BECS-184 Solved Assignment

Question 1

Compute Karl Pearson’s correlation for the given paired data and explain how to test whether the correlation is statistically significant

Rewritten question: For the six paired observations of X and Y given below, calculate Karl Pearson’s correlation coefficient (r), interpret the direction/strength, and then outline the standard hypothesis-testing steps used to check whether the population correlation is significant.

(a) Karl Pearson’s correlation coefficient (r) and interpretation

Given data (paired observations):

$$ r=\frac{n\sum XY-(\sum X)(\sum Y)}{\sqrt{\left[n\sum X^2-(\sum X)^2\right]\left[n\sum Y^2-(\sum Y)^2\right]}} $$

Substitution: n = 6, ΣXY = 1292, ΣX = 68, ΣY = 112, ΣX² = 786, ΣY² = 2128.

$$ \text{Numerator}=6(1292)-68(112)=7752-7616=136 $$

$$ \text{Denominator}=\sqrt{\left[6(786)-68^2\right]\left[6(2128)-112^2\right]} =\sqrt{(4716-4624)(12768-12544)}=\sqrt{92\cdot 224}=\sqrt{20608}\approx 143.56 $$

$$ r=\frac{136}{143.56}\approx 0.947 $$

Interpretation (student-friendly): r ≈ 0.947 indicates a very strong positive linear relationship: as X increases, Y tends to increase. Since -1 < r < +1, a value close to +1 implies strong positive association.

Extra practical insight: If you square the correlation, you get r² ≈ 0.897. That suggests roughly about 89.7% of the variation in Y is associated with variation in X in a linear sense (often discussed as “explained variation” in regression context).

(b) Procedure to test whether the correlation is statistically significant

The course approach is to test whether the population correlation (ρ) is zero using a t-based test.

1. State hypotheses H₀: ρ = 0 (no linear correlation in the population)
   H₁: ρ ≠ 0 (a linear correlation exists)
2. Choose significance level Commonly α = 0.05 (two-tailed), unless the question specifies otherwise.
3. Compute test statistic $$ t=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}},\quad \text{d.f.}=n-2 $$ This is the standard significance test for a correlation coefficient in the bivariate analysis treatment.
4. Decision rule Find the critical value t_α/2 with d.f. = n – 2. Reject H₀ if |t| > t_α/2.
5. Conclusion in words If significant, you report that the correlation is unlikely to be due to random sampling fluctuation at the chosen α level.

Applied quickly to this dataset (for completeness): With r ≈ 0.947 and n = 6, t ≈ 5.92 and d.f. = 4, which would typically exceed the usual 5% two-tailed critical value. Therefore, the correlation would be judged statistically significant in the usual classroom setting.

Question 2

Explain mathematical modelling (meaning + stages) and explain the concept of logic

Rewritten question: Define mathematical modelling in the context of data analysis, describe the main stages used to build a model, and then explain what “logic” means (how reasoning is structured).

(a) Meaning of mathematical modelling

In data analysis, a mathematical (or statistical) model is a simplified representation of a real-world situation using variables and relationships (often equations). Practically, we treat observed data as having a “systematic part” (signal/structure) plus an “error part” (noise). This is why modelling is useful: it helps separate the meaningful pattern from random variation.

(b) Stages in mathematical modelling (model-building flow)

A standard, student-friendly modelling cycle includes:

• Problem definition: clarify the objective (what you want to explain or predict) and identify relevant variables.
• Assumptions and conceptual model: decide what simplifications are reasonable (linearity, independence, etc.).
• Model formulation: write relationships mathematically (equations or statistical form).
• Data collection and preparation: gather data, clean it, and code variables consistently.
• Parameter estimation: use data to estimate unknown parameters (e.g., regression coefficients).
• Model checking/diagnostics: check whether assumptions are reasonable and whether fit is acceptable.
• Validation: test performance on new data or by cross-checking results; refine if needed.
• Interpretation and communication: explain what the model implies in real terms, including limits.

This staged approach aligns with the course’s structured model-building process (define objective, plan analysis, verify assumptions, estimate model/fit, interpret, validate).

