Business Statistics — Nepal Bank Deputy Manager (Level 7)

1. Research Design

Meaning: A research design is the overall plan or blueprint that specifies the methods and procedures for collecting, measuring and analysing the needed information. It is the framework that links the research problem to the data and conclusions.

Types of research design:

Steps in research design: (i) define the research problem, (ii) review literature, (iii) formulate objectives and hypotheses, (iv) select design type, (v) determine sampling design, (vi) design data-collection instruments, (vii) plan data analysis, (viii) prepare budget and time schedule.

Features of a good design: flexibility, appropriateness, efficiency, economy, reliability and validity, and freedom from bias.

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2. Sampling

Meaning: Sampling is the process of selecting a part (sample) of the population to estimate the characteristics of the whole population. A population (universe) is the entire group of items under study; a sampling frame is the list of all units from which the sample is drawn; a sampling unit is one element of that list.

Advantages: lower cost, less time, greater accuracy (fewer non-sampling errors), suitability for destructive testing, and feasibility when the population is infinite.

Limitations: sampling error, need for expert knowledge, possibility of bias, and risk of misleading results if the sample is not representative.

Sampling methods

A. Probability (random) sampling — every unit has a known, non-zero chance of selection:

1. Simple random sampling — lottery method or random-number table. Example: selecting 200 of 5,000 loan accounts by random numbers.
2. Systematic sampling — select every k-th unit after a random start, where k = N/n. Example: every 25th account from a list of 5,000 to get 200.
3. Stratified sampling — population divided into homogeneous strata; units drawn randomly from each stratum. Example: bank customers stratified by deposit size (small, medium, large).
4. Cluster sampling — population divided into clusters (usually geographical); whole clusters selected. Example: selecting 10 branches and surveying all customers of those branches.
5. Multi-stage sampling — sampling in successive stages (branches → departments → employees).

B. Non-probability sampling — selection depends on the researcher's judgement:

1. Convenience sampling — units chosen because they are easily available.
2. Judgement (purposive) sampling — researcher selects units believed to be representative.
3. Quota sampling — fixed quotas for different categories, filled at the interviewer's discretion.
4. Snowball sampling — existing respondents refer further respondents.

Sample size determination

$n = \frac{z^2 \cdot p \cdot q}{e^2}$

where $z$ = standard normal value at the confidence level (1.96 for 95%), $p$ = estimated proportion, $q = 1-p$, $e$ = allowable error.

Example: With 95% confidence, $p = 0.5$, $e = 0.05$:

$n = \frac{(1.96)^2 (0.5)(0.5)}{(0.05)^2} = \frac{3.8416 \times 0.25}{0.0025} = 384.16 \approx 385$

For a finite population, apply the correction:

$n_0 = \frac{n}{1 + \dfrac{n-1}{N}}$

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3. Data Collection

Meaning: Data collection is the systematic process of gathering facts, figures and information relevant to the research problem.

Sources of data:

• Primary data — collected first-hand by the researcher. Methods: observation, interview (personal, telephone), questionnaire (mail, e-mail, online), schedule, and focus-group discussion.
• Secondary data — already collected by others. Sources: NRB publications (Banking and Financial Statistics, Annual Report), Ministry of Finance Economic Survey, CBS statistical yearbook, bank annual reports, journals, websites.

Qualities of a good questionnaire: short and simple questions, unambiguous wording, no leading or personal questions, logical sequence (easy to difficult), pre-tested (pilot survey), and provision for cross-checking.

Precautions in data collection: avoid bias, ensure confidentiality, use trained enumerators, maintain consistency in definitions and units, and cross-verify secondary data for reliability.

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4. Data Analysis — Summarizing Data

Meaning: Data analysis is the process of editing, coding, classifying, tabulating and presenting data so that meaningful conclusions can be drawn.

Steps: editing (checking completeness and accuracy) → coding → classification → tabulation → presentation (tables, graphs) → interpretation.

Measures of central tendency

Measures of dispersion

• Range $= L - S$
• Quartile deviation $= \dfrac{Q_3 - Q_1}{2}$
• Mean deviation $= \dfrac{\sum |X - \bar{X}|}{n}$
• Standard deviation: $\sigma = \sqrt{\dfrac{\sum (X-\bar{X})^2}{n}}$ (ungrouped), $\sigma = \sqrt{\dfrac{\sum fm^2}{N} - \left(\dfrac{\sum fm}{N}\right)^2}$ (grouped)
• Coefficient of variation: $CV = \dfrac{\sigma}{\bar{X}} \times 100\%$

Example (grouped):

$\bar{X} = \frac{620}{30} = 20.67$

$\sigma = \sqrt{\frac{15{,}550}{30} - \left(\frac{620}{30}\right)^2} = \sqrt{518.33 - 427.11} = \sqrt{91.22} = 9.55$

$CV = \frac{9.55}{20.67} \times 100 = 46.2\%$

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5. Trend Analysis

Meaning: Trend analysis measures the general long-term tendency of a time series — the secular trend — and is used for forecasting.

