Hypothesis Testing in Statistics

Hypothesis Testing

Definition

• A procedure used to test a hypothesis based on a sample of data
• Involves comparing the data to a known probability distribution or a hypothesis about the population

Types of Hypotheses

• Null Hypothesis (H0): a statement of no effect or no difference
  • Typically denoted by μ (mu) or p
  • Example: "There is no significant difference in the means of two groups"
• Alternative Hypothesis (H1): a statement of an effect or difference
  • Typically denoted by μ (mu) or p
  • Example: "There is a significant difference in the means of two groups"

Steps in Hypothesis Testing

1. State the hypothesis: Clearly define the null and alternative hypotheses
2. Select a significance level: Choose a level of significance (α) to determine the probability of rejecting the null hypothesis when it is true (Type I error)
   • Typically set at 0.05
3. Collect and analyze the data: Collect a sample of data and calculate the test statistic
4. Determine the critical region: Identify the region of the test statistic distribution where the null hypothesis is rejected
5. Compare the test statistic to the critical value: Determine if the test statistic falls in the critical region
6. Make a decision: Reject the null hypothesis if the test statistic falls in the critical region, otherwise fail to reject the null hypothesis

Errors in Hypothesis Testing

• Type I error: Rejecting the null hypothesis when it is true (α)
• Type II error: Failing to reject the null hypothesis when it is false (β)

Common Test Statistics

• t-statistic: Used for small sample sizes and unknown population standard deviation
• z-statistic: Used for large sample sizes and known population standard deviation
• F-statistic: Used for comparing multiple group means (ANOVA)

Interpretation of Results

• p-value: The probability of observing the test statistic (or a more extreme value) assuming the null hypothesis is true
  • If p-value < α, reject the null hypothesis
  • If p-value ≥ α, fail to reject the null hypothesis

Hypothesis Testing

Definition

• A procedure used to test a hypothesis based on a sample of data
• Involves comparing the data to a known probability distribution or a hypothesis about the population

Hypotheses

• Null Hypothesis (H0): a statement of no effect or no difference
  • Typically denoted by μ (mu) or p
  • Example: "There is no significant difference in the means of two groups"
• Alternative Hypothesis (H1): a statement of an effect or difference
  • Typically denoted by μ (mu) or p
  • Example: "There is a significant difference in the means of two groups"

Steps in Hypothesis Testing

• State the hypothesis: Clearly define the null and alternative hypotheses
• Select a significance level: Choose a level of significance (α) to determine the probability of rejecting the null hypothesis when it is true (Type I error)
  • Typically set at 0.05
• Collect and analyze the data: Collect a sample of data and calculate the test statistic
• Determine the critical region: Identify the region of the test statistic distribution where the null hypothesis is rejected
• Compare the test statistic to the critical value: Determine if the test statistic falls in the critical region
• Make a decision: Reject the null hypothesis if the test statistic falls in the critical region, otherwise fail to reject the null hypothesis

Errors in Hypothesis Testing

• Type I error: Rejecting the null hypothesis when it is true (α)
• Type II error: Failing to reject the null hypothesis when it is false (β)

Common Test Statistics

• t-statistic: Used for small sample sizes and unknown population standard deviation
• z-statistic: Used for large sample sizes and known population standard deviation
• F-statistic: Used for comparing multiple group means (ANOVA)

Interpretation of Results

• p-value: The probability of observing the test statistic (or a more extreme value) assuming the null hypothesis is true
  • If p-value < α, reject the null hypothesis
  • If p-value ≥ α, fail to reject the null hypothesis