Rejecting a null hypothesis that is in fact true is:

A Type I error, with probability equal to the level of significance

A sampling error, with probability equal to the standard error

To test whether brand preference is independent of the buyer's age group, using counts from a survey, the appropriate test is:

Chi-square test of independence

A test yields a p-value of 0.08 at a significance level of 0.05. The correct conclusion is that:

There is insufficient evidence to reject the null hypothesis

The null hypothesis has been proved true

The null hypothesis is rejected at 5%

The alternative hypothesis has a probability of 0.08 of being true

Business Statistics II: Research, Sampling and Hypothesis Testing

The second half of Unit 5 is the research process from the question to the report: what kind of study to run, how to collect and classify data, how to draw a sample and how far to trust it, and — the part the paper leans on hardest — which test answers which question. The hypothesis-testing list is seven tests long and the examiner's favourite move is to describe a situation and ask which one fits. That decision turns on three things: how many groups, what kind of data, and whether the data can be assumed normal. This chapter is built so that those three questions can be answered in that order.

1. Research: concept, types and designs

Research is a systematic inquiry to establish facts or reach new conclusions. Its types are classified along several axes at once, and a question may use any of them: by purpose, into basic or pure research that extends knowledge and applied research that solves a practical problem; by approach, into quantitative work that measures and qualitative work that interprets; by objective, into descriptive, exploratory, explanatory and experimental studies.

Research designs by what they set out to do
Design	Purpose	Distinguishing feature
Exploratory	Clarify a vague problem and generate hypotheses	Flexible and unstructured; literature survey, expert interviews, case study
Descriptive	Describe characteristics of a population or phenomenon as they are	Structured; answers who, what, where and how much, not why
Causal, or experimental	Establish that a change in one variable produces a change in another	Manipulates the independent variable and controls the rest, ideally with a control group and random assignment

⚠️ Descriptive research does not explain

A survey that finds 40% of customers dissatisfied is descriptive; a study that shows dissatisfaction falls when delivery time falls is causal. The words that mark the boundary are 'effect', 'influence', 'causes' and 'leads to' — any of them in the research objective means the design has to be experimental, and a question describing a plain survey while asking about cause is testing exactly this.

2. Data: collection and classification

Primary data are collected for the purpose in hand — by observation, interview, questionnaire or experiment — and are current and specific but costly. Secondary data already exist, collected by someone else for some other purpose — published statistics, company records, earlier studies — and are cheap and quick but may not fit the definitions or the period the present study needs. A questionnaire is filled in by the respondent; a schedule is filled in by the enumerator, which is why schedules work where literacy is low and questionnaires do not.

Data are classified by the scale on which they are measured, and the scale decides which statistics are legitimate — which is why the four scales are examinable rather than decorative.

The four scales of measurement
Scale	What it records	Example	Average that makes sense
Nominal	Categories with no order	Gender, region, brand chosen	Mode only
Ordinal	Categories with an order but no fixed gap	Rankings, satisfaction from poor to excellent	Median and mode
Interval	Equal gaps but an arbitrary zero	Temperature in Celsius, calendar year	Mean, median, mode — but ratios are meaningless
Ratio	Equal gaps and a true zero	Income, weight, sales volume	Everything, including the geometric mean

3. Sampling: probability and non-probability methods

A census examines every unit; a sample examines some and infers the rest. Sampling is used because it is cheaper, quicker, sometimes the only possibility — testing every bulb to destruction leaves none to sell — and often more accurate, since a small well-run study makes fewer non-sampling errors than a vast careless one. The division that matters is whether every unit has a known, non-zero chance of selection.

