Source: ENGR 216 Practice Challenges, Texas A&M
Tags: normal distribution, z-score, z-table, probability, bell curve, standard normal, cumulative probability, ENGR 216, experimental physics
The normal (Gaussian) distribution is the workhorse of engineering statistics. Converting a raw value to a z-score tells you how many standard deviations it sits from the mean, and a z-table converts that into a probability. The mid-term tests your ability to go from a word problem to a z-score to a looked-up probability, often expressed as a percentage.
Normal distribution (Gaussian distribution)
A symmetric, bell-shaped probability distribution defined entirely by its mean (μ) and standard deviation (σ). Most values cluster near the mean, with tails extending equally in both directions.
Standard normal distribution
A normal distribution with μ = 0 and σ = 1. All normal distributions can be converted to it via the z-score formula.
Z-score (standard score)
The number of standard deviations a value lies above or below the mean: z = (x − μ) / σ.
Z-table (standard normal table)
A lookup table giving the cumulative probability P(Z ≤ z), the area under the standard normal curve to the left of a given z-value.
Cumulative probability
The probability that a randomly selected value falls at or below a specified value. Equivalently, the area under the curve from the far left up to that value.
Complementary probability
P(Z > z) = 1 − P(Z ≤ z). Used when you need the probability of exceeding a value.
z = (x − μ) / σ
Where x is the value of interest, μ is the population mean, and σ is the population standard deviation.
Compute z.
Look up z in the table. The row gives the ones and tenths digit (e.g., 1.2), the column gives the hundredths digit (e.g., 0.05 for z = 1.25).
The table value is P(Z ≤ z), the area to the left.
"Less than" or "shorter than" or "fails before":
P(X < x) = table lookup at z. The table gives this directly.
"Greater than" or "longer than" or "survives beyond":
P(X > x) = 1 − P(Z ≤ z)
"Between two values":
P(a < X < b) = P(Z ≤ z_b) − P(Z ≤ z_a)
Multiply the decimal probability by 100. If the exam asks for a number between 0.0 and 100.0, give the percentage, not the raw decimal.
Z-score: z = (x − μ) / σ
Left-tail probability (directly from table): P(X < x) = P(Z ≤ z)
Right-tail probability: P(X > x) = 1 − P(Z ≤ z)
Between two values: P(a < X < b) = P(Z ≤ z_b) − P(Z ≤ z_a)
Repair time: μ = 120 min, σ = 4 min. Find P(X ≥ 125).
z = (125 − 120) / 4 = 5 / 4 = 1.25
From z-table: P(Z ≤ 1.25) = 0.8944
P(X ≥ 125) = 1 − 0.8944 = 0.1056
As a percentage: 10.56%. As a decimal rounded to two places: 0.11 (or 10.56 if the answer is expected as a percentage between 0 and 100).
Check your exam prompt for which format is expected.
Battery life: μ = 600 days, σ = 60 days.
Survive beyond 690 days:
z = (690 − 600) / 60 = 90 / 60 = 1.50
P(Z ≤ 1.50) = 0.9332
P(X > 690) = 1 − 0.9332 = 0.0668
Fraction surviving beyond 690 days: 0.07 (rounded to two decimal places)
Fail before 555 days:
z = (555 − 600) / 60 = −45 / 60 = −0.75
P(Z ≤ −0.75) = 0.2266
Fraction failing before 555 days: 0.23 (rounded to two decimal places)
Heights: μ = 160 cm, σ = 5 cm. Find P(X < 155) as a percentage.
z = (155 − 160) / 5 = −5 / 5 = −1.00
P(Z ≤ −1.00) = 0.1587
As a percentage: 15.87%
Salaries: μ = $4,000, σ = $500. Find P(X > $4,500) as a percentage.
z = (4500 − 4000) / 500 = 500 / 500 = 1.00
P(Z ≤ 1.00) = 0.8413
P(X > 4500) = 1 − 0.8413 = 0.1587
As a percentage: 15.87%
⚠️ Always check whether the question asks for "less than" or "greater than." This determines whether you use the table value directly or subtract from 1.
⚠️ Check the answer format. Some questions want a decimal between 0 and 1. Others want a percentage between 0 and 100. Read the instructions carefully.
⚠️ Negative z-scores are normal. They mean the value is below the mean. The z-table for negative values gives small cumulative probabilities.
⚠️ For "or longer" / "or more" type phrasing, use the complementary probability: 1 minus the table value.
⚠️ Memorise a few key z-table values for sanity checks: z = 1.00 → 0.8413, z = 1.96 → 0.9750, z = 2.00 → 0.9772.
Q: What does a z-score of 2.0 mean?
A: The value is exactly 2 standard deviations above the mean.
Q: If z = −1.50, what is the cumulative probability P(Z ≤ −1.50)?
A: From the z-table, P(Z ≤ −1.50) = 0.0668.
Q: A process has μ = 50 and σ = 10. What percentage of output exceeds 65?
A: z = (65 − 50)/10 = 1.50. P(Z ≤ 1.50) = 0.9332. P(X > 65) = 1 − 0.9332 = 0.0668 = 6.68%.
Q: Two values have the same absolute z-score but opposite signs. How do their probabilities relate?
A: P(Z ≤ −z) = 1 − P(Z ≤ z). The standard normal distribution is symmetric about zero.
normal distribution, Gaussian distribution, bell curve, z-score, z-table, standard score, standard normal distribution, cumulative probability, left tail, right tail, complementary probability, area under the curve, probability percentage, ENGR 216, experimental physics lab, Texas A&M