📊 Test of Significance and Hypothesis 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.

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:

1. state the null and alternative hypotheses
2. choose the level of significance
3. select the appropriate test statistic
4. compute the test statistic
5. compare it with the critical value or p-value rule
6. decide whether to accept or reject H0
7. 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