Statistical Research Techniques using Softwares

1.1 Research design; qualitative and quantitative research

A research design is the plan for answering a research question: who or what will be studied, what information will be collected, and how it will be analysed.

A study can use more than one kind of information, but the method must match the question. “What proportion…?” usually needs quantitative data; “How do students describe…?” may need qualitative data.

1.2 Data collection and sampling

Data can be collected using questionnaires, interviews, observation, records or other suitable methods. Sampling selects units from a target population. Use the methods taught in earlier courses where relevant, and explain how the selection process relates to the research question.

A sampling frame is the practical list or source from which a sample is selected. A mismatch between the target population and the frame can leave some people out or over-represent others.

1.3 Research reporting and proposal writing

A research proposal explains what the study will ask and how it will be carried out. A clear proposal commonly contains:

1. Title and short background to the issue.
2. Specific problem statement and research question(s).
3. Objectives; define key terms or study variables.
4. Target population, sampling approach and proposed sample.
5. Data collection instrument/procedure.
6. Planned summaries or statistical analysis.
7. Practical schedule, limitations and references as directed.

A research report communicates what was done and found. Present tables/diagrams with titles and units; distinguish results from interpretation; state limitations; keep the conclusion tied to the evidence.

1.4 Apply methods to a social issue

2.1 Parameter and statistic

A parameter is a numerical feature of a population, such as population mean μ. A statistic is calculated from a sample, such as sample mean x̄, and is used to learn about the parameter.

2.2 Statistical hypotheses

A statistical hypothesis is a claim about a population distribution or parameter. A simple hypothesis specifies the distribution/parameter completely for the test; a composite hypothesis allows a set of possible values.

The null hypothesis H₀ is the reference claim tested. The alternative hypothesis H₁ describes a departure of interest. For a mean, a two-sided test might use H₀: μ=μ₀ against H₁: μ≠μ₀. A one-sided alternative must be selected from the research question before examining the result.

2.3 Errors and significance level

The significance level α is the chosen maximum long-run Type I error probability under the test procedure. It is selected before seeing the result; for example α=0.05 is a common convention, not a guarantee about an individual result.

Power is the probability the test rejects H₀ for a specified alternative when that alternative is true: power=1−β. Power depends on effect size, variability, sample size and α.

2.4 p-value, testing procedure and critical region

The p-value is the probability, assuming H₀ and the test model are true, of obtaining a test statistic at least as incompatible with H₀ as the observed one, in the direction(s) specified by H₁.

A critical region is the set of test-statistic values leading to rejection of H₀ at the chosen α. The p-value method and critical-region method should give the same decision when applied consistently.

1. State H₀ and H₁ in context.
2. Choose α and the appropriate test, with assumptions.
3. Calculate the test statistic and p-value, or compare with the critical region.
4. Reject H₀ or do not reject H₀ according to the rule.
5. State a contextual conclusion; distinguish statistical evidence from practical importance.

*The attached syllabus lists Module 2 as 10 hours; its sub-unit hours total 15. Confirm the intended allocation with the lecturer.

3.1 Large-sample z test for one mean and two means

For one population mean with known population SD σ (or the large-sample approximation taught in class), compare the sample mean with the hypothesised μ₀.

x̄ is sample mean; μ₀ is the null value; σ is population SD; n is sample size. The standard error σ/√n describes sample-mean variation under the model.

For equality of two independent means, the large-sample z statistic is commonly:

Δ₀ is the hypothesised difference (usually 0 for equality); use sample SDs in the large-sample version if that is the class convention. The groups must be independent.

3.2 Small-sample t tests

One-sample t test: tests a population mean when σ is unknown using sample SD s, under the model assumptions taught (commonly an approximately normal population for small samples).

Two independent means: when using the pooled equal-variance procedure, calculate pooled variance and standard error:

Degrees of freedom are n₁+n₂−2 for this pooled procedure. Use it only when the equal-variance assumption and course procedure are appropriate; follow the method prescribed by your lecturer.

Paired t test: use when the same units are measured twice or observations are naturally paired. Calculate differences dᵢ first, then test their mean.

3.3 One-way ANOVA

Analysis of variance (ANOVA) compares means of three or more groups by comparing variation between groups with variation within groups. The one-way version has one grouping factor.

k is number of groups; N is total observations. H₀ says all group means are equal; H₁ says at least one differs. A significant result does not by itself identify which groups differ. Typical assumptions: independent observations, approximately normal errors within groups and similar variances.

3.4 Chi-square test for association of attributes

For categorical data in a contingency table, the chi-square test compares observed cell counts O with counts E expected if the two attributes are independent.

A large discrepancy between observed and expected counts gives evidence against independence. Check that observations are counted once and expected counts are suitable for the approximation as taught in class.

3.5 Sign, median and Wilcoxon ranked tests

These non-parametric procedures use signs, ranks or counts rather than relying on the same mean/normal model as a t test. The syllabus limits these to simple problems.

• Sign test: for paired differences, count positive and negative signs; ignore zero differences. Under a no-median-difference null, positive/negative signs are equally likely (p=0.5), subject to the test setup.
• Median test: compare groups by classifying observations above/below a pooled median and examining the resulting counts.
• Wilcoxon ranked test: rank-based procedure. The syllabus does not specify which Wilcoxon variant; common introductory paired use is the signed-rank test. Confirm the exact variant and steps with the lecturer, and practise only the simple problems required.

3.6 Applications across fields

Tests can support decisions in education, health, agriculture, business and social research. State the population, outcome and comparison clearly. A test result does not replace study design, subject knowledge or an effect-size/context discussion.

Survey and project workflow

1. Choose a feasible social issue and write a specific question and objective.
2. Define target population, unit of observation and sampling method.
3. Prepare a clear questionnaire; pilot/check wording and response options.
4. Collect and enter data consistently; document coding and missing responses.
5. Summarise with descriptive statistics, tables, diagrams and geographical representation as appropriate.
6. Choose a statistical analysis that matches the question and variables.
7. Interpret the output in context, note limitations and prepare the report.

Minimum five practical record ideas

1. Create a questionnaire and codebook for a small social survey.
2. Enter survey data and produce frequency tables and descriptive statistics.
3. Create labelled diagrams and a geographical representation suited to the data available.
4. Use a suitable z or t procedure on a provided or collected data set; document assumptions and interpretation.
5. Analyse a paired or multiple-group question using a paired t test or one-way ANOVA as appropriate.
6. Analyse categorical association using a contingency table and chi-square procedure.
7. Complete a simple sign, median or Wilcoxon rank-based problem.
8. Prepare a concise project report explaining research question, method, results and limitations.

Choose at least five in coordination with your course requirements. For every record problem include: question, data, method, software/function or calculation, output, check, interpretation and conclusion.

Software-independent checking habits

• Keep a clean copy of raw data and document recoding.
• Check sample size, missing values, units and group labels.
• Match the test to independent/paired design and outcome type.
• Verify test direction, degrees of freedom and significance level.
• Report the statistic, degrees of freedom where relevant, p-value/decision, and context.
• Reproduce key values with a hand calculation or second method where practical.

Practical/viva prompts

• What was your target population and how was the sample chosen?
• How did you code responses and handle missing values?
• Why is the selected test appropriate for the design?
• What are H₀ and H₁ for your analysis?
• What conclusion follows from the output, and what does not follow?
• What limitation should a reader know?