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Random sampling aims for a subset reflecting the population to generalize results.
Chi-square tests compare observed vs. expected frequencies under a model.
It tests if the distribution of one variable is independent of another.
Positive correlation shows tandem movement in the same direction.
Dependent variable depends on predictors in the model.
Small p-values suggest data unlikely under H0.
Used for frequency data in categories.
Linearity (or appropriate transformation) supports valid inference.
Systematic random sampling orders all units in the sampling frame and then selects every nth unit, where n is the sampling interval (total population ÷ desired sample size).
Sampling is the process of selecting a subset (sample) from a larger population to make valid inferences about that population. It is used when studying the entire population is impractical, too costly, or impossible. A well-drawn sample should adequately represent the population so that findings can be generalised.
In cluster sampling, the population is divided into naturally occurring groups or clusters (e.g., households, villages, schools, or wards). A random sample of clusters is selected, and then all or a random subset of individuals within those clusters are studied.
A test of significance is a mathematical method used to determine the probability (P value) that an observed difference between groups occurred purely by chance (i.e., due to sampling variation). If this probability is very small (P < 0.05), we conclude the difference is statistically significant and unlikely to be due to chance alone.
The null hypothesis (H₀) always starts with the assumption that there is NO difference, NO association, or NO effect — it represents the 'status quo'.
Karl Pearson (1857–1936) invented the Chi-square distribution and the Chi-square test, publishing his landmark paper in Philosophical Magazine in 1900. His work introduced the concept of the goodness-of-fit test and test of independence for categorical data.
The chi-square test is a non-parametric test used specifically for categorical (qualitative) data — data that can be counted and sorted into categories.
A fundamental assumption of the chi-square test is that the expected frequency in each cell of the contingency table must be at least 5 (and no cell should have a zero count). If expected counts are less than 5, the chi-square approximation becomes unreliable and may lead to incorrect conclusions.
The Chi-square formula is χ² = Σ(Oᵢ − Eᵢ)²/Eᵢ. For each cell: (1) subtract the expected count from the observed count; (2) square the difference — this removes negative values; (3) divide by the expected count — this normalises for sample size. The sum of all these values across cells gives the χ² statistic. A larger χ² indicates a greater discrepancy between observed and expected frequencies.
The Pearson coefficient of correlation (r) always ranges from −1.00 to +1.00. A value of +1 indicates a perfect positive linear relationship (as X increases, Y increases proportionally); −1 indicates a perfect negative linear relationship (as X increases, Y decreases proportionally).
When r is close to 0, it indicates little to no linear association between the two variables. This does not mean the variables are unrelated — they might have a strong non-linear (e.g., curvilinear) relationship that Pearson r cannot detect. Generally: |r| < 0.3 = weak, 0.3–0.7 = moderate, and |r| > 0.7 = strong correlation.
By convention in correlation and regression analysis, the independent variable (predictor variable, X) is always plotted on the horizontal X-axis, and the dependent variable (outcome variable, Y) is plotted on the vertical Y-axis. This is consistent with the regression equation Y = a + bX, where X is the input used to predict Y. In a scatter diagram, each point represents a single observation with its (X, Y) coordinate pair.
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