IEE 380 Probability & Statistics for Engineering Final Exam
Actual Exam 2026/2027 – Complete Cheatsheet with Detailed
Rationales | 100% Verified | Pass Guaranteed – A+ Graded
Section A: Descriptive Statistics & Data Visualization
Q1: An engineering team measures the tensile strength of 50 steel samples. The data is
right-skewed with a few extremely high values due to heat treatment variations. Which
measure of central tendency best represents the typical tensile strength for this
dataset?
A. Mean, because it incorporates every data point in the calculation
B. Median, because it is resistant to the influence of extreme values [CORRECT]
C. Mode, because it identifies the most frequently occurring strength value
D. Geometric mean, because it is designed for all skewed distributions
Correct Answer: B
Rationale: The median is robust to outliers and extreme values, making it the preferred
measure of central tendency for skewed distributions. The mean (A) is pulled toward
the tail by extreme values, misrepresenting typical performance. The mode (C) may not
exist or be meaningful for continuous data. The geometric mean (D) is appropriate for
multiplicative processes and log-normal data, not general right-skewed data.
Q2: A quality control engineer calculates the sample variance of 20 measurements of
bolt diameters. If the sum of squared deviations from the sample mean is 190, what is
the sample variance?
,A. 9.5
B. 10.0 [CORRECT]
C. 10.5
D. 19.0
Correct Answer: B
Rationale: Sample variance s² = Σ(xi − x̄)² / (n − 1) = = 10.0. Option A incorrectly
divides by n (190/20). Option C uses an incorrect denominator. Option D reports the
sum of squared deviations rather than the variance.
Q3: A manufacturing process produces components with diameters normally
distributed with mean μ = 100 mm and standard deviation σ = 5 mm. According to the
empirical rule, approximately what percentage of components will have diameters
between 90 mm and 110 mm?
A. 68%
B. 95% [CORRECT]
C. 99.7%
D. 75%
Correct Answer: B
Rationale: The interval 90 to 110 mm represents μ ± 2σ (100 ± 10). The empirical rule
states that approximately 95% of data in a normal distribution falls within two standard
deviations of the mean. Option A corresponds to μ ± 1σ (95 to 105 mm). Option C
corresponds to μ ± 3σ (85 to 115 mm).
, Q4: A dataset of part weights has a mean of 50 grams and a standard deviation of 4
grams. Using Chebyshev's theorem, what is the minimum percentage of data that must
fall within 2.5 standard deviations of the mean?
A. 75%
B. 84% [CORRECT]
C. 89%
D. 95%
Correct Answer: B
Rationale: Chebyshev's theorem states that at least 1 − 1/k² of data falls within k
standard deviations of the mean. For k = 2.5: 1 − 1/(2.5)² = 1 − 0.16 = 0.84 or 84%.
Option A uses k = 2 (1 − 1/4 = 75%). Option C uses k = 3 (≈89%). Option D incorrectly
applies the empirical rule.
Q5: A boxplot of assembly times shows Q₁ = 12 minutes, Q₃ = 18 minutes, and median =
15 minutes. Which statement about this distribution is correct?
A. The interquartile range is 3 minutes
B. The distribution is positively skewed
C. Any value above 27 minutes is considered an outlier [CORRECT]
D. The range is exactly 6 minutes
Correct Answer: C
Actual Exam 2026/2027 – Complete Cheatsheet with Detailed
Rationales | 100% Verified | Pass Guaranteed – A+ Graded
Section A: Descriptive Statistics & Data Visualization
Q1: An engineering team measures the tensile strength of 50 steel samples. The data is
right-skewed with a few extremely high values due to heat treatment variations. Which
measure of central tendency best represents the typical tensile strength for this
dataset?
A. Mean, because it incorporates every data point in the calculation
B. Median, because it is resistant to the influence of extreme values [CORRECT]
C. Mode, because it identifies the most frequently occurring strength value
D. Geometric mean, because it is designed for all skewed distributions
Correct Answer: B
Rationale: The median is robust to outliers and extreme values, making it the preferred
measure of central tendency for skewed distributions. The mean (A) is pulled toward
the tail by extreme values, misrepresenting typical performance. The mode (C) may not
exist or be meaningful for continuous data. The geometric mean (D) is appropriate for
multiplicative processes and log-normal data, not general right-skewed data.
Q2: A quality control engineer calculates the sample variance of 20 measurements of
bolt diameters. If the sum of squared deviations from the sample mean is 190, what is
the sample variance?
,A. 9.5
B. 10.0 [CORRECT]
C. 10.5
D. 19.0
Correct Answer: B
Rationale: Sample variance s² = Σ(xi − x̄)² / (n − 1) = = 10.0. Option A incorrectly
divides by n (190/20). Option C uses an incorrect denominator. Option D reports the
sum of squared deviations rather than the variance.
Q3: A manufacturing process produces components with diameters normally
distributed with mean μ = 100 mm and standard deviation σ = 5 mm. According to the
empirical rule, approximately what percentage of components will have diameters
between 90 mm and 110 mm?
A. 68%
B. 95% [CORRECT]
C. 99.7%
D. 75%
Correct Answer: B
Rationale: The interval 90 to 110 mm represents μ ± 2σ (100 ± 10). The empirical rule
states that approximately 95% of data in a normal distribution falls within two standard
deviations of the mean. Option A corresponds to μ ± 1σ (95 to 105 mm). Option C
corresponds to μ ± 3σ (85 to 115 mm).
, Q4: A dataset of part weights has a mean of 50 grams and a standard deviation of 4
grams. Using Chebyshev's theorem, what is the minimum percentage of data that must
fall within 2.5 standard deviations of the mean?
A. 75%
B. 84% [CORRECT]
C. 89%
D. 95%
Correct Answer: B
Rationale: Chebyshev's theorem states that at least 1 − 1/k² of data falls within k
standard deviations of the mean. For k = 2.5: 1 − 1/(2.5)² = 1 − 0.16 = 0.84 or 84%.
Option A uses k = 2 (1 − 1/4 = 75%). Option C uses k = 3 (≈89%). Option D incorrectly
applies the empirical rule.
Q5: A boxplot of assembly times shows Q₁ = 12 minutes, Q₃ = 18 minutes, and median =
15 minutes. Which statement about this distribution is correct?
A. The interquartile range is 3 minutes
B. The distribution is positively skewed
C. Any value above 27 minutes is considered an outlier [CORRECT]
D. The range is exactly 6 minutes
Correct Answer: C