Probability methods, where selection chances are known
Method	How units are chosen	When it is right
Simple random	Every unit and every combination equally likely, by lottery or random numbers	A homogeneous population with a complete list
Stratified	Divide into homogeneous strata, sample randomly within each	A heterogeneous population; guarantees every stratum is represented and reduces error
Systematic	A random start, then every kth unit	A long ordered list; fails if the list has a cycle matching k
Cluster	Divide into heterogeneous clusters, sample whole clusters at random	A dispersed population with no list; cheaper, at the cost of precision
Multi-stage	Sample clusters, then units within the chosen clusters	Large national surveys

⚠️ Stratified and cluster are opposites, not cousins

Both divide the population into groups, and there the resemblance ends. In stratified sampling the groups are made as homogeneous as possible — all urban, all rural — and units are drawn from EVERY group. In cluster sampling the groups are as heterogeneous as possible — each a miniature of the whole — and only SOME groups are drawn, in their entirety. Stratifying buys precision; clustering buys economy. A question naming a district survey that picks a few villages and interviews everyone in them is describing cluster sampling.

Non-probability methods give no unit a known chance, so no standard error can be computed and results cannot be generalised with a stated confidence. Convenience sampling takes whoever is at hand; judgement or purposive sampling takes whoever the researcher thinks typical; quota sampling fixes the number from each category and fills it by convenience; snowball sampling asks each respondent to name the next, which is how hidden populations are reached at all.

4. Sampling distribution, the central limit theorem and estimation

Take every possible sample of size n from a population and compute the mean of each; the distribution of those means is the sampling distribution of the mean. Its own mean equals the population mean, and its standard deviation — called the standard error — is σ/√n. The standard error is the whole basis of inference: it says how far a sample mean is likely to sit from the truth, and it shrinks with the square root of the sample size, so quadrupling the sample halves the error.

🎯 The central limit theorem, and why it carries the unit

Whatever shape the population has — skewed, bimodal, anything — the sampling distribution of the mean approaches the normal as n grows, and is close enough for practical purposes at about thirty. That is why the z-test and the confidence interval can use normal tables without knowing what the population looks like, and it is the fact that makes the whole of parametric inference usable on real business data, which is almost never normal itself.

Estimation uses the sample to say something about the population parameter. A point estimate is a single figure — the sample mean for the population mean. An interval estimate gives a range with a stated confidence: sample mean ± z × standard error, where z is 1.96 for 95% confidence and 2.58 for 99%. The interval widens as confidence rises and narrows as the sample grows, and a good estimator is unbiased, consistent, efficient and sufficient — four words the paper asks for by name.

5. Hypothesis testing: the logic, then the seven tests

A hypothesis test starts from a null hypothesis H₀, usually that there is no difference or no effect, and asks whether the sample evidence is too unlikely under H₀ to be put down to chance. The level of significance α is the probability of rejecting a true H₀ — a Type I error — and is fixed in advance, commonly at 5%. Failing to reject a false H₀ is a Type II error, with probability β, and the power of the test is 1 − β. The p-value is the probability of evidence at least this extreme if H₀ were true; reject when it falls below α.

⚠️ Not rejecting is not accepting

A test that fails to reject H₀ has found the evidence insufficient, not the hypothesis true — a small sample fails to reject almost anything. The conclusion is 'there is not enough evidence of a difference', never 'there is no difference'. Questions test this wording directly, and the option that says the null hypothesis is 'proved' or 'accepted as true' is the one to avoid.

Which test, decided by groups, data and distribution
Test	Question it answers	Data and assumptions
z-test	Does a mean or proportion differ from a claimed value, or do two differ?	Large sample (n ≥ 30) or known population σ; normal by the central limit theorem
t-test	The same, for a small sample	n < 30, σ unknown and estimated from the sample; normal population; paired form for before-and-after on the same units
ANOVA (F-test)	Do THREE or more group means differ?	Normal populations with equal variances; compares variance between groups with variance within them
Chi-square	Do observed frequencies fit expected ones, or are two attributes independent?	Categorical counts in a contingency table; expected cell frequencies of at least five
Mann-Whitney U	Do TWO independent groups differ?	Ordinal data or non-normal populations; the non-parametric substitute for the independent-samples t-test
Kruskal-Wallis H	Do THREE or more independent groups differ?	Ordinal or non-normal; the non-parametric substitute for one-way ANOVA
Rank correlation	Are two rankings associated?	Ordinal data; Spearman's ρ tested for significance

🧠 Three questions, in order, pick the test

First: are the data counts in categories? Then it is chi-square, whatever else is true. Second: how many groups — two, or three and more? Third: can the data be assumed normal and measured on an interval scale? If yes, two groups is a t-test or z-test and three or more is ANOVA; if no, two groups is Mann-Whitney and three or more is Kruskal-Wallis. Mann-Whitney is to the t-test exactly what Kruskal-Wallis is to ANOVA, and knowing that pairing answers half the questions in the block.

