STAT-II GUIDE FOR BOARD AND QT BY UJJU

Ultimate Statistics II Exam Guide: Score 60/60

Preparing for your Statistics II board exam and aiming for a perfect 60/60? This comprehensive guide is tailored to help you ace the exam with confidence. Based on the syllabus and past question patterns, I’ve compiled key topics, possible questions with answers, and sure-shot questions to ensure you’re fully prepared. Let’s dive in!

Exam Structure Overview

Total Marks: 60
Group A: Long-answer questions (10 marks each, attempt 2 out of 3) – Total: 20 marks
Group B: Short-answer questions (5 marks each, attempt 8 out of 10 or 11) – Total: 40 marks
Goal: Answer both Group A questions (20/20) and all eight Group B questions (40/40) correctly.

Syllabus Breakdown and Key Topics

The syllabus includes:

Basics of Statistics: Definitions, scope, data types, measures of central tendency and dispersion.
Descriptive Statistics: Mean, median, mode, variance, standard deviation, skewness, kurtosis.
Probability: Basic rules, conditional probability, Bayes’ theorem, independence.
Sampling: Sampling methods, sampling distributions, Central Limit Theorem (CLT).
Random Variables: Discrete and continuous random variables, probability mass and density functions.
Mathematical Expectations: Expected value, variance, moments.
Probability Distributions: Binomial, Poisson, normal, exponential, uniform, etc.
Correlation: Pearson’s correlation coefficient, rank correlation.
Regression: Simple and multiple linear regression, interpretation of coefficients, goodness of fit.

Additional high-weightage topics from past papers:

Hypothesis Testing (z-test, t-test, chi-square, non-parametric tests)
Design of Experiments (ANOVA, Latin Square Design)
Confidence Intervals
Queuing Theory (M/M/1 model)

Detailed Guide to Important Topics

Key Concepts

Statistics: Science of collecting, analyzing, and interpreting data.
Data types: Qualitative (nominal, ordinal), Quantitative (discrete, continuous).
Measures: Mean (average), median (middle value), mode (most frequent), range, variance, standard deviation.

Possible Questions

Q1: Define statistics and explain its scope in real-life applications.

Answer: Statistics is the science of data collection, organization, analysis, interpretation, and presentation. Its scope includes business (market analysis), science (experiment design), medicine (drug trials), and government (census data). It aids decision-making under uncertainty.

Q2: Calculate mean, median, and mode for the data: 5, 7, 7, 8, 10.

Answer: Mean = (5+7+7+8+10)/5 = 37/5 = 7.4; Median = 7 (middle value); Mode = 7 (appears twice).

Key Concepts

Variance: s² = Σ(xᵢ - x̄)²/(n-1) (sample), measures spread.
Standard Deviation: s = √s².
Skewness: Measures asymmetry (positive, negative, or zero).
Kurtosis: Measures tailedness (leptokurtic, platykurtic, mesokurtic).

Possible Questions

Q1: Compute the variance and standard deviation for: 2, 4, 6, 8, 10.

Answer: Mean = (2+4+6+8+10)/5 = 6. Deviations squared: (2-6)² = 16, (4-6)² = 4, (6-6)² = 0, (8-6)² = 4, (10-6)² = 16. Sum = 40. Variance = 40/(5-1) = 10. Standard deviation = √10 ≈ 3.16.

Q2: Explain skewness and its types with examples.

Answer: Skewness measures distribution asymmetry. Positive skew: tail on right (e.g., income data). Negative skew: tail on left (e.g., time to failure). Zero skew: symmetric (e.g., normal distribution).

Key Concepts

Probability: P(A) = favorable outcomes / total outcomes, 0 ≤ P(A) ≤ 1.
Addition Rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) (if mutually exclusive).
Conditional Probability: P(A|B) = P(A ∩ B) / P(B).
Bayes’ Theorem: P(A|B) = P(B|A) P(A) / P(B).

Possible Questions

Q1: A bag has 3 red and 5 blue balls. Find the probability of drawing 2 red balls without replacement.

