Study guide
5.3 Engineering Data Analysis II — Confidence Intervals, Hypothesis Testing & Regression
PRC Registered Electrical Engineer REE
Track your progress as you read
Finished sections get a check mark.
1 Chapter 1: Overview, RA 7920 & Electrical Engineering Ethics
1.1 PRC Board of Electrical Engineering & REE Examination Overview 1.2 Republic Act No. 7920 (New Electrical Engineering Law) & License Classes 1.3 Code of Ethics & Professional Practice for Electrical Engineers
2 Chapter 2: Math — Algebra, Trig & Geometry
2.1 Algebraic Equations, Polynomials & Complex Numbers 2.2 Trigonometry — Identities, Triangles & Spherical Trig 2.3 Analytic & Solid Geometry
3 Chapter 3: Engineering Mathematics — Calculus & Differential Equations
3.1 Differential Calculus, Limits, Derivatives & Applications 3.2 Integral Calculus, Integration Techniques & Area/Volume Applications 3.3 Ordinary Differential Equations & Initial-Value Problems
4 Chapter 4: Engineering Mathematics — Advanced Math & Transform Methods
4.1 Laplace Transforms, Inverse Transforms & Circuit Applications 4.2 Fourier Series, Fourier Transforms & Frequency-Domain Analysis 4.3 Vector Analysis, Linear Algebra, Matrices & Complex Analysis
5 Chapter 5: Engineering Mathematics — Probability, Statistics & Engineering Data Analysis
5.1 Probability Theory, Counting Techniques & Bayes' Theorem 5.2 Engineering Data Analysis I — Descriptive Statistics & Probability Distributions 5.3 Engineering Data Analysis II — Confidence Intervals, Hypothesis Testing & Regression
6 Chapter 6: Engineering Mathematics — Numerical Methods & Analysis
6.1 Numerical Root-Finding & Solution of Linear Systems 6.2 Interpolation, Numerical Integration & Numerical Solution of ODEs
7 Chapter 7: ESAS — Physics, Chemistry & Computer Fundamentals
7.1 Physics for Engineers 7.2 General Chemistry & Materials Science 7.3 Computer Fundamentals, Programming & Data Logic 7.4 Logic Circuits, Switching Theory & Microprocessor Systems
8 Chapter 8: ESAS — Mechanics & Thermal-Fluid Engineering
8.1 Engineering Mechanics — Statics & Dynamics 8.2 Strength of Materials & Mechanics of Deformable Bodies 8.3 Thermal-Fluid Engineering — Thermodynamics & Fluid Mechanics
9 Chapter 9: ESAS — Engineering Economics, Management & Contracts
9.1 Engineering Economics & Depreciation 9.2 Engineering Management & Ethics 9.3 Philippine Electrical Engineering Law, Code & Contracts 9.4 Technopreneurship 101 & Management of Engineering Projects
10 Chapter 10: PEC 1 — Safety, Sizing & Protection
10.1 PEC Fundamentals, Safety Standards & Grounding/Bonding 10.2 Conductor Sizing, Ampacity Adjustment & Voltage Drop 10.3 Overcurrent Protection, Branch Circuit & Feeder Sizing 10.4 Basic Occupational Safety & Health (BOSH) and Electrical Safety Standards
11 Chapter 11: PEC 1 & 2 — Equipment & Utility Systems
11.1 PEC Equipment Installation & Wiring Methods 11.2 Utility Systems, Transformers & Switchgear 11.3 Special Occupancies & Hazardous Locations
12 Chapter 12: Electrical Professional — DC & AC Circuit Fundamentals
12.1 DC Circuit Analysis & Network Theorems 12.2 Single-Phase AC Circuits & Phasor Analysis 12.3 Balanced & Unbalanced Three-Phase AC Circuits
13 Chapter 13: Electrical Professional — EM Fields & Transients
13.1 Electrostatics, Magnetostatics & Maxwell's Equations 13.2 Electromagnetic Waves & Transmission Line Fields 13.3 First-Order & Second-Order Circuit Transients
14 Chapter 14: Electrical Professional — Electronics, Industrial Electronics & Communications
14.1 Electronics 1 & 2 — Semiconductor Devices & Analog Circuits 14.2 Electrical Apparatus & Devices — Industrial Electronics & Power Converters 14.3 Fundamentals of Electronic Communications
15 Chapter 15: Electrical Professional — Transformers & DC Machines
15.1 Single-Phase & Three-Phase Transformers 15.2 Transformer Connections & Special Transformers 15.3 DC Generators & DC Motors
16 Chapter 16: Electrical Professional — AC Synchronous & Induction Machinery
16.1 Synchronous Generators (Alternators) 16.2 Synchronous Motors & Special AC Machines 16.3 Three-Phase & Single-Phase Induction Motors
