Source: ENGR 216 Comprehensive Exam Practice Bank | Texas A&M University
Tags: confidence interval, sample size, margin of error, z-critical, population mean, UAE, universal accounting equation, mass balance, steady state, ENGR 216
Confidence intervals give a range around a sample mean that is likely to contain the true population mean. The width depends on confidence level, standard deviation, and sample size. The Universal Accounting Equation (UAE) is a general bookkeeping tool: accumulation equals input minus output plus generation minus consumption. It applies to mass, people, money, or any conserved quantity.
Confidence interval
A range of values, calculated from sample data, that is expected to contain the true population parameter with a specified probability (the confidence level).
Margin of error (E)
The half-width of the confidence interval: E = z · σ / √N. A smaller margin requires a larger sample or a lower confidence level.
Z-critical value (z)*
The number of standard deviations from the mean that corresponds to a given confidence level. Common values: 90% → 1.645, 95% → 1.960, 96% → 2.054, 99% → 2.576.
Sample size determination
Rearranging the margin of error formula: N = (z · σ / E)². Always round up to the next whole number, because you cannot sample a fraction of an observation.
Universal Accounting Equation (UAE)
Accumulation = Input − Output + Generation − Consumption. This is the master balance equation for any system. For quantities that are neither created nor destroyed (like mass in non-reactive systems, or people), the generation and consumption terms drop out.
Steady state
A condition where accumulation is zero. The amount stored in the system does not change over time, so input equals output (plus or minus any generation/consumption).
For a population with known standard deviation σ and a sample of size N with mean x̄:
CI = x̄ ± z* · (σ / √N)
The quantity z* · (σ / √N) is the margin of error.
We want E = 0.2, confidence = 99% (so z* = 2.576), σ = 0.75.
N = (z · σ / E)² = (2.576 × 0.75 / 0.2)² = (1.932 / 0.2)² = (9.66)² = 93.3
Round up: N = 94 samples required.
Sample size N = 30, sample mean x̄ = 780 hours, population σ = 40 hours, confidence level = 96%.
The z-critical value for 96% confidence is 2.054.
Margin of error: E = 2.054 × (40 / √30) = 2.054 × 7.303 ≈ 15.0 hours
Confidence interval: 780 ± 15.0, which gives (765.0, 795.0) hours.
We are 96% confident that the true mean bulb life falls between 765 and 795 hours.
Accumulation = Input − Output + Generation − Consumption
For non-reactive mass balances, generation and consumption are both zero, so:
Accumulation = Input − Output
For steady-state problems, accumulation is also zero:
0 = Input − Output, meaning Input = Output
Apple juice at 10 kg/min (29% sugar) and orange juice at 18 kg/min (15% sugar) flow in. Mixture drains at 20 kg/min.
First, check the overall mass balance. Total inflow = 10 + 18 = 28 kg/min. Outflow = 20 kg/min. Since outflow < inflow, the tank accumulates 8 kg/min of total mixture. But the problem says to assume steady state for sugar mass specifically.
Sugar inflow rate:
From apple juice: 0.29 × 10 = 2.9 kg/min
From orange juice: 0.15 × 18 = 2.7 kg/min
Total sugar in: 5.6 kg/min
At steady state for sugar, sugar in = sugar out:
5.6 = C_out × 20
C_out = 5.6 / 20 = 0.28, or 28% sugar concentration in the outflow.
A town had 3,428 residents in 1980 and 2,783 in 1990. There were 178 births and 87 deaths. People are not "generated" or "consumed" in the chemical sense, but births act as generation and deaths as consumption.
Accumulation = Final − Initial = 2,783 − 3,428 = −645
Generation (births) = 178
Consumption (deaths) = 87
UAE: Accumulation = Input − Output + Generation − Consumption
−645 = (In − Out) + 178 − 87
−645 = (In − Out) + 91
In − Out = −645 − 91 = −736
Net migration = −736 people (a net outflow of 736 people over the decade).
Confidence interval: CI = x̄ ± z* · (σ / √N)
Required sample size: N = (z* · σ / E)² (round up)
Common z-critical values: 90% → 1.645, 95% → 1.960, 96% → 2.054, 99% → 2.576
UAE: Accumulation = Input − Output + Generation − Consumption
Steady state simplification: Input = Output (when generation and consumption are zero)
⚠️ Always round sample size up. N = 93.3 means you need 94, not 93.
⚠️ Know your z-values. The exam may give a z-table, but memorising the common ones (1.645, 1.960, 2.576) saves time.
⚠️ The 96% confidence level is less common and its z-value (2.054) is easy to mix up with 95% (1.960). Double-check.
⚠️ In UAE problems, clearly identify what quantity you are balancing (total mass, sugar mass, people, etc.) before writing the equation. Many mistakes come from mixing up the overall balance with a component balance.
⚠️ For the population problem, births count as generation (not input) and deaths count as consumption (not output). Input and output refer to migration.
Q: If you want to cut the margin of error in half without changing the confidence level, what happens to the required sample size?
A: It quadruples. Since N = (z · σ / E)², halving E means multiplying N by 4.
Q: A 95% confidence interval is (12.3, 15.7). What are the sample mean and the margin of error?
A: The sample mean is the midpoint: (12.3 + 15.7) / 2 = 14.0. The margin of error is the half-width: (15.7 − 12.3) / 2 = 1.7.
Q: In the juice mixing problem, why can the overall mass not be at steady state?
A: Because total inflow (28 kg/min) exceeds total outflow (20 kg/min), so the tank is accumulating 8 kg/min of mixture. Only the sugar component is assumed to be at steady state.
Q: In the population census problem, what does a negative net migration value mean?
A: More people left the town than moved in during the decade.
confidence interval, margin of error, z-critical value, z-star, sample size, population mean, standard deviation, normal distribution, z-table, universal accounting equation, UAE, mass balance, component balance, steady state, accumulation, generation, consumption, input, output, migration, ENGR 216, Texas A&M