Answer: P(1st red) = 3/8, P(2nd red | 1st red) = 2/7. Total probability = (3/8) * (2/7) = 6/56 = 3/28.

Q2: Given P(A) = 0.3, P(B|A) = 0.4, P(B) = 0.5, find P(A|B).

Answer: P(A|B) = (P(B|A) P(A)) / P(B) = (0.4 * 0.3) / 0.5 = 0.24.

Key Concepts

Sampling Methods: Simple random, stratified, cluster, systematic.
Sampling Distribution: Distribution of a statistic (e.g., mean).
Central Limit Theorem (CLT): For large n, sample mean x̄ ~ N(μ, σ/√n).

Possible Questions

Q1: Explain the Central Limit Theorem with an example.

Answer: CLT states that for a large sample size, the sampling distribution of the mean is approximately normal, regardless of population distribution. Example: Rolling a die (uniform), mean of 100 rolls approximates normal.

Q2: A population has μ = 50, σ = 10. For n = 100, find P(x̄ > 51).

Answer: σ_x̄ = 10/√100 = 1. z = (51-50)/1 = 1. P(Z > 1) = 0.1587 (from z-table).

Key Concepts

Discrete: Finite or countable (e.g., number of students).
Continuous: Infinite within a range (e.g., height).
PMF (discrete): P(X = x).
PDF (continuous): f(x), where ∫ f(x) dx = 1.

Possible Questions

Q1: For a discrete RV X with P(X=1) = 0.4, P(X=2) = 0.6, find E(X).

Answer: E(X) = 1*0.4 + 2*0.6 = 0.4 + 1.2 = 1.6.

Q2: For a continuous RV with PDF f(x) = 2x, 0 < x < 1, verify it’s a valid PDF.

Answer: ∫₀¹ 2x dx = [x²]₀¹ = 1-0 = 1, so valid.

Key Concepts

Expected Value: E(X) = Σ x P(x) (discrete), E(X) = ∫ x f(x) dx (continuous).
Variance: Var(X) = E(X²) - [E(X)]².

Possible Questions

Q1: Find E(X) and Var(X) for X with P(X=1) = 0.5, P(X=2) = 0.5.

Answer: E(X) = 1*0.5 + 2*0.5 = 1.5. E(X²) = 1²*0.5 + 2²*0.5 = 0.5 + 2 = 2.5. Var(X) = 2.5 - 1.5² = 2.5 - 2.25 = 0.25.

Q2: For f(x) = 3x², 0 < x < 1, find E(X).

Answer: E(X) = ∫₀¹ x * 3x² dx = ∫₀¹ 3x³ dx = [3x⁴/4]₀¹ = 3/4.

Key Concepts

Binomial: P(X=k) = (n choose k) pᵏ (1-p)ⁿ⁻ᵏ.
Poisson: P(X=k) = (λᵏ e⁻λ) / k!.
Normal: f(x) = (1/√(2πσ²)) e⁻(x-μ)²/(2σ²).
Exponential: f(x) = λ e⁻λx, x ≥ 0.

Possible Questions

Q1: In 10 trials, p=0.3, find P(X=2) (Binomial).

Answer: P(X=2) = (10 choose 2) (0.3)² (0.7)⁸ = 45 * 0.09 * 0.057648 = 0.233.

Q2: For Poisson with λ = 4, find P(X ≥ 3).

Answer: P(X < 3) = P(0) + P(1) + P(2) = e⁻⁴ (1 + 4 + 8) = 0.0183 * 13 = 0.238. So, P(X ≥ 3) = 1 - 0.238 = 0.762.

Key Concepts

Pearson’s r = Σ(xᵢ - x̄)(yᵢ - ȳ) / √(Σ(xᵢ - x̄)² Σ(yᵢ - ȳ)²).
Range: -1 to 1; 0 means no linear relationship.

Possible Questions

Q1: Compute r for (1,2), (2,3), (3,5).

Answer: x̄ = 2, ȳ = 3.33. Σ(xᵢ - x̄)(yᵢ - ȳ) = (1-2)(2-3.33) + (2-2)(3-3.33) + (3-2)(5-3.33) = 1.33 + 0 + 1.67 = 3. √(Σ(xᵢ - x̄)²) = √2, √(Σ(yᵢ - ȳ)²) = √4.66. r = 3 / (√2 * √4.66) = 0.98.

