Hypothesis Testing: Null and Alternative Hypothesis

• Statistics and probability are fundamental tools for analyzing and interpreting data, forming the basis for hypothesis testing.
• Hypothesis testing is a statistical method used to make decisions or inferences about a population based on sample data.
• The core of hypothesis testing involves formulating two mutually exclusive statements: the null hypothesis and the alternative hypothesis.

Null Hypothesis (H0)

• The null hypothesis is a statement of no effect, no difference, or no relationship in the population.
• It represents the status quo or a commonly accepted belief.
• Researchers aim to disprove or reject the null hypothesis.
• The null hypothesis always includes an equality sign (=, ≥, or ≤).
• Examples of null hypotheses:
  • The average height of men and women is the same.
  • There is no correlation between smoking and lung cancer.
  • The new drug has no effect on reducing blood pressure.

Alternative Hypothesis (Ha or H1)

• The alternative hypothesis contradicts the null hypothesis.
• It represents the claim or effect the researcher is investigating or trying to prove.
• It suggests that there is a significant difference, effect, or relationship in the population.
• The alternative hypothesis can be one-tailed (directional) or two-tailed (non-directional).
• The alternative hypothesis never includes an equality sign, instead using ≠, >, or <.

One-Tailed vs. Two-Tailed Tests

• A one-tailed test is used when the alternative hypothesis specifies the direction of the effect or relationship.
• It focuses on one side of the distribution.
• Examples of one-tailed alternative hypotheses:
  • The average height of men is greater than the average height of women.
  • The new drug reduces blood pressure.
• A two-tailed test is used when the alternative hypothesis does not specify the direction of the effect or relationship.
• It considers both sides of the distribution.
• Examples of two-tailed alternative hypotheses:
  • The average height of men and women is different.
  • The new drug has an effect on blood pressure.

Formulating Hypotheses

• The process of formulating hypotheses involves:
  • Identifying the research question or claim.
  • Stating the null hypothesis, which represents no effect or no difference.
  • Stating the alternative hypothesis, which represents the effect or difference the researcher is investigating.
• Ensure hypotheses are clear, specific, and testable.

Errors in Hypothesis Testing

• In hypothesis testing, there are two types of errors that can occur: Type I error (false positive) and Type II error (false negative).
• Type I Error: Rejecting the null hypothesis when it is actually true.
  • Also known as a false positive.
  • The probability of making a Type I error is denoted by α (alpha), also known as the significance level.
  • Researchers typically set α to a small value (e.g., 0.05), meaning there is a 5% chance of rejecting a true null hypothesis.
• Type II Error: Failing to reject the null hypothesis when it is actually false.
  • Also known as a false negative.
  • The probability of making a Type II error is denoted by β (beta).
  • The power of a test (1 - β) represents the probability of correctly rejecting a false null hypothesis.

Significance Level (α)

• The significance level (α) is the probability of rejecting the null hypothesis when it is true (Type I error).
• It is a pre-determined threshold for statistical significance.
• Common values for α are 0.05 (5%) and 0.01 (1%).
• A smaller α reduces the risk of a Type I error but increases the risk of a Type II error.

P-value

• The p-value is the probability of obtaining results as extreme as, or more extreme than, the observed results, assuming the null hypothesis is true.
• It is used to assess the evidence against the null hypothesis.
• If the p-value is less than or equal to the significance level (α), the null hypothesis is rejected.
• If the p-value is greater than the significance level (α), the null hypothesis is not rejected.
• A small p-value indicates strong evidence against the null hypothesis.
• A large p-value indicates weak evidence against the null hypothesis.

Steps in Hypothesis Testing

• State the null hypothesis (H0) and alternative hypothesis (Ha).
• Choose a significance level (α).
• Select an appropriate statistical test (e.g., t-test, chi-square test).
• Calculate the test statistic.
• Determine the p-value.
• Compare the p-value to the significance level (α).
• Make a decision: reject or fail to reject the null hypothesis.
• Draw a conclusion in the context of the research question.

Example

• Research Question: Does a new fertilizer increase crop yield?
• Null Hypothesis (H0): The new fertilizer has no effect on crop yield.
• Alternative Hypothesis (Ha): The new fertilizer increases crop yield.
• Significance Level (α): 0.05
• Statistical Test: t-test
• Results:
  • Calculated t-statistic: 2.5
  • P-value: 0.02
• Decision: Since the p-value (0.02) is less than the significance level (0.05), we reject the null hypothesis.
• Conclusion: The new fertilizer significantly increases crop yield.

Learn about hypothesis testing, a statistical method for making inferences about a population. Understand the null hypothesis, a statement of no effect, and the alternative hypothesis, which contradicts the null hypothesis. Explore examples of null and alternative hypotheses.