(c) Concept of logic (in simple terms)

Logic is the discipline of correct reasoning. In data analysis and research, logic helps you move from statements (premises) to a conclusion in a way that is consistent and testable. A common way to express logic is through propositions (statements that are true/false) and rules for combining them (AND, OR, NOT, IF–THEN). This supports clear hypothesis statements, correct interpretation of results, and avoidance of contradictory conclusions.

Question 3

Differentiate between census and survey, and list the stages used to plan and organise them

Rewritten question: Explain how a census differs from a sample survey, and then outline the key stages involved in planning and organising both types of data collection.

(a) Census vs. sample survey

• Census: data are collected from every unit in the population (complete enumeration). It can give very detailed coverage but is usually expensive and time-consuming, and harder to repeat frequently.
• Sample survey: data are collected from a subset (sample) of the population. If sampling is well-designed, it provides reliable estimates faster and at lower cost, and it is more practical for frequent studies.

(b) Typical stages in planning and organising a census/survey

A practical sequence (used in many course-aligned research designs) is:

1. Set objectives and scope: decide what information is needed and why.
2. Define universe and units: specify target population, sampling units, and coverage area/time.
3. Decide census vs. sampling plan: if survey, choose sampling design, sample size, and frame.
4. Design instruments: questionnaire/schedule, definitions, coding instructions, and pre-test/pilot.
5. Fieldwork planning: recruit/train investigators, supervision plan, logistics, timeline and budget.
6. Data collection: execute enumeration/interviews/measurement with quality checks.
7. Editing, coding, and entry: clean data, handle missing values, code open responses.
8. Tabulation and analysis: summaries, tables, and statistical analysis aligned to objectives.
9. Reporting and dissemination: communicate results, limitations, and recommendations.

Question 4

Define (i) Z score, (ii) snowball sampling, (iii) Type I and Type II errors, and (iv) the normal distribution curve

Rewritten question: Provide short, correct definitions (with formula where needed) for Z score, snowball sampling, Type I/Type II errors, and the normal curve.

(i) Z score

A Z score tells how many standard deviations a value x is away from the mean.

$$ Z=\frac{x-\mu}{\sigma} $$

In practice, students often use it to standardise values so that different scales become comparable (for example, comparing test scores from two different exams).

(ii) Snowball sampling

Snowball sampling is a non-probability sampling method where existing participants help recruit additional participants (the sample “grows” through referrals). It is especially useful for hard-to-reach or hidden populations, but it can introduce bias because the network of referrals may not represent the full population.

(iii) Type I and Type II errors

• Type I error: rejecting a true H₀ (a “false positive”). Its probability is α.
• Type II error: failing to reject a false H₀ (a “false negative”). Its probability is β.

Power of a test is (1 – β), meaning the ability to detect a real effect when it exists.

(iv) Normal distribution curve

The normal distribution is a continuous, bell-shaped, symmetric distribution where the mean, median, and mode coincide. Many inferential procedures assume normality (or approximate normality), and areas under the curve correspond to probabilities.

Question 5

(a) Explain why correlation does not automatically mean causation

Rewritten question: Using clear reasoning, explain why a high correlation between two variables is not enough to claim that one variable causes the other.

Answer (with a practical feel): Correlation measures the strength of association, not a cause-and-effect mechanism. Two variables can move together because:

• A third factor influences both (confounding),
• Reverse causality is possible (Y might influence X, not the other way),
• Coincidence can occur in small samples or short time periods,
• Common trend effects can create correlation even without direct linkage.

In real analysis work, a good habit is to ask: “What is the plausible mechanism?” and “Can I rule out alternative explanations using design (experiment) or controls (regression)?”

(b) One-way ANOVA: do the mean retail prices differ across three cities?

Rewritten question: Given three small samples of retail prices from Mumbai, Kolkata, and Delhi, use one-way ANOVA at the 5% level to test whether the population means are equal. The critical value is given as F = 5.14.

