Portage Learning Statistics Module 4 The
Z-Test (2026) Questions & Complete
Solutions with Detailed Rationales | New
Version
This "Portage Learning Statistics Module 4 The Z-Test: 20 Questions and Answers" is a
comprehensive study resource used to prepare for the course exam. It covers fundamental
normal curve probability, hypothesis testing, and calculating sample \(z\)-scores.
Exam Content & Key Concepts
The 20 questions primarily evaluate your understanding of the following concepts:
• The Normal Distribution: Questions often cover the 68-95-99.7 rule and how the
mean, median, and mode interact in a bell-shaped curve.
• Understanding Z-Scores: How to transform raw scores into standard \(z\)-scores.
The \(z\)-test formula for a sample mean is:
\(z = \frac{\overline{x} - \mu}{\frac{\sigma}{\sqrt{n}}}\)
• Hypothesis Testing Steps: The 4 steps of hypothesis testing are commonly tested:
1. State null and alternative hypotheses (\(H_{0}\) and \(H_{a}\)).
2. Determine critical values based on alpha levels and tails.
3. Compute the sample \(z\)-score.
4. Compare with the critical value to make a decision (Reject or Fail to Reject
\(H_{0}\)).
Study Resource
To test your knowledge and practice calculating probabilities on the
standard normal table, you can utilize this dedicated & curated,
downloadable pdf set for a small fee
Q1. Explain why the area under a normal curve equals 100% and what that area
represents. [Short Answer]
1
, Answer: The area under the curve equals 100% because it represents the total relative
frequency (or total probability) of all possible outcomes in the distribution.
Explanation: A probability density curve assigns portions of the total probability to ranges of
values; integrating over all values must give the total probability. For relative frequencies this
total is 100%, so the entire area under the normal curve sums to 100%.
Q2. Given a normal distribution with mean = 73 and σ = 4, which interval
contains approximately 95% of the values? [Multiple Choice]
A) 95% between 65 and 81
B) 68% between 69 and 77
C) 99.7% between 61 and 85
D) 95% between 69 and 77
Answer: 95% between 65 and 81
Explanation: With mean 73 and σ = 4, two standard deviations from the mean is 73 ± 2×4 = 73 ±
8, giving the interval 65 to 81, which contains about 95% of values by the empirical rule.
Distractors: "68% between 69-77" is true but refers to 1σ (not the 95% interval asked here).
"99.7% between 61-85" is the 3σ interval, not the 95% one. "95% between 69-77" mixes the
wrong percentage with the 1σ interval.
Q3. Explain why, in a normal curve, an equal number of observations fall above
and below the mean. [Short Answer]
Answer: Because a normal distribution is symmetric about the mean, exactly half the
total area (and thus half the observations) lies above the mean and half below it.
Explanation: Symmetry means the left and right sides are mirror images centered on the mean.
That mirror property makes areas on either side equal, so equal numbers of observations fall
above and below the mean.
Q4. Which set of percentages correctly expresses the empirical 68-95-99.7 rule
for a normal distribution? [Multiple Choice]
A) Approximately 68% within 1σ, 95% within 2σ, 99.7% within 3σ
B) About 50% within 1σ, 75% within 2σ, 90% within 3σ
C) About 34% within 1σ, 47.5% within 2σ, 49.85% within 3σ
D) 100% within 1σ, 100% within 2σ, 100% within 3σ
Answer: Approximately 68% within 1σ, 95% within 2σ, 99.7% within 3σ
2
Z-Test (2026) Questions & Complete
Solutions with Detailed Rationales | New
Version
This "Portage Learning Statistics Module 4 The Z-Test: 20 Questions and Answers" is a
comprehensive study resource used to prepare for the course exam. It covers fundamental
normal curve probability, hypothesis testing, and calculating sample \(z\)-scores.
Exam Content & Key Concepts
The 20 questions primarily evaluate your understanding of the following concepts:
• The Normal Distribution: Questions often cover the 68-95-99.7 rule and how the
mean, median, and mode interact in a bell-shaped curve.
• Understanding Z-Scores: How to transform raw scores into standard \(z\)-scores.
The \(z\)-test formula for a sample mean is:
\(z = \frac{\overline{x} - \mu}{\frac{\sigma}{\sqrt{n}}}\)
• Hypothesis Testing Steps: The 4 steps of hypothesis testing are commonly tested:
1. State null and alternative hypotheses (\(H_{0}\) and \(H_{a}\)).
2. Determine critical values based on alpha levels and tails.
3. Compute the sample \(z\)-score.
4. Compare with the critical value to make a decision (Reject or Fail to Reject
\(H_{0}\)).
Study Resource
To test your knowledge and practice calculating probabilities on the
standard normal table, you can utilize this dedicated & curated,
downloadable pdf set for a small fee
Q1. Explain why the area under a normal curve equals 100% and what that area
represents. [Short Answer]
1
, Answer: The area under the curve equals 100% because it represents the total relative
frequency (or total probability) of all possible outcomes in the distribution.
Explanation: A probability density curve assigns portions of the total probability to ranges of
values; integrating over all values must give the total probability. For relative frequencies this
total is 100%, so the entire area under the normal curve sums to 100%.
Q2. Given a normal distribution with mean = 73 and σ = 4, which interval
contains approximately 95% of the values? [Multiple Choice]
A) 95% between 65 and 81
B) 68% between 69 and 77
C) 99.7% between 61 and 85
D) 95% between 69 and 77
Answer: 95% between 65 and 81
Explanation: With mean 73 and σ = 4, two standard deviations from the mean is 73 ± 2×4 = 73 ±
8, giving the interval 65 to 81, which contains about 95% of values by the empirical rule.
Distractors: "68% between 69-77" is true but refers to 1σ (not the 95% interval asked here).
"99.7% between 61-85" is the 3σ interval, not the 95% one. "95% between 69-77" mixes the
wrong percentage with the 1σ interval.
Q3. Explain why, in a normal curve, an equal number of observations fall above
and below the mean. [Short Answer]
Answer: Because a normal distribution is symmetric about the mean, exactly half the
total area (and thus half the observations) lies above the mean and half below it.
Explanation: Symmetry means the left and right sides are mirror images centered on the mean.
That mirror property makes areas on either side equal, so equal numbers of observations fall
above and below the mean.
Q4. Which set of percentages correctly expresses the empirical 68-95-99.7 rule
for a normal distribution? [Multiple Choice]
A) Approximately 68% within 1σ, 95% within 2σ, 99.7% within 3σ
B) About 50% within 1σ, 75% within 2σ, 90% within 3σ
C) About 34% within 1σ, 47.5% within 2σ, 49.85% within 3σ
D) 100% within 1σ, 100% within 2σ, 100% within 3σ
Answer: Approximately 68% within 1σ, 95% within 2σ, 99.7% within 3σ
2
