Notes, summaries, assignments, exams, and problems for Baccalaureate

Problem 9: Z-Test for Two Means

A researcher wants to know the intelligence of students in a school. He selected two groups of students. In the first group, there are 150 students having a mean IQ of 75 with a S.D. of 15. In the second group, there are 250 students having a mean IQ of 70 with a S.D. of 20. Is there a significant difference between the means of the two groups?

Solution

Given

For Group I:

• Sample size, n₁ = 150
• Sample mean, x̄₁ = 75
• Standard deviation, s₁ = 15

For Group II:

• Sample size, n₂ = 250
• Sample mean, x̄₂ = 70
• Standard deviation, s₂ = 20

Level of significance: α = 0.05

Since both sample sizes are large (n₁, n₂ > 30), use the Z-test for the difference between two means.

Step 1: State the Hypotheses

• Null Hypothesis (H₀): μ₁ = μ₂

Calculating Mean and Standard Deviation

The marks obtained in a statistics examination are normally distributed. If 15% of students scored ≥ 60 marks and 40% scored < 30 marks, find the mean (μ) and standard deviation (σ).

Given

• P(X ≥ 60) = 0.15
• P(X < 30) = 0.40

We need to find the values of μ and σ.

Step 1: Convert to Z-values

(i) P(X ≥ 60) = 0.15

Since P(X ≥ 60) = 0.15, then P(X < 60) = 0.85. From the standard normal table, P(Z < 1.04) = 0.85. Therefore:

(60 − μ) / σ = 1.04
μ = 60 − 1.04σ (Equation 1)

(ii) P(X < 30) = 0.40

From the standard normal table, P(Z < -0.25) = 0.40. Therefore:

(30 − μ) / σ = -0.25
μ = 30 + 0.25σ (Equation 2)

Step 3: Solve the Equations

Equating (1) and (2):

60 − 1.04σ = 30 + 0.25σ
30

1. Find the GCD of (143, 227)

The Euclidean Algorithm is used to find the Greatest Common Divisor (GCD) of two integers by repeatedly dividing the larger number by the smaller number until the remainder becomes 0.

• Step 1: 227 = 143 × 1 + 84
• Step 2: 143 = 84 × 1 + 59
• Step 3: 84 = 59 × 1 + 25
• Step 4: 59 = 25 × 2 + 9
• Step 5: 25 = 9 × 2 + 7
• Step 6: 9 = 7 × 1 + 2
• Step 7: 7 = 2 × 3 + 1
• Step 8: 2 = 1 × 2 + 0

Since the remainder becomes 0, the last non-zero remainder is 1.

Answer: GCD(143, 227) = 1

2. Find m and n such that gcd(a, b) = ma + nb for a = 56 and b = 72

Given:

a = 56, b = 72. We need to find integers m and n such that: gcd(56, 72) = 56m + 72n.

Step 1: Find the GCD using the Euclidean Algorithm

72 = 56 × 1 + 16
56 = 16 × 3 + 8
16 = 8 × 2 + 0
Hence,... Continue reading "Number Theory Problems: GCD, Congruences, and Fermat Numbers" »

Quick revision table for Mathematics and Statistics exams.

Unit I: Integration and Data Presentation

Unit II: Central Tendency and

Research Ethics and Methodology

3. 1C, 2A, 3F, 4G, 5H, 6B, 7D, 8E

1. A rule, policy, or… 2 GUI
2. DELINE
3. A person who is the… 6 HUMAN SUBJECT
4. Relating to someone’s… 1 MORAL
5. A study in which the… 7 OBSERVATIONAL STUDY
6. The process of using… 8 EXPERIMENTAL DATA
7. The positive or helpful… 3 BENEFIT
8. Relating to boundaries 4 DEFINED
9. A child… 5 MINOR

4. 1. Field experiment, 2. Interview, 3. Animal study, 4. Consent, 5. Ethical, 6. Regulations

Data Collection and Measurement

3. 1D, 2B, 3C, 4A, 5E

1. A digital program… 4 COMPUTER SIMULATION
2. The process… 2 DATA COLLECTION
3. A subset of… 3 RANDOM SAMPLE MEASUREMENT
4. A real-life event… 1 CASE STUDY
5. A characterization of data… 5 NOMINAL

4.

