Source: ENGR 216 Practice Challenges, Texas A&M
Tags: confidence interval, confidence level, z-critical value, margin of error, upper bound, lower bound, normal distribution, ENGR 216, experimental physics
A confidence interval gives a range around a sample mean where you expect the true population mean to lie, at a stated confidence level. The formula combines the sample mean, the z-critical value for the chosen confidence level, the known standard deviation, and the sample size. This is a high-frequency mid-term topic.
Confidence interval (CI)
A range of values, computed from sample data, that is expected to contain the true population mean with a specified probability (confidence level).
Confidence level
The probability that the interval contains the true mean. Common levels: 90%, 95%, 96%, 99%.
Z-critical value (z_α/2)
The z-score that cuts off the tail area corresponding to the confidence level. For a 95% CI, z = 1.96. For 99%, z = 2.576.
Margin of error (E)
The half-width of the confidence interval: E = z_α/2 · (σ / √n).
Known standard deviation (σ)
These problems give you the population standard deviation directly. When σ is known, use the z-interval formula. (When σ is unknown, you would use a t-interval, but ENGR 216 mid-term problems provide σ.)
CI = x̄ ± z_α/2 · (σ / √n)
x̄ = sample mean
z_α/2 = critical z-value for the desired confidence level
σ = population standard deviation (given)
n = sample size
Lower bound = x̄ − z_α/2 · (σ / √n)
Upper bound = x̄ + z_α/2 · (σ / √n)
90% confidence: z = 1.645
95% confidence: z = 1.960
96% confidence: z = 2.054
99% confidence: z = 2.576
96% confidence leaves 4% in the tails, split equally: 2% in each tail. The cumulative area to the left of the upper critical value is 0.98. Looking up 0.98 in the z-table body gives z ≈ 2.054.
Some problems ask for the width or range of the interval:
Range = Upper bound − Lower bound = 2 · z_α/2 · (σ / √n)
Confidence interval: CI = x̄ ± z_α/2 · (σ / √n)
Margin of error: E = z_α/2 · (σ / √n)
Interval range (width): Range = 2E = 2 · z_α/2 · (σ / √n)
Light bulbs: σ = 25 hours, n = 20, x̄ = 1012 hours, 95% confidence.
z_α/2 = 1.960
E = 1.960 × (25 / √20) = 1.960 × (25 / 4.4721) = 1.960 × 5.5902 = 10.96
Lower bound = 1012 − 10.96 = 1001.04
Upper bound = 1012 + 10.96 = 1022.96
Piston rings: σ = 0.001 mm, n = 15, x̄ = 74.036 mm, 99% confidence. Find the range between boundaries.
z_α/2 = 2.576
E = 2.576 × (0.001 / √15) = 2.576 × (0.001 / 3.8730) = 2.576 × 0.0002582 = 0.000665
Lower bound = 74.036 − 0.000665 = 74.035335
Upper bound = 74.036 + 0.000665 = 74.036665
Range = 2 × 0.000665 = 0.0013 (rounded to four decimal places)
Cereal boxes: σ = 0.8 oz, n = 49, x̄ = 24.1 oz, 96% confidence.
z_α/2 = 2.054 (from looking up 0.98 in the z-table, since 96% CI leaves 2% in each tail)
E = 2.054 × (0.8 / √49) = 2.054 × (0.8 / 7) = 2.054 × 0.11429 = 0.23
Lower bound = 24.1 − 0.23 = 23.87
Upper bound = 24.1 + 0.23 = 24.33
⚠️ Use the z-critical value, not the z-score from a data point. These are different things. The z-critical value comes from the confidence level, not from the data.
⚠️ The denominator is √n, not n. Forgetting the square root is one of the most common calculation errors.
⚠️ For 96% confidence, the z-critical value is 2.054 (or approximately 2.05). This is less commonly memorised than 1.96 or 2.576, so know how to derive it from the z-table.
⚠️ Read whether the problem asks for the bounds (two numbers) or the range between them (one number, which is twice the margin of error).
⚠️ These problems give σ as known. If the problem said "estimated standard deviation" or "sample standard deviation," you would use a t-distribution instead, but that distinction is less likely on the ENGR 216 mid-term.
Q: What happens to the confidence interval width if you increase the sample size?
A: The interval gets narrower. The margin of error is proportional to 1/√n, so larger samples reduce uncertainty.
Q: What happens to the confidence interval width if you increase the confidence level?
A: The interval gets wider. Higher confidence requires a larger z-critical value, which increases the margin of error.
Q: For a 95% confidence interval, what percentage of the area is in each tail?
A: 2.5% in each tail. The central 95% is bounded by z = ±1.96.
Q: A sample of 36 has x̄ = 50 and σ = 6. What is the 95% CI?
A: E = 1.96 × (6/√36) = 1.96 × 1 = 1.96. CI: (48.04, 51.96).
confidence interval, CI, confidence level, z-critical value, z-alpha-over-2, margin of error, upper bound, lower bound, interval estimate, point estimate, sample mean, population standard deviation, known sigma, normal distribution, ENGR 216, experimental physics lab, Texas A&M