Q2: Interpret r = -0.8.

Answer: Strong negative linear relationship; as one variable increases, the other decreases.

Key Concepts

Simple: y = b₀ + b₁x, b₁ = Σ(xᵢ - x̄)(yᵢ - ȳ) / Σ(xᵢ - x̄)², b₀ = ȳ - b₁x̄.
Multiple: y = b₀ + b₁x₁ + b₂x₂ + ⋯.
R²: Proportion of variance explained.

Possible Questions

Q1: Fit a simple regression for (1,2), (2,4), (3,6).

Answer: x̄ = 2, ȳ = 4. b₁ = [(1-2)(2-4) + (2-2)(4-4) + (3-2)(6-4)] / [(1-2)² + (2-2)² + (3-2)²] = (2 + 0 + 2) / (1 + 0 + 1) = 2. b₀ = 4 - 2*2 = 0. So, y = 2x.

Q2: Interpret b₁ = 3 in multiple regression.

Answer: For a unit increase in x₁, y increases by 3, holding other variables constant.

Key Concepts

Null (H₀) vs. Alternative (H₁).
Test Statistics: z = (x̄ - μ) / (σ/√n) (known σ), t = (x̄ - μ) / (s/√n) (unknown σ).
P-value: Probability of observing data if H₀ is true.

Possible Questions

Q1: Test if μ = 50 with n=25, x̄=52, s=5, α=0.05 (two-tailed).

Answer: t = (52-50)/(5/√25) = 2/1 = 2. df=24, t₀.₀₂₅,₂₄ = 2.064. Since 2 < 2.064, fail to reject H₀.

Q2: Test μ₁ = μ₂ with n₁=50, x̄₁=75, s₁=10; n₂=60, x̄₂=80, s₂=12, α=0.05 (one-tailed).

Answer: SE = √(10²/50 + 12²/60) = √(2 + 2.4) = 2.097. z = (80-75)/2.097 = 2.384. z₀.₀₅ = 1.645. Since 2.384 > 1.645, reject H₀; μ₂ > μ₁.

Key Concepts

One-way ANOVA: Compare means across groups.
Two-way ANOVA: Includes blocking.
Latin Square Design: Controls two sources of variation.
F-test: F = MS_treatment / MS_error.

Possible Questions

Q1: For a 3x3 Latin Square (A:10,13,18; B:12,15,16; C:14,17,20), test treatment differences at α=0.05.

Answer: ȳ = 15. TSS = Σ(yᵢⱼ - 15)² = 78. SSR = 3 * [(12-15)² + (15-15)² + (18-15)²] = 54. SSC = 3 * [(15-15)² + (15.67-15)² + (14.33-15)²] = 8/3. SST = 3 * [(13.67-15)² + (14.33-15)² + (17-15)²] = 56/3. SSE = 78 - 54 - 8/3 - 56/3 = 8/3. ANOVA: F_treatment = (28/3) / (4/3) = 7. F₀.₀₅,₂,₂ = 19. 7 < 19, so no significant difference.

Q2: Explain ANOVA assumptions.

Answer: Normality, equal variances, independence of observations.

Key Concepts

Chi-Square: χ² = Σ (O-E)²/E (tests independence or goodness of fit).
Mann-Whitney U: Compares two independent samples.

Possible Questions

Q1: Test independence (Males: 35X, 25Y; Females: 20X, 20Y) at α=0.05.

Answer: E₁₁ = (60*55)/100 = 33, etc. χ² = (35-33)²/33 + ⋯ = 0.673. χ²₀.₀₅,₁ = 3.841. 0.673 < 3.841, so independent.

Q2: Explain when to use non-parametric tests.

Answer: Use when data is non-normal or ordinal (e.g., ranks, Likert scales).

Key Concepts

Mean (known σ): x̄ ± z (σ/√n).
Mean (unknown σ): x̄ ± t (s/√n).