Components of a time series: secular trend ($T$), seasonal variation ($S$), cyclical variation ($C$) and irregular variation ($I$). The multiplicative model is $Y = T \times S \times C \times I$.

Methods of measuring trend:

1. Freehand (graphic) method — a smooth curve drawn by eye.
2. Semi-average method — data divided into two equal halves; averages joined.
3. Moving-average method — averages of fixed overlapping periods; for even periods, centring is required.
4. Least-squares method — fitting $Y_c = a + bX$, with

$a = \frac{\sum Y}{n}, \qquad b = \frac{\sum XY}{\sum X^2}$

(taking deviations from the middle year so that $\sum X = 0$).

Example: Fitting a straight line to yearly sales 100, 110, 125, 140, 155 (coded $X = -2,-1,0,1,2$):

$\sum Y = 630,\ n = 5 \Rightarrow a = \frac{630}{5} = 126$

$\sum XY = (-2)(100)+(-1)(110)+0+1(140)+2(155) = 140,\quad \sum X^2 = 10$

$b = \frac{140}{10} = 14$

$Y_c = 126 + 14X$

Forecast for the next year ($X = 3$): $Y_c = 126 + 14(3) = 168$.

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6. Test of Hypothesis

Meaning: A hypothesis is a tentative statement about a population parameter. Hypothesis testing is a procedure to decide, on the basis of sample evidence, whether to accept or reject it.

• Null hypothesis ($H_0$): statement of no difference or no effect, e.g. $H_0: \mu = 50$.
• Alternative hypothesis ($H_1$): the claim to be established, e.g. $H_1: \mu \neq 50$ (two-tailed) or $\mu > 50$ (right-tailed).
• Level of significance ($\alpha$): probability of Type I error, usually 5% or 1%.
• Type I error: rejecting a true $H_0$. Type II error: accepting a false $H_0$.
• Critical value: table value of $z$ or $t$ at the given $\alpha$.
• Decision rule: if $|z_{cal}| > z_{tab}$, reject $H_0$; otherwise do not reject.

Test statistics

$z = \frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \quad (\sigma \text{ known, or } n \geq 30)$

$t = \frac{\bar{X} - \mu}{s/\sqrt{n}}, \quad df = n-1 \quad (\sigma \text{ unknown, } n < 30)$

$z = \frac{\hat{p} - p}{\sqrt{pq/n}} \quad (\text{proportion})$

$\chi^2 = \sum \frac{(O-E)^2}{E} \quad (\text{goodness of fit / independence})$

Example: A bank claims the average waiting time is 12 minutes. A sample of 25 customers gives $\bar{X} = 13.5$ minutes and $s = 2.5$ minutes. Test at 5% level.

$H_0: \mu = 12,\quad H_1: \mu \neq 12,\quad \alpha = 0.05,\quad t_{0.025,\,24} = 2.064$

$t = \frac{13.5 - 12}{2.5/\sqrt{25}} = \frac{1.5}{0.5} = 3.0$

Since $|t_{cal}| = 3.0 > 2.064$, reject $H_0$. The average waiting time differs significantly from 12 minutes.

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7. Preparation of a Research Proposal

Meaning: A research proposal is a written plan of the proposed study submitted for approval or funding. It states what is to be studied, why, how, where and when.

Contents:

1. Title of the study — clear, specific and researchable.
2. Background / introduction — context of the problem.
3. Statement of the problem — what gap exists and why it matters.
4. Objectives — general and specific, in measurable terms.
5. Research questions / hypotheses.
6. Rationale / significance — academic and practical contribution.
7. Literature review — past studies and theoretical framework.
8. Research methodology — design, population, sample size and technique, data-collection tools, data-analysis tools (mean, SD, $t$-test, $\chi^2$, regression).
9. Scope and limitations.
10. Organization of the study — chapter scheme.
11. Time schedule (Gantt chart) and budget.
12. References in standard format.

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8. Preparation of Research Reports

Meaning: A research report is the systematic, written presentation of the findings of a study in a form that can be understood and used by others.

Structure:

Qualities of a good report: clarity, conciseness, objectivity, logical organization, accuracy, use of tables and graphs, and proper citation.

Steps in report writing: (i) plan the report, (ii) collect and organize the material, (iii) make an outline, (iv) write the first draft, (v) edit and revise, (vi) prepare the final draft, (vii) submit.

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Key Formulas at a Glance