17 Chapter 17: Electrical Professional — Power Plants & Energy Conversion
17.1 Hydroelectric, Thermal & Combined Cycle Power Plants 17.2 Renewable Energy Systems & Geothermal Power Plants 17.3 Power Plant Economics, Load Curves & Energy Conversion
18 Chapter 18: Electrical Professional — Transmission & Distribution
18.1 Transmission Line Parameters & Performance Models 18.2 Mechanical Overhead Line Design, Underground Cables & Insulators 18.3 Distribution Systems, Substation Layouts & Power Flow Control
19 Chapter 19: Electrical Professional — Power System Analysis
19.1 The Per-Unit System, Base Conversion & Impedance Diagrams 19.2 Symmetrical Components & Unbalanced Fault Analysis 19.3 Load Flow, Economic Dispatch & Power System Stability
20 Chapter 20: Electrical Professional — Protection, Control, Instrumentation & Illumination
20.1 Power System Protection & Protective Relaying 20.2 Industrial Motor Control, Logic Circuits & PLC Automation 20.3 Electrical Instrumentation, Transducers & Measurement Systems 20.4 Illumination Engineering, Lighting Design & Photometrics 20.5 Feedback Control Systems & Research Methods
5.3 Engineering Data Analysis II — Confidence Intervals, Hypothesis Testing & Regression
Key Takeaways Explain This Topic AI
Use the z distribution when σ is known or n > 30; use the t distribution with (n − 1) degrees of freedom when σ is unknown and the sample is small.
A 95% confidence interval for a mean is x̄ ± 1.96·(σ/√n) when σ is known; the margin of error shrinks with √n, so quadrupling the sample halves the error.
Reject the null hypothesis when the p-value is less than the significance level α, or equivalently when the test statistic falls in the rejection region.
Type I error (α) is rejecting a true null hypothesis; Type II error (β) is failing to reject a false one. Test power equals 1 − β.
The least-squares slope is b = [nΣxy − ΣxΣy]/[nΣx² − (Σx)²], and the coefficient of determination r² gives the fraction of variation explained by the fitted line.
Last updated: August 2026
5.3 Engineering Data Analysis II — Confidence Intervals, Hypothesis Testing & Regression
The second half of the PRC Enhanced TOS Engineering Data Analysis topic moves from describing a data set to drawing defensible conclusions from it. These are the tools behind load-forecasting studies, meter-accuracy audits, insulation-test acceptance decisions and reliability benchmarking under the Philippine Distribution Code.
1. Sampling Distributions & the Central Limit Theorem
If repeated samples of size n n n are drawn from a population with mean μ \mu μ and standard deviation σ \sigma σ , the sample means themselves form a distribution with μ x ˉ = μ σ x ˉ = σ n \mu_{\bar{x}} = \mu \qquad\qquad \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} μ x ˉ = μ σ x ˉ = n σ
The quantity σ / n \sigma/\sqrt{n} σ / n is the standard error of the mean . The Central Limit Theorem states that for sufficiently large n n n (conventionally n ≥ 30 n \ge 30 n ≥ 30 ) the distribution of x ˉ \bar{x} x ˉ is approximately normal regardless of the shape of the parent population . This is why normal-based methods work on decidedly non-normal quantities such as outage durations.
Note the n \sqrt{n} n : to halve the standard error you must quadruple the sample size.
2. Confidence Intervals
When σ \sigma σ is known (or n > 30 n > 30 n > 30 ) x ˉ ± z α / 2 σ n \bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}} x ˉ ± z α /2 n σ
When σ \sigma σ is unknown and n n n is small x ˉ ± t α / 2 , ν s n , ν = n − 1 degrees of freedom \bar{x} \pm t_{\alpha/2,\;\nu} \frac{s}{\sqrt{n}}, \qquad \nu = n - 1 \text{ degrees of freedom} x ˉ ± t α /2 , ν n s , ν = n − 1 degrees of freedom
- Confidence level z α / 2 z_{\alpha/2} z α /2
- 90 % 90\% 90% 1.645 1.645 1.645
- 95 % 95\% 95% 1.960 1.960 1.960
- 98 % 98\% 98% 2.326 2.326 2.326
- 99 % 99\% 99% 2.576 2.576 2.576
The t t t distribution is wider than the normal, converging to it as ν → ∞ \nu \to \infty ν → ∞ . Choosing z z z when the problem states "sample standard deviation" and n n n is small is a graded error.
