| Concept | Description |
|---|---|
| Population, Sample, Parameter, and Statistic | |
| Population and Sample | A population is the entire relevant set of elements; a sample is a subset actually measured |
| Parameter and Statistic | Parameters describe a population; statistics are calculated from a sample and used to estimate parameters |
| Hypothesis Testing | |
| Hypothesis Testing | A statistical method for making decisions about a population using only sample data |
| Fundamental Concepts of Hypothesis Testing | Null and alternative hypotheses, significance level, p-value, and test statistic are the building blocks of every hypothesis test |
| Steps in Hypothesis Testing | Formulate hypotheses, choose alpha, select a test, calculate the test statistic, determine the p-value, and decide |
| Assumptions of the Test | Most tests assume normality, independence, or homogeneity of variance; violating these affects the test's validity |
34 Fundamentals of Statistical Tests
Population and sample (George W. Snedecor & William G. Cochran, 1989)
- A population is the entire set of elements relevant to a study — every farm in a district, say.
- The list of all elements or individuals in the study is called the frame.
- Studying every element of a population is called a census.
- Selecting only a subset of the population is called a sample — the 40 farms a survey team actually visited.
Parameter and statistic
- Descriptive measures calculated from an entire population are called parameters; measures calculated from a sample are called statistics. A district’s true average yield (if every farm were measured) is a parameter; the average yield computed from 40 surveyed farms is a statistic used to estimate it.
This topic builds on the summary statistics already covered in Descriptive Analytics — in particular, Measures of Dispersion, which covers standard deviation in depth with a worked R example. Rather than repeat that here, this topic moves on to what standard deviation and the other descriptive measures are used for: testing whether a pattern seen in a sample is likely to hold for the population it came from.
34.1 Hypothesis Testing
Hypothesis testing is a statistical method for making decisions about a population based on sample data. It lets a researcher, agronomist, or analyst test an assumption or claim — a parameter of a population, such as a mean or a proportion — using only a sample, rather than having to measure every member of the population.
34.1.1 Fundamental Concepts of Hypothesis Testing
Hypotheses: hypothesis testing formulates two opposing statements:
- Null hypothesis (\(H_0\)): assumes no effect or no difference — the “status quo.” For example, “a new bio-fertilizer has no effect on yield compared to the standard fertilizer.”
- Alternative hypothesis (\(H_a\) or \(H_1\)): what the researcher is trying to demonstrate — a new effect, difference, or change. For example, “the new bio-fertilizer increases yield compared to the standard fertilizer.”
Significance level (\(\alpha\)): the threshold for rejecting the null hypothesis, typically set at 0.05 (5%). It’s the probability of rejecting a true null hypothesis — a Type I error (covered next).
P-value: the probability of observing the sample data, or something more extreme, if the null hypothesis were true. A small p-value (typically ≤ 0.05) is read as strong evidence against the null hypothesis.
Test statistic: a single value calculated from the sample data, used to judge how compatible the data is with the null hypothesis. Its exact form depends on the test — a t-statistic for a t-test, an F-statistic for ANOVA, and so on.
34.1.2 Steps in Hypothesis Testing
- Formulate hypotheses: define \(H_0\) and \(H_a\) based on the research question — does this new seed variety out-yield the standard one?
- Choose the significance level: set \(\alpha\) (commonly 0.05).
- Select the appropriate test: based on the data type and the hypothesis (see Choosing the Right Statistical Test).
- Calculate the test statistic: compute it from the sample data.
- Determine the p-value: the probability of the observed result (or something more extreme) under \(H_0\).
- Make a decision: compare the p-value to \(\alpha\). If the p-value is smaller, reject \(H_0\); otherwise, fail to reject it.
34.1.3 Assumptions of the Test
Most statistical tests carry assumptions about the data — normality, independence, homogeneity of variance (homoscedasticity). Violating these assumptions can undermine the validity of the test’s conclusions. Checking the assumptions, and switching to a non-parametric alternative when they don’t hold, is part of choosing the right test rather than an afterthought (see Choosing the Right Statistical Test).