Possible Questions

Q1: n=25, x̄=100, s=15, find 95% CI.

Answer: t₀.₀₂₅,₂₄ = 2.064. CI = 100 ± 2.064 * (15/5) = 100 ± 6.192 = (93.81, 106.19).

Q2: Find n for σ=10, margin of error=3, 95% confidence.

Answer: n = (1.96 * 10 / 3)² = (6.533)² ≈ 43.

Key Concepts

λ: Arrival rate, μ: Service rate.
Traffic intensity: ρ = λ/μ < 1.
L = λ / (μ - λ) (number in system).

Possible Questions

Q1: λ=10/hr, service time=5min, find L.

Answer: μ = 60/5 = 12/hr. L = 10/(12-10) = 5.

Q2: Find W (waiting time in system).

Answer: W = L/λ = 5/10 = 0.5 hours = 30 minutes.

Sure-Shot Questions to Score 60/60

These questions cover all key topics and are designed to prepare you for any variation the board might throw at you.

Group A (Long-Answer, 10 Marks Each)

1. Multiple Linear Regression: Given n=5, Σx₁=10, Σx₂=15, Σy=20, Σx₁²=30, Σx₂²=55, Σy²=90, Σx₁y=40, Σx₂y=60, Σx₁x₂=35, fit y = b₀ + b₁x₁ + b₂x₂ and interpret.

Answer: Normal equations: 5b₀ + 10b₁ + 15b₂ = 20; 10b₀ + 30b₁ + 35b₂ = 40; 15b₀ + 35b₁ + 55b₂ = 60. Solve: b₀ = 4, b₁ = 0, b₂ = 0 (data-specific, typically non-zero). y = 4. Interpret: No linear effect of x₁ or x₂ (special case).

2. ANOVA (Latin Square): For data (Row1: A=10, B=12, C=14; Row2: B=15, C=17, A=13; Row3: C=20, A=18, B=16), perform ANOVA at 5%.

Answer: TSS=78, SSR=54, SSC=8/3, SST=56/3, SSE=8/3. F_treatment = 7 < 19, no significant difference.

Group B (Short-Answer, 5 Marks Each)

1. Hypothesis Testing: n₁=50, x̄₁=75, s₁=10; n₂=60, x̄₂=80, s₂=12. Test μ₂ > μ₁ at 5%.

Answer: z = 2.384 > 1.645, reject H₀; μ₂ > μ₁.

2. Chi-Square Test: Males (35X, 25Y), Females (20X, 20Y), test independence at 5%.

Answer: χ² = 0.673 < 3.841, independent.

3. Confidence Interval: n=25, x̄=100, s=15, 95% CI.

Answer: (93.81, 106.19).

4. Queuing Theory: λ=10/hr, service=5min, find L.

Answer: L = 5.

5. Sample Size: σ=10, E=3, 95%, find n.

Answer: n = 43.

6. Central Limit Theorem: μ=50, σ=10, n=100, P(x̄ > 51).

Answer: 0.1587.

7. Correlation: Compute r for (1,2), (2,3), (3,5).

Answer: r = 0.98.

8. Probability: 3 red, 5 blue balls, P(2 red without replacement).

Answer: 3/28.

Additional Practice Problems with Solutions

1. Binomial: n=10, p=0.3, P(X=2).

Answer: P = 0.233.

2. Normal: μ=100, σ=15, P(X < 110).

Answer: z = (110-100)/15 = 0.67, P = 0.7486.

3. Short Note: Type I vs. Type II errors.

Answer: Type I: Reject true H₀ (false positive). Type II: Accept false H₀ (false negative).

Final Tips for 60/60

Time Management: Allocate ~30 min per Group A question, ~10 min per Group B question.
Presentation: Show all steps, use headings, and box final answers.
Review: Double-check calculations and interpretations.
Memorize: Key formulas (e.g., z, t, χ², ANOVA tables) and critical values (e.g., z₀.₀₅=1.645, t₀.₀₂₅,₂₄=2.064).

This guide covers every syllabus topic, aligns with past trends, and provides robust preparation. Master these concepts and questions, and you’ll be set for a perfect score! Good luck!