6. Report writing

A research report communicates what was asked, how it was investigated, what was found and what follows from it, in a form a reader can check. Its standard structure runs from the preliminary pages — title, acknowledgements, table of contents, list of tables, an abstract or executive summary — through the body, into the end matter of references and appendices.

Introduction — the problem, its background, the objectives and the hypotheses.
Review of literature — what is already known and where the gap is that this study fills.
Methodology — design, population, sampling method and size, instruments, procedure and the tests used, in enough detail that the study could be repeated.
Analysis and findings — the results, with tables and figures, reported without interpretation.
Discussion and conclusions — what the findings mean against the objectives and the earlier literature, the limitations, and the recommendations.
References and appendices — every source cited, in one consistent style, and the questionnaire, raw tables and computations that would clutter the body.

ℹ️ Findings and discussion are kept apart on purpose

The findings section says what the data showed; the discussion says what the researcher makes of it. Keeping them separate lets a reader who disagrees with the interpretation still use the results, and it is the convention a question on report structure is most likely to test — an option that puts recommendations inside the findings has put opinion where the reader expects fact.

Key takeaways

Descriptive research describes and cannot explain; the words effect, influence and cause in an objective demand an experimental design.
Stratified sampling draws from every homogeneous group for precision; cluster sampling draws whole heterogeneous groups for economy. They divide the population for opposite reasons.
The standard error σ/√n is the basis of all inference, and the central limit theorem is why normal tables apply to sample means whatever the population looks like.
Failing to reject H₀ means insufficient evidence, never that H₀ is true; a Type I error rejects a true null and its probability is the significance level.
Counts in categories mean chi-square; then count the groups and ask whether the data are normal — Mann-Whitney is to the t-test what Kruskal-Wallis is to ANOVA.