Data:

Step 1: Hypotheses

H₀: μ₁ = μ₂ = μ₃ (all city means are equal)
H₁: At least one mean differs

Step 2: Compute group means and grand mean

• Mumbai mean = (643 + 655 + 702) / 3 = 666.67
• Kolkata mean = (469 + 427 + 525) / 3 = 473.67
• Delhi mean = (484 + 456 + 402) / 3 = 447.33
• Grand mean (all 9 values) = 529.22

Step 3: Compute sums of squares

$$ SS_B=\sum n_i(\bar{x}i-\bar{x})^2,\quad SS_W=\sum\sum (x{ij}-\bar{x}_i)^2 $$

Using the standard one-way ANOVA structure (between-groups vs within-groups variation):

• SS_B = 82108.22
• SS_W = 14195.33
• SS_T = SS_B + SS_W = 96303.56

Step 4: ANOVA table and F statistic

Decision: F_cal = 25.18 > F_crit = 5.14, so reject H₀. Mean retail prices are not all the same across the three cities.

Practical note: ANOVA tells you “a difference exists somewhere.” If you needed to identify which pair differs (Mumbai vs Kolkata, etc.), you would run a post-hoc comparison (beyond this question’s requirement).

Question 6

(a) When to use t-test, F-test, and Z-test (conditions)

Rewritten question: State the typical situations/assumptions under which each of the following tests is appropriate: t, F, and Z.

• Z-test (normal test) Used mainly when the population standard deviation (σ) is known, or when the sample size is large so the sampling distribution is approximately normal; it is common for testing means or proportions under normal approximation.
• t-test Used when σ is unknown (so we use sample s), especially for small samples, and when the underlying population is approximately normal. It is widely used for testing a single mean or difference between two means.
• F-test Used for comparing variances and, more broadly, as the basis of ANOVA (testing equality of multiple means via variance decomposition). It is also used in regression contexts to test overall model significance.

(b) What is multivariate analysis, and what should you keep in mind while interpreting results?

Rewritten question: Define multivariate analysis and list key precautions/points for interpreting multivariate results so that findings are meaningful and not spurious.

Meaning: Multivariate analysis refers to statistical techniques that analyse multiple measurements/variables simultaneously on individuals or objects. In many practical problems, more than two variables interact, so multivariate methods (e.g., multiple regression, MANOVA, factor analysis, cluster analysis) are used to model complex relationships.

Important points while interpreting multivariate results (course-aligned, practical checklist)

• Separate statistical vs practical significance: a result can be statistically significant but not useful for decision-making if the effect is too small.
• Sample size matters: large samples can make tiny effects “significant,” while small samples may miss real effects.
• Know your data first: check outliers, missing values, and basic distributions before trusting multivariate outputs.
• Check assumptions: for regression-type models, look for normality, equal variance, linearity, independence, and non-multicollinearity.
• Prefer parsimonious models: do not add variables without reason; simpler models are often more stable and interpretable.
• Validate results: use hold-out checks, cross-validation, or replication; do not assume the sample result automatically generalises.

Question 7

Differentiate between paired concepts in research and inference

Rewritten question: Clearly distinguish between each of the following pairs: (i) quantitative vs qualitative research, (ii) phenomenology vs ethnography, (iii) observational vs experimental method, and (iv) point estimate vs interval estimate.

(i) Quantitative research vs qualitative research

• Quantitative: focuses on numerical measurement, statistical analysis, and hypothesis testing; aims at describing relationships mathematically and often uses larger samples.
• Qualitative: focuses on meanings, experiences, and interpretations in natural settings; uses non-numerical data (words, narratives, observations) and typically smaller, purposive samples.

(ii) Phenomenology vs ethnography

• Phenomenology: studies the lived experiences of individuals to understand the essence/meaning of a phenomenon (what it feels like, how it is experienced).
• Ethnography: studies a group/culture (“portrait of a people”), often using participant observation and fieldwork to understand norms, behaviours, and shared meanings.

(iii) Observational method vs experimental method

• Observational: the researcher observes and records what happens naturally, without controlling who receives what condition; useful when manipulation is unethical or impractical.
• Experimental: the researcher actively manipulates an independent variable (often with control and treatment groups) to study causal effects under controlled conditions.

(iv) Point estimate vs interval estimate

• Point estimate: a single best-value estimate of a population parameter (e.g., using the sample mean as an estimate of population mean).
• Interval estimate: a range of plausible values (confidence interval) that is likely to contain the true parameter with a stated confidence level (e.g., 95%).