Null and Alternative Hypotheses

Understanding Null and Alternative Hypotheses

• A hypothesis is a claim or statement about a parameter from a population.
• There are two complementary hypotheses in any test: the Null Hypothesis (H₀) and the Alternative Hypothesis (H₁).
• The Null Hypothesis (H₀) is a claim of no change or effect and is the hypothesis we test.
• The Alternative Hypothesis (H₁) is a claim that contradicts the null hypothesis. It’s what we possibly accept if we have strong evidence against H₀.

Formulating Null and Alternative Hypotheses

• Hypotheses are about population parameters, such as the population mean or proportion.
• These hypotheses are mutually exclusive. If the null hypothesis is true, the alternative must be false and vice versa.
• The null hypothesis often denotes the equality statement, where no differences or relationships exist.
• The alternative hypothesis represents the inequality statement or the existence of a relationship or difference.

Critical Regions and Acceptance Regions

• The Critical Region is the set of values for the test statistic that would lead us to reject the null hypothesis.
• The Acceptance Region is where we would accept the null hypothesis, and it is the set of values that are not in the critical region.
• The boundary that separates these regions is called the critical value.

Significance Levels and Critical Values

• The significance level of a statistical hypothesis test is a fixed probability of wrongly rejecting the null hypothesis H₀, if it is in fact true.
• Commonly used significance levels are 5% and 1%.
• A critical value is a line on a graph that splits the graph into regions. One or more of these regions is the “rejection region”; if your test value falls into that region, then you reject H0. If your value falls into the other region, you fail to reject H0. The two regions are separated by the critical value.

Hypothesis Testing Procedures

• Hypothesis testing begins by identifying a population parameter of interest such as the population mean, and creating a null and alternative hypotheses about that parameter.
• The statistical evidence comes from the data collected. This data will be used to determine whether or not to reject the null hypothesis.
• After calculations, a decision is made. This decision will either be to reject the null hypothesis and hence accept the alternative hypothesis, or to fail to reject the null hypothesis which implies that there isn’t enough evidence against the null hypothesis given the data.

Note: Failing to reject the null hypothesis doesn’t necessarily prove it true. It just suggests that the data at hand fails to provide sufficient evidence against it. This distinction is important for understanding the concept of statistical proof.

Remember, in hypothesis testing, you operate under the assumption that the null hypothesis is true. It remains so unless you have enough evidence against it.