Practice questions (9)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

A researcher wants to compare customer satisfaction scores, measured on a five-point ordinal scale, across four branches. The appropriate test is:
1. Kruskal-Wallis H test
2. One-way ANOVA
3. Mann-Whitney U test
4. Paired t-test
Show answer
Answer: A — Kruskal-Wallis H test
Apply the three questions in order. The data are scores rather than counts in categories, so chi-square is out. There are four groups, so a two-group test — Mann-Whitney or any t-test — is out. The scale is ordinal, so the data cannot be assumed normal and interval-measured, which rules out ANOVA and points to its non-parametric substitute for three or more independent groups: Kruskal-Wallis. Mann-Whitney is the two-group version of the same idea, and a paired t-test requires the same units measured twice, which four separate branches are not.
Rejecting a null hypothesis that is in fact true is:
1. A Type I error, with probability equal to the level of significance
2. A Type II error, with probability β
3. The power of the test
4. A sampling error, with probability equal to the standard error
Show answer
Answer: A — A Type I error, with probability equal to the level of significance
The two errors are defined by which way the decision goes wrong. Rejecting a true null hypothesis is a Type I error, and its probability is α, the level of significance the researcher fixes in advance — choosing 5% means accepting a one-in-twenty chance of this mistake. Failing to reject a false null hypothesis is a Type II error with probability β, and the power of the test is 1 − β, the chance of correctly detecting a real effect. The standard error is a measure of the spread of a sampling distribution, not the probability of any decision error, so the last option confuses two unrelated quantities.
A survey of a district selects ten villages at random and interviews every household in each. This is:
1. Cluster sampling
2. Stratified sampling
3. Systematic sampling
4. Quota sampling
Show answer
Answer: A — Cluster sampling
The population has been divided into groups — villages — and a few whole groups have been chosen at random with every unit inside them examined. That is cluster sampling, chosen for economy when the population is dispersed and no list of households exists. Stratified sampling would divide the district into homogeneous strata and draw households from every stratum, not skip most of the villages entirely. Systematic sampling takes every kth unit from an ordered list after a random start, and quota sampling is a non-probability method that fills fixed category counts by convenience, neither of which is described here.
A population has standard deviation 20. The standard error of the mean of a sample of 100 is:
1. 2
2. 0.2
3. 20
4. 200
Show answer
Answer: A — 2
The standard error of the mean is the population standard deviation divided by the square root of the sample size, σ/√n = 20/√100 = 20/10 = 2. Dividing by n itself rather than its square root gives 0.2, which is the standard slip and understates the error tenfold. Leaving the figure at 20 confuses the spread of individual values with the spread of sample means, which is always smaller because averaging cancels variation. Multiplying instead of dividing gives 200 and describes nothing. The square root is why quadrupling a sample only halves the error.
To test whether brand preference is independent of the buyer's age group, using counts from a survey, the appropriate test is:
1. Chi-square test of independence
2. Two-sample t-test
3. One-way ANOVA
4. Spearman's rank correlation
Show answer
Answer: A — Chi-square test of independence
Both variables here are categorical — a brand and an age group — and the data are counts of how many buyers fall into each combination, laid out in a contingency table. That is exactly the situation the chi-square test of independence handles: it compares the observed counts with the counts expected if the two attributes were unrelated. The t-test and ANOVA compare means of a measured variable across groups, and there is no measured variable here. Spearman's rank correlation measures association between two ordinal rankings, and brand preference is nominal with no order to rank.
Temperature measured in degrees Celsius is on which scale of measurement?
1. Interval
2. Ratio
3. Ordinal
4. Nominal
Show answer
Answer: A — Interval
Celsius has equal intervals — the gap from 10 to 20 degrees is the same as from 20 to 30 — but its zero is an arbitrary point, the freezing point of water, and not the absence of temperature. That makes it an interval scale, on which differences are meaningful but ratios are not: 20 degrees is not twice as hot as 10 degrees. A ratio scale requires a true zero, as income or weight has. Ordinal data carry an order without equal gaps, and nominal data are unordered categories; temperature in degrees is neither, since its intervals are equal and it is fully ordered.
A test yields a p-value of 0.08 at a significance level of 0.05. The correct conclusion is that:
1. There is insufficient evidence to reject the null hypothesis
2. The null hypothesis has been proved true
3. The null hypothesis is rejected at 5%
4. The alternative hypothesis has a probability of 0.08 of being true
Show answer
Answer: A — There is insufficient evidence to reject the null hypothesis
The rule is to reject the null hypothesis when the p-value falls below the chosen significance level. Here 0.08 exceeds 0.05, so the null is not rejected — and the only honest way to say that is that the evidence is insufficient, because a test that fails to reject has not shown the null to be true, merely failed to show it false. Claiming the null is proved is the standard misreading and is never correct in this framework. The p-value is the probability of evidence at least this extreme if the null were true; it is not the probability that either hypothesis is true, which is what the last option asserts.
Research undertaken to clarify a vaguely understood problem and generate hypotheses for later study is:
1. Exploratory research
2. Descriptive research
3. Experimental research
4. Applied research
Show answer
Answer: A — Exploratory research
Exploratory research comes first in a programme of inquiry: its purpose is to define the problem sharply enough that hypotheses can be stated, and it is therefore flexible and unstructured, relying on literature surveys, expert interviews and case studies. Descriptive research follows with a structured design that measures characteristics as they are, and experimental research tests a stated hypothesis by manipulating a variable — both presuppose that the problem has already been defined. Applied research is a classification by purpose rather than by stage, and an exploratory study may be either basic or applied.
In a research report, recommendations are properly placed in the:
1. Discussion and conclusions
2. Analysis and findings
3. Review of literature
4. Methodology
Show answer
Answer: A — Discussion and conclusions
The findings section reports what the data showed and is kept free of interpretation, so that a reader who disagrees with the researcher's reading can still use the results. Recommendations are the researcher's judgement about what should follow from the findings, and they belong with the discussion and conclusions, where the results are set against the objectives and the earlier literature. The review of literature surveys what was already known before the study, and the methodology describes how the study was carried out; neither is a place for advice about what to do next.