3. Hypothesis Testing
A hypothesis test compares a null hypothesis H 0 H_0 H 0 (the status quo — "the meters meet the accuracy class") against an alternative H 1 H_1 H 1 .
Test statistic for a mean z = x ˉ − μ 0 σ / n t = x ˉ − μ 0 s / n z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} \qquad\qquad t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} z = σ / n x ˉ − μ 0 t = s / n x ˉ − μ 0
Decision rule
Reject H 0 H_0 H 0 when the test statistic falls in the rejection region, equivalently when p < α p < \alpha p < α .
The two error types
H 0 H_0 H 0 actually true H 0 H_0 H 0 actually false
Reject H 0 H_0 H 0 Type I error , probability α \alpha α Correct decision, probability 1 − β 1-\beta 1 − β ( power )
Fail to reject H 0 H_0 H 0 Correct decision, probability 1 − α 1-\alpha 1 − α Type II error , probability β \beta β
In electrical acceptance testing, a Type I error rejects a good batch (producer's risk); a Type II error accepts a defective batch (consumer's risk). Reducing α \alpha α without increasing n n n necessarily increases β \beta β .
One-tailed vs. two-tailed
"Is the mean different from 220 V?" is two-tailed. "Is the loss greater than 150 W?" is one-tailed, and the whole of α \alpha α sits in one tail, so the critical value is smaller (1.645 rather than 1.96 at 95%).
4. Linear Regression & Correlation
The least-squares line y = a + b x y = a + bx y = a + b x minimises ∑ ( y i − y ^ i ) 2 \sum (y_i - \hat{y}_i)^2 ∑ ( y i − y ^ i ) 2 : b = n ∑ x y − ∑ x ∑ y n ∑ x 2 − ( ∑ x ) 2 a = y ˉ − b x ˉ b = \frac{n\sum xy - \sum x \sum y}{n\sum x^{2} - \left(\sum x\right)^{2}} \qquad\qquad a = \bar{y} - b\bar{x} b = n ∑ x 2 − ( ∑ x ) 2 n ∑ x y − ∑ x ∑ y a = y ˉ − b x ˉ
The Pearson correlation coefficient measures linear association: r = n ∑ x y − ∑ x ∑ y [ n ∑ x 2 − ( ∑ x ) 2 ] [ n ∑ y 2 − ( ∑ y ) 2 ] , − 1 ≤ r ≤ 1 r = \frac{n\sum xy - \sum x \sum y}{\sqrt{\left[n\sum x^{2} - (\sum x)^{2}\right]\left[n\sum y^{2} - (\sum y)^{2}\right]}}, \qquad -1 \le r \le 1 r = [ n ∑ x 2 − ( ∑ x ) 2 ] [ n ∑ y 2 − ( ∑ y ) 2 ] n ∑ x y − ∑ x ∑ y , − 1 ≤ r ≤ 1
The coefficient of determination r 2 r^{2} r 2 is the proportion of the variation in y y y explained by the regression on x x x . An r = 0.9 r = 0.9 r = 0.9 means r 2 = 0.81 r^{2} = 0.81 r 2 = 0.81 , so 81% of the variation is explained — not 90%.
Correlation is not causation, and r r r measures only linear association: a perfect parabola can have r ≈ 0 r \approx 0 r ≈ 0 .
5. Designed Experiments & Statistical Process Control
- Term Meaning in electrical practice
- Factor / level Controlled input (e.g. conductor temperature) and its settings
- Response Measured output (e.g. measured ampacity)
- Replication Repeating runs to estimate pure experimental error
- Randomisation Ordering runs randomly to neutralise lurking variables
- Blocking Grouping similar units to remove a known nuisance source
ANOVA Partitioning total variation into between-treatment and within-treatment components; the F F F ratio tests treatment significance
X ˉ \bar{X} X ˉ –R control chart Plots subgroup means and ranges against ± 3 σ \pm 3\sigma ± 3 σ control limits to detect process drift
Control limits are not specification limits: control limits describe what the process actually does, while specification limits state what the customer requires.
