📄 Test of Significance and Hypothesis Testing
Understand null and alternative hypotheses, standard error, critical region, decision errors, and the logic of statistical significance testing.
When an experiment shows different mean yields, the first question is not "Are they different?" but "Are they different enough that the difference is unlikely to be due to chance?" That is the central idea behind test of significance.
Why Hypothesis Testing Is Needed
In agricultural experiments and surveys, sample results always contain some random variation. Because of this, we cannot conclude from a sample difference alone that a treatment effect is real.
Hypothesis testing helps us decide whether an observed difference is:
- likely due to random sampling fluctuation, or
- strong enough to treat as statistically significant
Sampling Distribution and Standard Error
A statistic such as the sample mean changes from sample to sample. The probability distribution of that statistic over all possible samples is called its sampling distribution.
The standard deviation of a sampling distribution is called the standard error.
Why standard error matters
Standard error tells us how much a statistic is expected to vary from sample to sample. A small standard error means the statistic is more stable; a large standard error means more fluctuation.
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When an experiment shows different mean yields, the first question is not "Are they different?" but "Are they different enough that the difference is unlikely to be due to chance?" That is the central idea behind test of significance.
Why Hypothesis Testing Is Needed
In agricultural experiments and surveys, sample results always contain some random variation. Because of this, we cannot conclude from a sample difference alone that a treatment effect is real.
Hypothesis testing helps us decide whether an observed difference is:
- likely due to random sampling fluctuation, or
- strong enough to treat as statistically significant
Sampling Distribution and Standard Error
A statistic such as the sample mean changes from sample to sample. The probability distribution of that statistic over all possible samples is called its sampling distribution.
The standard deviation of a sampling distribution is called the standard error.
Why standard error matters
Standard error tells us how much a statistic is expected to vary from sample to sample. A small standard error means the statistic is more stable; a large standard error means more fluctuation.
Standard deviation describes variability among observations, while standard error describes variability of a statistic.
Hypothesis, Null Hypothesis, and Alternative Hypothesis
A hypothesis is a statement that can be tested statistically.
In most tests, we begin with a null hypothesis (H0). It usually states:
- no treatment difference
- no association
- no effect
The alternative hypothesis (H1) states that a real difference or effect exists.
Examples:
| Situation | Null Hypothesis | Alternative Hypothesis |
|---|---|---|
| Two paddy varieties | H0: mu1 = mu2 | H1: mu1 != mu2 |
| One-sided superiority test | H0: mu1 = mu2 | H1: mu1 > mu2 |
Critical Region and Level of Significance
The critical region is the part of the sampling distribution where we reject the null hypothesis.
The level of significance, usually denoted by alpha, is the probability of rejecting a true null hypothesis. Common levels are:
- 5%
- 1%
Meaning of 5% significance
If a result is significant at the 5% level, it means such an extreme result would occur by chance in about 5 out of 100 repeated situations if the null hypothesis were true.
Errors in Decision Making
A statistical decision can be right or wrong.
| True Situation | Decision | Result |
|---|---|---|
| H0 is true | Reject H0 | Type I error |
| H0 is false | Accept H0 | Type II error |
Key ideas
- Type I error: rejecting a true null hypothesis
- Type II error: failing to reject a false null hypothesis
- Power of a test: probability of correctly rejecting a false null hypothesis
One-Tailed and Two-Tailed Tests
The form of the alternative hypothesis decides the test type.
| Test Type | When Used |
|---|---|
| One-tailed test | Direction matters, such as mu1 > mu2 |
| Two-tailed test | Any difference matters, such as mu1 != mu2 |
Example:
- If the question is whether a new variety yields more than the old one, use a one-tailed test.
- If the question is whether the two varieties are different, use a two-tailed test.
Steps in Hypothesis Testing
The testing procedure is usually written in this order:
- state the null and alternative hypotheses
- choose the level of significance
- select the appropriate test statistic
- compute the test statistic
- compare it with the critical value or p-value rule
- decide whether to accept or reject H0
- write the final conclusion in words
This stepwise approach is what makes hypothesis testing exam-friendly and scientifically defensible.
Summary Cheat Sheet
| Topic | Key Point |
|---|---|
| Null hypothesis | Starting assumption of no difference or no effect |
| Alternative hypothesis | Competing claim that a difference or effect exists |
| Standard error | Variability of a statistic across samples |
| Level of significance | Probability of Type I error |
| Type I error | Rejecting a true H0 |
| Type II error | Accepting a false H0 |
| Main exam trap | Standard deviation and standard error are not the same |
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