• Collection Methods: Archival search, self-report, survey
• Examination: Content

Uveitis Signs and Symptoms

Anterior Uveitis

• Reduced visual acuity (VA)
• Ciliary injection/flush around the limbus
• Miosis due to pupillary sphincter spasm
• Formation of posterior synechiae
• Anterior chamber cells and flare
• 'Spillover' of cells into the vitreous
• Hypopyon (white blood cell collection)
• Keratic precipitates (large/greasy suggest systemic; small suggest AAU)

Intermediate Uveitis

• Cells and flare
• Scattered keratic precipitates
• Vitreous cell spillover
• Vitreous snowballs and snow banking (white cells seen on fundus)
• Neovascularization and cystoid macular edema (CMO)

Grading Systems

Cells (/1x1 mm)

• 0: <1 | 0.5+: 1–5 | 1+: 6–15 | 2+: 16–25 | 3+: 26–50 | 4+: >50

Flare

• 0: None | 1+: Faint | 2+: Moderate (iris/lens clear) | 3+: Marked (iris/lens hazy)

Environmental Stewardship: Lessons from The Lorax

Treat it with care. Give it clean water. And feed it fresh air. Grow a forest. Protect it from axes that hack. Then the Lorax and all of his friends may come back.

Trátalo con cuidado. Dale agua limpia. Y aliméntalo con aire fresco. Haz crecer un bosque. Protégelo de las hachas que talan. Entonces el Lorax y todos sus amigos podrán regresar.

“They loved living here. But I can't let them stay. They'll have to find food. And I hope that they may. Good luck, boys,” he cried. And he sent them away.

“Les encantaba vivir aquí. Pero no puedo dejarlos quedarse. Tendrán que buscar comida. Y espero que la encuentren. ‘Buena suerte, chicos’, gritó. Y los envió lejos.”

Professional Interpretation

Revolution White Solera

• Grape: Chardonnay, Scheurebe, Riesling
• Style: Natural, oxidative-influenced white with solera aging
• Fermentation: Spontaneous fermentation with native yeasts
• Aging: Multi-vintage solera system in large neutral oak barrels
• Notes: Dried apple, orange peel, herbal tea, light nuttiness
• Acidity: Medium, balanced by texture and oxidative depth

2024 Scheurebe

• Grape: Scheurebe
• Style: Aromatic, precise dry white
• Fermentation: Stainless steel to preserve aromatics
• Aging: Short lees contact
• Notes: Grapefruit, passionfruit, cassis, subtle mineral tones
• Acidity: Vibrant, fresh and structured

2023 Albariño »Do Ferreiro«

• Grape: Albariño
• Style: Coastal, mineral-driven white
• Fermentation: Stainless steel with controlled temperatures
• Aging: Lees aging

Verb Tense Summary

Present Simple (Pres.S)

am/is/are + participle. He/She/It: do $\rightarrow$ does/doesn't. Will not.

Present Continuous (Pres.Cont)

am/is/are + being + participle.

Past Simple (Past.S)

was/were + participle.

Past Continuous (Past.Cont)

was/were + being + participle.

Present Perfect (Pres.Perf)

have/has + been + participle.

Past Perfect (Past.Perf)

had + been + participle.

Future Simple (Future.S)

will + be + participle.

Going To Future

am/is/are + going to + be + participle.

Auxiliary Verbs Summary

have/has/past: had.

Modal Verbs (Second Conditional Context)

Second Modals: might/may/could.

Examples: not buy $\rightarrow$ didn't buy. He/She/It WERE.

Participles

Third Participle: verb + ed. Go $\rightarrow$ gone. Be $\rightarrow$ have been.

Conditional