- Solved Board Exam Examples
- Example 1: Confidence Interval on Feeder Voltage
Thirty-six voltage readings on a distribution feeder average 228.4 V with a known population standard deviation of 3.6 V. Construct the 95% confidence interval for the true mean voltage.
Solution. With σ \sigma σ known and n = 36 n = 36 n = 36 , use z 0.025 = 1.96 z_{0.025} = 1.96 z 0.025 = 1.96 : ME = 1.96 × 3.6 36 = 1.96 × 0.60 = 1.176 V \text{ME} = 1.96 \times \frac{3.6}{\sqrt{36}} = 1.96 \times 0.60 = 1.176 \text{ V} ME = 1.96 × 36 3.6 = 1.96 × 0.60 = 1.176 V 228.4 ± 1.176 ⟹ 227.22 V ≤ μ ≤ 229.58 V 228.4 \pm 1.176 \implies \boxed{227.22 \text{ V} \le \mu \le 229.58 \text{ V}} 228.4 ± 1.176 ⟹ 227.22 V ≤ μ ≤ 229.58 V
The whole interval lies inside the ± 10 % \pm 10\% ± 10% service-voltage band, so the feeder is compliant.
Example 2: Hypothesis Test on Transformer Efficiency
A manufacturer claims a mean efficiency of at least 98.5%. Sixteen units average 98.2% with a sample standard deviation of 0.4%. Test the claim at α = 0.05 \alpha = 0.05 α = 0.05 .
Solution. H 0 : μ ≥ 98.5 % H_0: \mu \ge 98.5\% H 0 : μ ≥ 98.5% against H 1 : μ < 98.5 % H_1: \mu < 98.5\% H 1 : μ < 98.5% (one-tailed, left). With σ \sigma σ unknown and n = 16 n = 16 n = 16 , use t t t with ν = 15 \nu = 15 ν = 15 : t = 98.2 − 98.5 0.4 / 16 = − 0.3 0.1 = − 3.00 t = \frac{98.2 - 98.5}{0.4/\sqrt{16}} = \frac{-0.3}{0.1} = -3.00 t = 0.4/ 16 98.2 − 98.5 = 0.1 − 0.3 = − 3.00
The one-tailed critical value is t 0.05 , 15 = − 1.753 t_{0.05,15} = -1.753 t 0.05 , 15 = − 1.753 . Since − 3.00 < − 1.753 -3.00 < -1.753 − 3.00 < − 1.753 , reject H 0 H_0 H 0 : at the 5% level the evidence contradicts the manufacturer's efficiency claim.
Example 3: Least-Squares Load Growth
Peak demand over four years is: year 1, 12.0 MW; year 2, 12.9 MW; year 3, 13.6 MW; year 4, 14.7 MW. Fit a least-squares line and forecast year 6.
Solution. With n = 4 n = 4 n = 4 : ∑ x = 10 \sum x = 10 ∑ x = 10 , ∑ y = 53.2 \sum y = 53.2 ∑ y = 53.2 , ∑ x y = 138.3 \sum xy = 138.3 ∑ x y = 138.3 , ∑ x 2 = 30 \sum x^2 = 30 ∑ x 2 = 30 . b = 4 ( 138.3 ) − ( 10 ) ( 53.2 ) 4 ( 30 ) − ( 10 ) 2 = 553.2 − 532 120 − 100 = 21.2 20 = 1.06 MW/yr b = \frac{4(138.3) - (10)(53.2)}{4(30) - (10)^{2}} = \frac{553.2 - 532}{120 - 100} = \frac{21.2}{20} = 1.06 \text{ MW/yr} b = 4 ( 30 ) − ( 10 ) 2 4 ( 138.3 ) − ( 10 ) ( 53.2 ) = 120 − 100 553.2 − 532 = 20 21.2 = 1.06 MW/yr a = 53.2 4 − 1.06 ( 10 4 ) = 13.30 − 2.65 = 10.65 MW a = \frac{53.2}{4} - 1.06\left(\frac{10}{4}\right) = 13.30 - 2.65 = 10.65 \text{ MW} a = 4 53.2 − 1.06 ( 4 10 ) = 13.30 − 2.65 = 10.65 MW y ^ ( 6 ) = 10.65 + 1.06 ( 6 ) = 17.01 MW \hat{y}(6) = 10.65 + 1.06(6) = \boxed{17.01 \text{ MW}} y ^ ( 6 ) = 10.65 + 1.06 ( 6 ) = 17.01 MW