesn't have a probability of falling in the
interval. The sample mean is the midpoint (49.0), but the interval is about the unknown population
mean, not the sample data distribution.
Correct Answer: B
Q3. A researcher wants to decrease the width of a confidence interval without decreasing the
confidence level. Which strategy is most effective?
A. Increase the sample size [CORRECT]
B. Decrease the sample size
C. Increase the confidence level to 99%
D. Use the z-distribution instead of the t-distribution
Rationale: The best answer is A. The margin of error is inversely related to the square root of the sample
size, so increasing n reduces the width of the confidence interval while keeping the confidence level the
same. Decreasing the sample size would widen the interval. Increasing the confidence level to 99%
would actually widen the interval because you'd need a larger critical value. Switching from t to z might
narrow it slightly, but the effect is minimal and doesn't address the fundamental relationship between
sample size and precision.
Correct Answer: A
Q4. A pollster wants to estimate the proportion of voters who support a new tax. She wants a 95%
confidence interval with a margin of error of no more than 3 percentage points. Using the conservative
estimate p = 0.5, what is the minimum required sample size?
A. 385
B. 1,068 [CORRECT]
, C. 2,401
D. 10,000
Rationale: The best answer is B. The formula for sample size for a proportion is n = (z*)² × p(1-p) / (ME)².
Using z* = 1.96 for 95% confidence, p = 0.5, and ME = 0.03: n = (1.96)² × 0.25 / (0.03)² = 3.8416 × 0.25 /
0.0009 = 0..0009 ≈ 1,067.1, which rounds up to 1,068. Using p = 0.5 gives the most conservative
(largest) sample size requirement because p(1-p) is maximized at 0.25 when p = 0.5.
Correct Answer: B
Q5. A 99% confidence interval for a population mean is wider than a 95% confidence interval based on
the same sample. Which explanation is most accurate?
A. The higher confidence level requires a larger critical value, which increases the margin of error
[CORRECT]
B. The sample size increased.
C. The standard deviation decreased.
D. The sample mean changed.
Rationale: The best answer is A. To be more confident that our interval captures the true parameter, we
need to cast a wider net. The critical value for 99% confidence (z* ≈ 2.576 or t* with appropriate df) is
larger than for 95% confidence (z* ≈ 1.96), which directly increases the margin of error and thus the
interval width. The sample size, standard deviation, and sample mean haven't changed—only our
demand for confidence has increased, and that comes at the cost of precision.
Correct Answer: A
Q6. A researcher constructs a 90% confidence interval for a population mean and obtains (18.5, 24.3).
What is the margin of error?
A. 2.9 [CORRECT]
B. 5.8
C. 21.4
D. 42.8
Rationale: The best answer is A. The margin of error is half the width of the confidence interval. The
width is 24.3 - 18.5 = 5.8, so the margin of error is 5. = 2.9. Alternatively, the point estimate (sample
mean) is the midpoint: (18.5 + 24.3) / 2 = 21.4, and the margin of error is the distance from the midpoint
to either endpoint: 24.3 - 21.4 = 2.9. Understanding how margin of error relates to interval width is
fundamental for interpreting confidence intervals.
Correct Answer: A
